Context:
Determine the number of different ways to set up the limits of integration for the integral $$\iiint_D xyz \, dx \, dy \, dz,$$ where $D=\{(x,y,z) : 0 \le x \le y \le z \le 1 \}$. Provide your answer as a single integer.
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Context:
Evaluate the limit: $$\lim_{n\to \infty} \frac{1}{n^{n}}\int_{0}^{n}x^n\arctan(\sqrt[n]{x})\,dx$$
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Context:
Evaluate the integral: $$ \int_{0}^{4\pi}\frac{x|\sin x|dx}{1 + |\cos x|} $$
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Context:
Calculate the surface integral of the vector field \( F = \frac{x - (0,0,-1)}{||x - (0,0,-1)||^3} \) over the unit sphere, excluding the point \((0,0,-1)\).
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Context:
In the prism $ABC A_1 B_1 C_1$, vertices $A$ and $B$ are located on the axis of a cylinder, while the remaining vertices lie on the lateral surface of the cylinder. Determine the dihedral angle formed with edge $AB$ in this prism.
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Context:
Determine the dimension of the algebraic dual space $V^*$ of an infinite-dimensional vector space $V$ over the field $\mathbb{F}$, where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$, given that the dimension of $V$ is a cardinal number $\alpha$. Express your answer in terms of $\alpha$. \( \text{(Hint: Consider the properties of dual spaces and cardinality.)} \)
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Context:
Determine the first non-zero digit from the right in the base-10 representation of the product \( \prod\limits_{i=1}^{n/2} (i)(n-i+1) \) for even numbers \( n \geq 6 \).
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Context:
10.284. A square and an equilateral triangle are described around a circle of radius $R$, with one side of the square lying on a side of the triangle. Calculate the area of the common part of the triangle and the square.
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Context:
What is the minimum number of colors needed to color a hexagonal tessellation grid, such that no two tiles sharing a vertex are the same color?
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Context:
Suppose $R>0$, $$D_R=\{(r\cos \theta, r\sin \theta):0\leq r\leq R,0\leq \theta\leq \frac{\pi}{R^2} \},$$ and $$I_R=\int\int_{D_R}\frac{x^2\cos y}{x^2+y^2+1}dxdy.$$ Calculate $\lim_{R\rightarrow \infty}I_R$.
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Context:
The graph of the equation $x+y=\left\lfloor x^{2}+y^{2}\right\rfloor$ consists of several line segments. Compute the sum of their lengths.
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Context:
The five-digit number  $12110$  is divisible by the sum of its digits  $1 + 2 + 1 + 1 + 0 = 5.$  Find the greatest five-digit number which is divisible by the sum of its digits
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Context:
In order to spot-check the implementation of vehicle exhaust emission standards in a city, vehicles with license plates ending in the digit 5 were selected for inspection on the main roads of the city. This sampling method is called __________.
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Context:
How many four-digit numbers greater than 5000 can be formed from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 if only the digit 4 may be repeated?
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Context:
Given complex numbers \( z_1 \) and \( z_2 \) that satisfy \( \left|z_1\right| = \left|z_1 + z_2\right| = 3 \) and \( \left|z_1 - z_2\right| = 3\sqrt{3} \), determine:

\[ \log_{2} \left| \left(z_1 \overline{z_2}\right)^{2000} + \left(\overline{z_1} z_2\right)^{2000} \right| \]
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Context:
Find the smallest solution to the equation \[\lfloor x^2 \rfloor - \lfloor x \rfloor^2 = 19.\]
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Context:
In convex hexagon $ABCDEF$, all six sides are congruent, $\angle A$ and $\angle D$ are right angles, and $\angle B$, $\angle C$, $\angle E$, and $\angle F$ are congruent. The area of the hexagonal region is $2116(\sqrt{2}+1)$. Find $AB$.
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Context:
Given the triangular pyramid $D-ABC$, where $\angle ACB=\angle ABD=90^{\circ}$, $CA=CB$, and $\angle BAD=30^{\circ}$. If the projection of point $C$ onto plane $ABD$ lies exactly on line $AD$, then find the sine of the dihedral angle between planes $CAB$ and $ABD$.
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Context:
A tree has 10 pounds of apples at dawn.  Every afternoon, a bird comes and eats x pounds of apples.  Overnight, the amount of food on the tree increases by 10%.  What is the maximum value of x such that the bird can sustain itself indefinitely on the tree without the tree running out of food?
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Context:
A cardboard box in the shape of a rectangular parallelepiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimeters and its base has a radius of $30$ centimeters, find the volume of the largest box that can be completely enclosed inside the container with the lid on.
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Context:
If there is exactly one acute triangle $\triangle ABC$ that satisfies $\angle A=60^{\circ}$, $BC=6$, and $AB=k$, then the range of the real number $k$ is _______.
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Context:
We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number?
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Context:
How many pairs $(m, n)$ of positive integers with $m < n$ fulfill the equation $\frac{3}{2008} = \frac{1}{m} + \frac{1}{n}$?
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Context:
Determine the type of cross-section of a cube by a plane passing through its center perpendicular to its diagonal.
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Context:
For a positive integer  $n$  not divisible by  $211$ , let  $f(n)$  denote the smallest positive integer  $k$  such that  $n^k - 1$  is divisible by  $211$ . Find the remainder when   $$ \sum_{n=1}^{210} nf(n) $$   is divided by  $211$ .

*Proposed by ApraTrip*
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Context:
The legs of a right triangle are $a$ and $b$. Find the length of the angle bisector drawn from the vertex of the right angle.
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Context:
$S=\int_{-\infty}^{\infty} \frac{1}{1+x^{2}+x^{4}} \mathrm{~d} x$, 则 $\left[100 S^{2}\right]=$ $\qquad$
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Context:
From the beginning to the end of 2007, the price of a stock rose 20 percent. In 2008, it dropped 25 percent. In 2009, it rose 25 percent. In 2010, it dropped 15 percent. In 2011, it rose 30 percent. In 2012, it dropped 10 percent. In 2013, it rose 15 percent. What percent of the stock's 2007 starting price was the price of the stock at the end of 2013?
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Context:
Reflect a cube in space in the planes of each of its six faces in some order. How many different transformations are produced by the succession of these six reflections?
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Context:
A, B, C, and D enter into a partnership. A subscribes 1/3 of the capital, B 1/4, C subscribes some fraction, and D the rest. A gets a share of Rs. 810 in a profit of Rs. 2430. What fraction of the capital did C subscribe?
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Context:
For every $m \geq 2$, let $Q(m)$ be the least positive integer with the following property: For every $n \geq Q(m)$, there is always a perfect cube $k^3$ in the range $n < k^3 \leq m \cdot n$. Find the remainder when \[\sum_{m = 2}^{2017} Q(m)\]is divided by 1000.
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Context:
Given the real number \(a\) such that there exists exactly one square with all four vertices lying on the curve \(y = x^3 + ax\). Determine the side length of this square.
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Context:
已知 $\omega=\cos \frac{2 \pi}{5}+\operatorname{isin} \frac{2 \pi}{5}, f(x)=x^{2}+x+2$, 求 $f(\omega) f\left(\omega^{2}\right) f\left(\omega^{3}\right) f\left(\omega^{4}\right)$ 的值。
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Context:
A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of each square is 24, and the perimeter of each small rectangle is 16. What is the perimeter of the large rectangle?
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Context:
Let $\triangle ABC$ be a triangle with $AB=85$, $BC=125$, $CA=140$, and incircle $\omega$. Let $D$, $E$, $F$ be the points of tangency of $\omega$ with $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ respectively. Furthermore, denote by $X$, $Y$, and $Z$ the incenters of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$, respectively. Find the circumradius of $\triangle XYZ$.
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Context:
The Razorback shop makes $76 dollars off each jersey, $204 off each t-shirt, $132 off each hoodie, and $48 off each hat. During the Arkansas and Texas Tech game, they sold 158 t-shirts, 2 jerseys, 75 hoodies, and 120 hats. How much money did they make from selling all these items?
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Context:
A signal soldier uses three flags of red, yellow, and blue, hung from top to bottom on a vertical flagpole to represent signals. Each time, 1, 2, or 3 flags can be hung in any order, and different sequences represent different signals. In total, there are ______ different signals that can be represented.
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Context:
The square shown is divided into 16 equal smaller squares, and each small square is further divided into two rectangles along one of its diagonals. If the perimeter of the larger square is 256 units, what is the perimeter of one of these smaller rectangles?
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Context:
Find the limit $\lim_{x\rightarrow -1}f(x)$ for the function $f(x) = \frac{x^2+2x-3|1-x^2|+1}{2|x+1|-|x^2+x|}$.
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Context:
Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $O$ be the circumcenter of $A B C$. Find the distance between the circumcenters of triangles $A O B$ and $A O C$.
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Context:
Determine the asymptotic upper bound for the recurrence relation $T(n) = T(\sqrt{n}) + T(n/2)$, with $T(n)=n$ for $n\leq 2$.
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Context:
Given the hyperbola $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1(a > 0,b > 0)$ with its left and right foci being $F_{1}(-c,0)$ and $F_{2}(c,0)$ respectively. If there exists a point $P$ on the hyperbola such that $\dfrac {\sin PF_{1}F_{2}}{\sin PF_{2}F_{1}}= \dfrac {a}{c}$, then the range of the eccentricity of this hyperbola is \_\_\_\_\_\_.
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Context:
A square is divided into 5 strips of equal width, with 3 black strips and 2 grey strips. Calculate the fraction of the perimeter of the square that is grey. Express your answer as a simplified fraction.
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Context:
If Jake loses 8 pounds, he will weigh twice as much as his sister. Jake's brother is currently 6 pounds heavier than twice Jake's weight. Together, all three of them now weigh 480 pounds. Additionally, the brother's weight is 125% of the sister's weight. What is Jake's present weight, in pounds?
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Context:
In a range of integers, there is 1 prime number. The range starts at 200 and ends at a certain number. What is the ending number of the range if there is only 1 prime number between 200 and the ending number?
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Context:
Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Find the value of $a+b.$
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Context:
Three congruent circles of radius $2$ are drawn in the plane such that each circle passes through the centers of the other two circles. The region common to all three circles has a boundary consisting of three congruent circular arcs. Let $K$ be the area of the triangle whose vertices are the midpoints of those arcs. If $K = \sqrt{a} - b$ for positive integers $a, b$, find $100a+b$.
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Context:
11. String 6 red balls, 1 white ball, and 8 yellow balls into a necklace, then the number of possible arrangements is $\qquad$ (balls of the same color are indistinguishable).
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Context:
King Tin writes the first $n$ perfect squares on the royal chalkboard, but he omits the first one. For example, if $n = 3$, he writes $4$ and $9$. His son, Prince Tin, comes along and repeats the following process until only one number remains:

He erases the two greatest numbers still on the board, calls them $a$ and $b$, and writes the value of $\frac{ab-1}{a+b-2}$ on the board.

Let $S(n)$ be the last number that Prince Tin writes on the board. Consider $\lim_{n\to \infty} S(n) = r$, meaning that $r$ is the unique number such that for every $\epsilon > 0$, there exists a positive integer $N$ so that $|S(n) - r| < \epsilon$ for all $n > N$. If $r$ can be written in simplest form as $\frac{m}{n}$, find $m + n$.
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Context:
One of the receipts for a math tournament showed that 72 identical trophies were purchased for \$99.9, where the first and last digits were illegible. How much did each trophy cost?
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Context:
2019 students are voting on the distribution of \(N\) items. For each item, each student submits a vote on who should receive that item, and the person with the most votes receives the item (in case of a tie, no one gets the item). Suppose that no student votes for the same person twice. Compute the maximum possible number of items one student can receive, over all possible values of \(N\) and all possible ways of voting.
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Context:
Given the function
\[ f(x) = x^2 - (k^2 - 5ak + 3)x + 7 \quad (a, k \in \mathbb{R}) \]
for any \( k \in [0, 2] \), if \( x_1, x_2 \) satisfy
\[ x_1 \in [k, k+a], \quad x_2 \in [k+2a, k+4a], \]
then \( f(x_1) \geq f(x_2) \). Find the maximum value of the positive real number \( a \).
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Context:
Given the quadratic equation with complex coefficients \((1+2\mathrm{i}) x^2 + m x + 1 - 2\mathrm{i} = 0\), determine the minimum value of the modulus of the complex number \(m\) such that the equation has a real root.
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Context:
Reimu invented a new number base system that uses exactly five digits. The number $0$ in the decimal system is represented as $00000$. Whenever a number is incremented, Reimu finds the leftmost digit (of the five digits) that is equal to the "units" (rightmost) digit, increments this digit, and sets all the digits to its right to 0.

For example, an analogous system that uses three digits would begin with $000$, $100$, $110$, $111$, $200$, $210$, $211$, $220$, $221$, $222$, $300$, $\ldots$.

Compute the decimal representation of the number that Reimu would write as $98765$.
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Context:
Sandhya must save 35 files onto disks, each with 1.44 MB space. 5 of the files take up 0.6 MB, 18 of the files take up 0.5 MB, and the rest take up 0.3 MB. Files cannot be split across disks. Calculate the smallest number of disks needed to store all 35 files.
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Context:
Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese?
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Context:
设有三个函数，第一个是 $y=\varphi(x)$ ，它的反函数是第二个函数，而第三个函数的图象与第二个函数的图象关于 $x+y=0$ 对称。第三个函数的表达式为 $y=ax+b\varphi(cx)$，请给出a+b+c的值。
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Context:
Find the angle $C$ of triangle $ABC$ if vertex $A$ is equidistant from the centers of the excircles that touch sides $AB$ and $BC$.
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Context:
In a group of 100 cars, some cars do not have air conditioning. If at least 51 cars have racing stripes, the greatest number of cars that could have air conditioning but not racing stripes is 49. How many cars do not have air conditioning?
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Context:
In circle \(\omega\), two perpendicular chords intersect at a point \( P \). The two chords have midpoints \( M_1 \) and \( M_2 \) respectively, such that \( P M_1 = 15 \) and \( P M_2 = 20 \). Line \( M_1 M_2 \) intersects \(\omega\) at points \( A \) and \( B \), with \( M_1 \) between \( A \) and \( M_2 \). Compute the largest possible value of \( B M_2 - A M_1 \).
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Context:
Let $\triangle ABC$ have median $CM$ ($M \in AB$) and circumcenter $O$. The circumcircle of $\triangle AMO$ bisects $CM$. Determine the least possible perimeter of $\triangle ABC$ if it has integer side lengths.
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Context:
Compute the following limit:

$$ \lim_{n \to \infty} \frac{\left(1 + \frac{1}{n} + \frac{1}{n^2}\right)^n - \left(1 + \frac{1}{n} - \frac{1}{n^2}\right)^n }{ 2 \left(1 + \frac{1}{n} + \frac{1}{n^2}\right)^n - \left(1 + \frac{1}{n} - \frac{1}{n^2 + 1}\right)^n - \left(1 + \frac{1}{n} - \frac{1}{n^2 (n^2 +1)}\right)^n } $$
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Context:
8. An $A M C$ exam consists of 30 questions, with the scoring rules specifying 5 points for each correct answer, 0 points for each wrong answer, and 2 points for each unanswered question.
The sum of the points a participant earns for each question is called the participant's total score. Assuming there are enough participants, the number of different possible total scores is $\qquad$
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Context:
Two cellphones and three laptops are connected to load a certain video. The first cellphone takes nine minutes, the second cellphone takes six minutes, the first laptop takes fifteen seconds, the second laptop takes eleven seconds, and the third laptop takes eight seconds to load that same video. If all devices are connected so that they operate in concert at their respective rates, how many seconds would it take them to load the video, rounded to the nearest thousandth?
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Context:
Consider a regular  $2n + 1$ -gon  $P$  in the plane, where n is a positive integer. We say that a point  $S$  on one of the sides of  $P$  can be seen from a point  $E$  that is external to  $P$ , if the line segment  $SE$  contains no other points that lie on the sides of  $P$  except  $S$ . We want to color the sides of  $P$  in  $3$  colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to  $P$ , at most  $2$  different colors on  $P$  can be seen (ignore the vertices of  $P$ , we consider them colorless). Find the largest positive integer for which such a coloring is possible.
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Context:
Two players alternately choose the sign for each of the numbers 1, 2, ..., 20. Once a sign has been chosen for a number, it cannot be changed. The first player tries to minimize the final absolute value of the total, and the second player tries to maximize it. What is the outcome, assuming both players play perfectly?
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Context:
If $\alpha=3$, then the terminal side of $\alpha$ lies in the $\boxed{\text{answer}}$ quadrant.
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Context:
Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that
$P(1) = P(3) = P(5) = P(7) = a$,  and
$P(2) = P(4) = P(6) = P(8) = -a$.
Find the smallest possible value of $a$.
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Context:
Given two real numbers \( p > 1 \) and \( q > 1 \) such that \( \frac{1}{p} + \frac{1}{q} = 1 \) and \( pq = 9 \), what is \( q \)?
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Context:
Mary earns $46 cleaning a home. After cleaning a number of homes, she had $276. However, she had to pay a 10% tax on her earnings and has an additional expense of $15 for cleaning supplies and transportation. How many homes did she clean?
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Context:
There are 12 students participating in a performance event. The event consists of $n$ programs. Each program is performed by 6 students, and any two programs have at most 2 common performers. Find the maximum value of $n$.
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Context:
A wheel of radius 8 rolls along the diameter of a semicircle of radius 25 until it bumps into this semicircle. Find the length of the portion of the diameter that cannot be touched by the wheel. Express your answer as a whole number.
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Context:
An inscribed circle is drawn inside isosceles trapezoid \(ABCD\) with \(AB = CD\). Let \(M\) be the point where the circle touches side \(CD\), \(K\) be the intersection point of the circle with segment \(AM\), and \(L\) be the intersection point of the circle with segment \(BM\). Calculate the value of \(\frac{AM}{AK} + \frac{BM}{BL}\).
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Context:
The tangents of the angles of a triangle are in the ratio $1: 2: 3$. How do the sides of the triangle compare to each other in length?
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Context:
In triangle $XYZ$, we have $\angle X = 90^\circ$, $YZ = 20$, and $\tan Z = 3 \sin Y$. Calculate $XY$.

---
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Context:
How do we find the value of $a$ such that the function $f_a(x) = \sin x + \frac{a}{3\pi}x$ has exactly three zeros in the interval $[0, 3\pi]$?
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Context:
A circle with a radius of 2 units rolls around the inside of a triangle with sides 9, 12, and 15 units. The circle is always tangent to at least one side of the triangle. Calculate the total distance traveled by the center of the circle when it returns to its starting position.
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Context:
If the quadratic trinomial $x^{2}+kx+5$ with respect to $x$ can be factored using the cross multiplication method, then the integer $k$ is ______.
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Context:
Find the exact value of the integral: $$\int_{x}^{+\infty}\frac{e^{-t}}{t}dt$$
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Context:
In a \( 29 \times 29 \) table, the numbers \( 1, 2, 3, \ldots, 29 \) are each written 29 times. It is found that the sum of the numbers above the main diagonal is three times the sum of the numbers below this diagonal. Find the number written in the central cell of the table.
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Context:
A rectangular piece of paper has corners labeled $A, B, C$, and $D$, with $BC = 80$ and $CD = 120$. Let $M$ be the midpoint of side $AB$. The corner labeled $A$ is folded along line $MD$ and the corner labeled $B$ is folded along line $MC$ until the segments $AM$ and $MB$ coincide. Let $S$ denote the point in space where $A$ and $B$ meet. If $H$ is the foot of the perpendicular from $S$ to the original plane of the paper, find $HM$.
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Context:
Let \( ABC \) be a triangle with \( AB = 6 \), \( AC = 7 \), and \( BC = 8 \). Let \( I \) be the incenter of \( ABC \). Points \( Z \) and \( Y \) lie on the interior of segments \( AB \) and \( AC \), respectively, such that \( YZ \) is tangent to the incircle. Given a point \( P \) such that
\[
\angle ZPC = \angle YPB = 90^\circ,
\]
find the length of \( IP \).
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Context:
Five teams participated in a commercial football tournament. Each team was supposed to play exactly one match against every other team. However, due to financial difficulties, some games were canceled.

By the end of the tournament, all teams had different scores, and no team had zero points. What is the minimum number of games that could have been played in this tournament, given that:
- A win awards 3 points,
- A draw awards 1 point,
- A defeat awards 0 points?
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Context:
We call a rectangle of the size  $1 \times 2$  a domino. Rectangle of the  $2 \times 3$  removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated.

It requires to tile rectangle of size  $2008  \times 2010$  by using dominoes and tetraminoes. What is the minimal number of dominoes should be used?
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Context:
Liz is preparing for a huge family Thanksgiving dinner and she decides to roast 4 turkeys. The turkeys weigh 16 pounds, 18 pounds, 20 pounds, and 22 pounds respectively. She can only roast one turkey at a time due to the size of her oven. The first two turkeys need to be roasted at a higher temperature, requiring an additional 20 minutes of preheating time. The other two turkeys do not require additional preheating time.

She roasts each turkey for 15 minutes per pound. If dinner is served at 6:00 pm, what is the latest time she can start preheating her oven and begin roasting the turkeys in order to finish on time?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A $12 \times 12$ grid is colored such that each cell is either black or white. Ensure that within any $3 \times 4$ or $4 \times 3$ subgrid, there is at least one black cell. Find the minimum number of black cells needed.
Please reason step by step, and put your final answer within \boxed{}.


Context:
To create a specific type of steel for a particular application, a certain amount of a chemical element must be added during the steelmaking process. It is known that the amount of this element to be added per ton of steel should be within the range of [1000, 2000] grams. Using the 0.618 method to determine the optimal amount to be added, let the amounts of the element added at the first, second, and third trial points be $x_1$, $x_2$, and $x_3$ respectively ($x_1 > x_2$). If the first trial point is better than the second, then find the amount $x_3$ to be added for the third trial point.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tom originally was only able to lift 100 kg farmer handles per hand. After some intensive training, he was able to increase this number by 150%. He then decided to specialize and was able to get an extra 25% improvement. In addition, he started using a new grip technique which further increased his lifting capacity by 10%.

He was later introduced to deadlifting, and he found that he could lift 400 pounds with ease. He then learned a new deadlift technique called "sumo style" which improved his lifting ability by 20%.

Given that 1 kg is approximately equal to 2.20462 pounds, calculate Tom's total lifting capacity (in kg) for the farmer handles and deadlift combined.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest natural  $ k $  such that among any  $ k $  distinct and pairwise coprime naturals smaller than  $ 2018, $  a prime can be found.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The negation of the proposition "$\forall x \gt 0$, $e^{x} \gt x+1$" is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways are there to color every integer either red or blue such that \( n \) and \( n+7 \) are the same color for all integers \( n \), and there does not exist an integer \( k \) such that \( k, k+1 \), and \( 2k \) are all the same color?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$ with angle $C$ equal to $30^{\circ}$, a median $AD$ is drawn. Angle $ADB$ equals $45^{\circ}$. Find angle $BAD$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a quadrilateral \(PQRS\) made from two similar right-angled triangles \(PQR\) and \(PRS\), where \(PQ = 3\), \(QR = 4\), and \(\angle PRQ = \angle PSR\), calculate the perimeter of quadrilateral \(PQRS\). Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
You have infinitely many boxes, and you randomly put 3 balls into them. The boxes are labeled \(1, 2, \ldots\). Each ball has a probability \(\frac{1}{2^n}\) of being put into box \(n\). The balls are placed independently of each other. What is the probability that some box will contain at least 2 balls?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\alpha$ and $\beta$ be such that $\pi<\alpha-\beta<3\pi$, and $cos\alpha+cos\beta=-27/65$ and $sin\alpha+sin\beta= -21/65$, find $cos(\alpha-\beta)/2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 8 football teams playing a round-robin tournament. The winning team gets 1 point, the losing team gets 0 points, and in the case of a draw, each team gets 0.5 points. After the tournament, the scores of the teams are ranked from highest to lowest, and it is found that no two teams have the same score. Additionally, the score of the second-place team is equal to the total score of the last four teams combined. Determine the score of the team that came in second place in this tournament.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(m\) be a positive integer, and let \(T\) denote the set of all subsets of \(\{1, 2, \ldots, m\}\). Call a subset \(S\) of \(T\) \(\delta\)-good if for all \(s_1, s_2 \in S\), \(s_1 \neq s_2\), \(|\Delta(s_1, s_2)| \geq \delta m\), where \(\Delta\) denotes symmetric difference (the symmetric difference of two sets is the set of elements that are in exactly one of the two sets). Find the largest possible integer \(s\) such that there exists an integer \(m\) and a \(\frac{1024}{2047}\)-good set of size \(s\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x)=\frac{|x|-\sin{x}+1}{|x|+1}$ ($x\in\mathbb{R}$), its maximum value is $M$ and its minimum value is $m$. Find the value of $M+m$ $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a parabolic curve $y^{2}=2x$ with a point $P$ on it, points $B$ and $C$ are on the y-axis. A circle described by $(x-1)^{2}+y^{2}=1$ is inscribed within the triangle $PBC$. Find the minimum area of triangle $PBC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three lines are drawn parallel to the sides of a triangle through a point inside it, dividing the triangle into six parts: three triangles and three quadrilaterals. The areas of all three inner triangles are equal. Determine the range within which the ratio of the area of each inner triangle to the area of the original triangle can lie.
Please reason step by step, and put your final answer within \boxed{}.


Context:
某校三个年级举办乒乓球比赛，每个年级选派 4 名选手参加比赛. 组委会随机将这 12 名选手分成 6 组，每组 2 人，则在上述分组方式中每组的 2 人均来自不同年级的概率为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ x$ be a real number such that the five numbers $ \cos(2 \pi x)$, $ \cos(4 \pi x)$, $ \cos(8 \pi x)$, $ \cos(16 \pi x)$, and $ \cos(32 \pi x)$ are all nonpositive.  What is the smallest possible positive value of $ x$?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the odd integer that contains the digit 5, is divisible by 3, and lies between $12^2$ and $13^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum:
\[ \sum_{x=0}^\infty x(x-1) {2+x \choose x}(0.008)(0.8)^x \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
求最小的实数 C>1，满足如下条件：对任意整数 n\geq 2 和满足 \df{1}{a_1}+\df{1}{a_2}+\cdots+\df{1}{a_n}=1 的非整正实数 a_1, a_2, \cdots, a_n，都可以找到正整数 b_i，使得 (1) 对 1\leq i\leq n，b_i=[a_i] 或 [a_i]+1；(2) 1<\df{1}{b_1}+\df{1}{b_2}+\cdots+\df{1}{b_n}\leq C.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate $$\int_{0}^{\infty}\frac{x}{1+x^3}dx.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Marie has 75 raspberry lollipops, 132 mint lollipops, 9 blueberry lollipops, and 315 coconut lollipops. She decides to distribute these lollipops equally among her 13 friends, distributing as many as possible. How many lollipops does Marie end up keeping for herself?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a,$ $b,$ $c$ be real numbers such that
\[a + 3b + 4c = a^2 + b^2 + c^2 = 25.\]Find the largest possible value of $a.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The display has a row of 4 parallel holes, each of which can display 0 or 1. If each time it displays two of the holes, but two adjacent holes cannot be displayed at the same time, the total number of signals that the display can show is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The train schedule in Hummut is hopelessly unreliable. Train A will enter Intersection X from the west at a random time between 9:00 am and 2:30 pm; each moment in that interval is equally likely. Train B will enter the same intersection from the north at a random time between 9:30 am and 12:30 pm, independent of Train A; again, each moment in the interval is equally likely. If each train takes 45 minutes to clear the intersection, what is the probability of a collision today?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Someone arranged a 10-volume collection of works in a random order. Let's call a "disorder" a pair of volumes where the volume with the higher number is to the left. For a given arrangement of volumes, the number \( S \) of all disorders is calculated. What values can \( S \) take?
Please reason step by step, and put your final answer within \boxed{}.


Context:
4. Determine the minimum value of the quantity $|x+y|$ given that the numbers $x$ and $y$ satisfy the relation $5 \cos (x+4 y)-3 \cos (x-4 y)-4 \sin (x-4 y)=10$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are infinitely many boxes - initially, one of them contains $n$ balls, and all others are empty. In a single move, we take some balls from a non-empty box and put them into an empty box. On a sheet of paper, we write down the product of the resulting number of balls in the two boxes. After some moves, the sum of all numbers on the sheet of paper became $2023$. What is the smallest possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square of perimeter 20 is inscribed in a square of perimeter 28. What is the greatest distance between a vertex of the inner square and a vertex of the outer square?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of counterexamples to the statement:
\[ \text{If } N \text{ is an odd positive integer the sum of whose digits is 4 and none of whose digits is 0, then } N \text{ is prime.} \]
Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f(x) = \frac{1}{x^3 + 3x^2 + 2x} \). Determine the smallest positive integer \( n \) such that
\[ f(1) + f(2) + f(3) + \cdots + f(n) > \frac{503}{2014}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
When Qiuqiu serves beer, one bottle can fill a certain number of cups. Each cup always has half a cup of foam, and the volume of the beer foam expands to 3 times its original volume. If one bottle of beer can fill 4 cups when there is no foam, how many cups can Qiuqiu fill with one bottle of beer? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The number $734,851,474,594,578,436,096$ is equal to $n^6$ for some positive integer $n$. What is the value of $n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dimension of the nullspace of the matrix $B_n$ over the field $\mathbb{F}_2$, where:
$$ B_n = \begin{bmatrix} J_n & J_n & \dots & J_n \\ J_n & J_n & \ddots & \vdots \\ \vdots & \ddots & \ddots & J_n \\ J_n & \dots & J_n & J_n \end{bmatrix} $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In trapezoid \(ABCD\), \(BC\) is parallel to \(AD\), \(AB = AD\), \(\angle ABC = \frac{2\pi}{3}\), and \(\angle BCD = \frac{\pi}{2}\). \(\triangle ABD\) is folded along \(BD\) such that point \(A\)'s projection onto plane \(BCD\) is point \(P\). Given that the cosine of the angle between \(AB\) and \(CD\) is \(\frac{\sqrt{3}}{6}\), find the cosine of the angle between \(BP\) and \(CD\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A point \(P\) is chosen at random in the interior of equilateral triangle \(ABC\). What is the probability that \(\triangle ABP\) has a greater area than each of \(\triangle ACP\) and \(\triangle BCP\)? Express your answer in the form \(\frac{k}{m}\), where \(k\) and \(m\) are coprime positive integers. Find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximal positive integer $p$ such that $5^7$ is the sum of $p$ consecutive positive integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A small airplane has $4$ rows of seats with $3$ seats in each row. Eight passengers have boarded the plane and are distributed randomly among the seats. A married couple is next to board. What is the probability there will be 2 adjacent seats in the same row for the couple?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( a \) and \( b \) be nonnegative real numbers such that
\[ \cos (ax + b) = \cos 31x \]
for all integers \( x \). Find the smallest possible value of \( a \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The letter T is formed by placing a $2\:\text{inch} \times 6\:\text{inch}$ rectangle vertically and a $2\:\text{inch} \times 4\:\text{inch}$ rectangle horizontally across the top center of the vertical rectangle. What is the perimeter of the T, in inches?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using the method of Lagrange Multipliers, find the maximum value of $a + b + c + abc$ given that $0< a,b,c < 1$ and $ab + bc + ca = 1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
袋中装有 2 个红球、 3 个白球、 4 个黄球，从中任取 4 个球，则其中三种颜色的球都有的概率是 $\qquad$。原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mr. and Mrs. Seventh have 7 children, all born on April 1, actually within six consecutive April 10. This year, for their birthdays, Mrs. Seventh made a cake with candles for each one - the number of candles equal to the number of years of each. João Seventh, the child who likes Mathematics the most, noticed that this year the total number of candles is double what it was 2 years ago and that there are 2 more cakes. How many candles will be lit this year?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many sequences of 5 positive integers $(a, b, c, d, e)$ satisfy $a b c d e \leq a+b+c+d+e \leq 10$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the greatest positive integer $m$ such that $n^2(1+n^2-n^4) \equiv 1 \pmod{2^m}$ for all odd integers $n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a $p\times q$ matrix $A$ of rank $\alpha$ and an $r\times s$ matrix $B$ of rank $\beta$, consider the set $V$ of all $q\times r$ matrices $C$ such that $ACB=0_{p\times s}$. Determine the dimension of the vector space $V$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Farmer John has $ 5$ cows, $ 4$ pigs, and $ 7$ horses. How many ways can he pair up the animals so that every pair consists of animals of different species? (Assume that all animals are distinguishable from each other.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that point A has coordinates (4, 3), and F is the focus of the parabola $y^2=4x$, if point P moves on the parabola, then when $|PA|+|PF|$ takes the minimum value, the coordinates of point P are ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Cyclic quadrilateral $ABCD$ satisfies $\angle ADC = 2 \cdot \angle BAD = 80^\circ$ and $\overline{BC} = \overline{CD}$. Let the angle bisector of $\angle BCD$ meet $AD$ at $P$. What is the measure, in degrees, of $\angle BPD$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
From the set $\{1,2,3, \cdots, 100\}$, find the subset with the largest number of elements such that no element is double any other element in the subset.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A stack contains 300 cards: 100 white, 100 black, and 100 red. For each white card, the number of black cards below it is counted; for each black card, the number of red cards below it is counted; and for each red card, the number of white cards below it is counted. Find the maximum possible value of the sum of these 300 counts.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Points \( A, B, C \) lie on a circle \(\omega\) such that \( BC \) is a diameter. \( AB \) is extended past \( B \) to point \( B' \) and \( AC \) is extended past \( C \) to point \( C' \) such that line \( B'C' \) is parallel to \( BC \) and tangent to \(\omega\) at point \( D \). If \( B'D = 4 \) and \( C'D = 6 \), compute \( BC \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two congruent circles $\odot A$ and $\odot B$ intersecting at points $E$ and $F$, with each circle passing through the center of the other, let line $AB$ intersect the two circles at points $C$ and $D$. Determine the measure of angle $\angle CED$ in degrees.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the hyperbola $x^{2}-y^{2}=a^{2}$ ($a \gt 0$) with left and right foci $F_{1}$ and $F_{2}$ respectively, and the circle $F_{2}$: $(x-c)^{2}+y^{2}=1$ intersecting the hyperbola and a line $x-y=0$ at point $M$, calculate the area of $\triangle F_{1}MF_{2}$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the least positive integer $k$ such that, in every convex 101 -gon, the sum of any $k$ diagonals is greater than or equal to the sum of the remaining diagonals?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $x$ is a real number, let $\lfloor x \rfloor$ be the greatest integer that is less than or equal to $x$. If $n$ is a positive integer, let $S(n)$ be defined by
\[
  S(n)
    = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
        + 10 \left( n - 10^{\lfloor \log n \rfloor}
               \cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
             \right) \, .
\]
(All the logarithms are base 10.) How many integers $n$ from 1 to 2011 (inclusive) satisfy $S(S(n)) = n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit $\lim_{n \rightarrow \infty} \left(\prod_{k=1}^{n} \sin \frac{k}{n}\right)^{\frac{1}{n}}$ using the fact that $\lim_{n \rightarrow \infty} \frac{n!}{n^n} = \frac{1}{e}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sequence $\{a_i\}_{i \ge 1}$ is defined by $a_1 = 1$ and \[ a_n = \lfloor a_{n-1} + \sqrt{a_{n-1}} \rfloor \] for all $n \ge 2$. Compute the eighth perfect square in the sequence.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two circles $C_1: x^2 + y^2 = m^2$ and $C_2: x^2 + y^2 - 2x - 4y - 20 = 0$ have exactly two common tangents, the range of real number $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three volleyballs with a radius of 18 lie on a horizontal floor, each pair touching one another. A tennis ball with a radius of 6 is placed on top of them, touching all three volleyballs. Find the distance from the top of the tennis ball to the floor. (All balls are spherical in shape.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\) with the right focus at \(F(c, 0)\), there exists a line \(l\) passing through point \(F\) that intersects the ellipse at points \(A\) and \(B\) such that \(OA \perp OB\). Find the range of the eccentricity of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left[\frac{\sqrt{5} - 1}{2}, 1\right)}


Context:
Given the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) with the left vertex \(A\) and the right focus \(F\), let \(P\) be an arbitrary point on the hyperbola in the first quadrant. If it is always true that \(\angle PFA = 2 \angle FAP\), find the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A regular triangle and a regular nonagon are coplanar and share a common side $\overline{AD}$. What is the degree measure of the exterior angle $BAC$?

[asy]
for(int i=0; i <=9; ++i) {
draw(dir(360*i/9+90)--dir(360*(i+1)/9+90));
}
pair A = dir(360*5/9+90);
pair D = dir(360*6/9+90);
pair C = A+dir(-30)*(D-A);
pair B = dir(360*4/9+90);

draw(A--C--D);

label("$A$",A,NE);
label("$B$",B,W);
label("$C$",C,S);
label("$D$",D,NE);

[/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Codger is a sloth with a certain number of feet. He has a challenging time buying shoes because the stores only sell the shoes in pairs. He already owns the set of shoes he is wearing. He needs to buy 6 pairs of shoes to have 5 complete sets of shoes. How many feet does Codger have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A bus, a truck, and a car are driving in the same direction on a straight road at constant speeds. At a certain moment, the bus is in front, the car is behind, and the truck is exactly in the middle between the bus and the car. After 10 minutes, the car catches up with the truck; another 5 minutes later, the car catches up with the bus. How many more minutes will it take for the truck to catch up with the bus?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Points $A$, $B$, $Q$, $D$, and $C$ lie on the circle shown and the measures of arcs $BQ$ and $QD$ are $42^\circ$ and $38^\circ$, respectively.  Find the sum of the measures of angles $P$ and $Q$, in degrees.

[asy]
import graph;

unitsize(2 cm);

pair A, B, C, D, P, Q;

A = dir(160);
B = dir(45);
C = dir(190);
D = dir(-30);
P = extension(A,B,C,D);
Q = dir(0);

draw(Circle((0,0),1));
draw(B--P--D);
draw(A--Q--C);

label("$A$", A, NW);
label("$B$", B, NE);
label("$C$", C, SW);
label("$D$", D, SE);
label("$P$", P, W);
label("$Q$", Q, E);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral $$\int_{-\infty}^{\infty} \frac{1}{(1 + x^2)^{n+1}} dx.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the side \( CD \) of the trapezoid \( ABCD \) with \( AD \parallel BC \), a point \( M \) is marked. From vertex \( A \), a perpendicular \( AH \) is dropped to the segment \( BM \). It is known that \( AD = HD \). Find the length of segment \( AD \), given that \( BC = 16 \), \( CM = 8 \), and \( MD = 9 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
When the roots of the polynomial

\[P(x)  = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10}\]

are removed from the number line, what remains is the union of $11$ disjoint open intervals. Find the number of these intervals on which \( P(x) \) is positive.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are \( n \) colored balls, of which some are red and some are black. Out of every 50 balls, 49 are red, and thereafter, out of every 8 balls, 7 are red. Given that at least 90% of all the balls are red, what is the maximum value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In quadrilateral $\mathrm{ABCD}$, $\angle B=90^{\circ}$, and $A B=B C=5$, $C D=2 \sqrt{15}$, $D A=\sqrt{10}$. What is the measure of $\angle B A D$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
从圆 \(x^{2}+y^{2}=4\) 上的点向椭圆 \(C: \frac{x^{2}}{2}+y^{2}= 1\) 引切线，两个切点间的线段称为切点弦。求所有的切点弦所在的直线围成区域的面积。原始答案的形式为\(\frac{k\pi}{m}\)，请给出k + m的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the Fibonacci sequence $\{a_n\}$, where each number from the third one is equal to the sum of the two preceding numbers, find the term of the Fibonacci sequence that corresponds to $\frac{{a_1}^2 + {a_2}^2 + {a_3}^2 + … + {a_{2017}}^2}{a_{2017}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
One day, School A bought 56 kilograms of fruit candy, each kilogram costing 8.06 yuan. Several days later, School B also needed to buy the same 56 kilograms of fruit candy, but happened to catch a promotional offer, reducing the price per kilogram by 0.56 yuan, and also offering an additional 5% of the same fruit candy for free with any purchase. How much less in yuan will School B spend compared to School A?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an arithmetic sequence $\{a_n\}$, the sum of the first $m$ terms is 30, and the sum of the first $2m$ terms is 100. Find the sum of the first $3m$ terms.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n \to \infty} \int_{\epsilon}^1\frac {n+1}{2^{n+1}}\frac{(1+t)^{n+1}-(1-t)^{n+1}}{t}\,dt \] where \(\epsilon \in (0, 1)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify a non-cyclic group such that the intersection of all its non-trivial subgroups is a non-trivial subgroup.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \iiint_A x^2 y^2 z \, dx \, dy \, dz \) where the region \( A \) is defined by \( A = \{ (x, y, z) \in \mathbb{R}^3 \mid 0 \leq z \leq x^2 + y^2, \ |x| + |y| = 1 \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the least significant digit of the product \( \prod\limits_{i=1}^{n/3} (i)(n-2i+1) \) for multiples of 3, where \( n \geq 9 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest natural number $n$ such that for any subset of $A = \{1, 2, 3, \ldots, 2003\}$ with $n$ elements, the complementary set contains at least one pair of consecutive numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle has a radius of 1. A rhombus and a triangle are described around this circle. Two sides of the triangle are parallel to the diagonals of the rhombus, and the third side is parallel to one of the sides of the rhombus and measures 5 units. Determine the side length of the rhombus.
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Context:
Express the antipodal map as a composition of one, two, or three reflections. Determine the minimum number of reflections needed.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \iint_{D} xy\sqrt{1-x-y} \, dx \, dy \) where the domain is \( D=\{(x,y): x\ge0, y\ge0, x+y\le1\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of states in a DFA that accepts the language \(L_n \subseteq \{0,1\}^*\) defined by:

\[L_n=\{s\in \{0,1\}^*\, |\,\, \text{for every substring } t \text{ of } s, \text{abs}(N_0(t)-N_1(t)) \leq n \}\]

where \(\text{abs}\) denotes the absolute value and \(N_0(t),N_1(t)\) denote the number of \(0\)'s and \(1\)'s in \(t\) respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Lebesgue measure of the set \( C = \left(\bigcup_{n=1}^\infty \left(-\frac{1}{2^n}, \frac{1}{2^n}\right)\right) \cap \mathbb{R} \setminus \mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct elements in the quotient group \(\mathbb{R}/\mathbb{Q}\), where \(\mathbb{R}\) is the set of real numbers and \(\mathbb{Q}\) is the set of rational numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of the constant \( c \) such that the random variable \( Y = \frac{c(X_1 + X_2)}{\sqrt{(X_3)^2 + (X_4)^2}} \) follows a t-distribution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a one-kilometer race, the completion times of three individuals are modeled by random variables. Let $X_1$, $X_2$, and $X_3$ represent the completion times in minutes for each person, where:
- $X_1$ is uniformly distributed over [2.9, 3.1]
- $X_2$ is uniformly distributed over [2.7, 3.1]
- $X_3$ is uniformly distributed over [2.9, 3.3]

Assuming the completion times are independent, calculate the expected value of the latest completion time, $Y = \max\{X_1, X_2, X_3\}$, to the nearest 0.1 minute.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$A = \lim_{n \to \infty} \left(\binom{n}{0} \binom{n}{1} \binom{n}{2}\cdots \binom{n}{n}\right)^{1/((n)(n+1))}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Provide an example of a decreasing function \( f(x) \) such that \( \lim_{x\to +\infty} f(x) = e. \) The function \( f(x) \) should have a simple form.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \( f(z) = \sqrt[3]{z^2 - 1} \), where \( f(3) = 2 \), find the value of \( f(-3) \) by considering the analytic continuation of \( f \) along the semicircle \( |z| = 3 \) in the lower half-plane.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive number \( \lambda \) such that for any sequence of real numbers \( \{x_n\} \) and for all positive integers \( m \), the inequality \( \frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i} \le \sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} \) holds, where \( \{y_n\} \) is defined by \( y_1 = x_1 \) and \( y_{n+1} = x_{n+1} - (\sum\limits_{i = 1}^{n} {x^2_i})^{\frac{1}{2}} \) for \( n \ge 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Trapezoid \( {ABCD} \) ,with \( {AB} \parallel  {CD} \) ,has side lengths \( {AB} = {11},{BC} = 8,{CD} = {19} \) ,and \( {DA} = 4 \) . Compute the area of
 the convex quadrilateral whose vertices are the circumcenters of \( \bigtriangleup  {ABC}, \bigtriangleup  {BCD}, \bigtriangleup  {CDA}, \) and \( \bigtriangleup  {DAB} \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( G \) be a non-abelian group of order \( p^3 \), where \( p \) is a prime. Determine the number of distinct conjugacy classes in \( G \).
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Context:
Determine the minimum number of times a rectangular object with faces labeled A, B, C, and D must be tossed so that the probability of the ratio of the number of D's to the total number of tosses being between 0.15 and 0.35 is at least 0.95.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of values for $a > 0$ such that the curves $C_1: y = e^{6x}$ and $C_2: y = ax^2$ have exactly two common tangents.
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Context:
Determine whether there is a closed-form expression for the integral: $$\frac{1}{2\pi i}\oint_{x=0} e^{\sqrt{z}(x+\frac{1}{x})} \frac{dx}{x}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a geometric sequence with the first term a_1 = 1536 and the common ratio q = -1/2, find the value of n that maximizes the product of the first n terms, denoted as Π_n. Express your answer as an integer value of n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
All digits in the 6-digit natural numbers $a$ and $b$ are even, and any number between them contains at least one odd digit. Find the largest possible value of the difference $b-a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
two numbers have a h . c . f of 21 and a product of two numbers is 138567 . find the l . c . m of the two numbers ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
5. [5 points] On a plane with a given rectangular Cartesian coordinate system, a square is drawn with vertices at points $(0 ; 0),(0 ; 63),(63 ; 63)$, and $(63 ; 0)$. Find the number of ways to choose two grid nodes inside this square (not including its boundary) such that at least one of these nodes lies on one of the lines $y=x$ or $y=63-x$, but neither of the chosen nodes lies on any line parallel to any of the coordinate axes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the parade ground, 2018 soldiers are lined up in a single row. The commander can order either all soldiers standing in even positions or all soldiers standing in odd positions to leave the formation. After this order, the soldiers remaining in formation close ranks into a single row. In how many ways can the commander issue a series of 8 orders so that there are exactly 7 soldiers left in formation?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the point \( P \) lies in the plane of the right triangle \( \triangle ABC \) with \( \angle BAC = 90^\circ \), and \( \angle CAP \) is an acute angle. Also given are the conditions:
\[ |\overrightarrow{AP}| = 2, \quad \overrightarrow{AP} \cdot \overrightarrow{AC} = 2, \quad \overrightarrow{AP} \cdot \overrightarrow{AB} = 1. \]

Find the value of \( \tan \angle CAP \) when \( |\overrightarrow{AB} + \overrightarrow{AC} + \overrightarrow{AP}| \) is minimized.
Please reason step by step, and put your final answer within \boxed{}.


Context:
David and Brenda are playing Scrabble. In the first turn, Brenda scores 18 points, and David scores 10 points. In the second turn, Brenda scores 25 points and David scores 35 points. In the third turn, Brenda is ahead by 22 points when she makes a 15-point play. David responds with a 32-point play. By how many points is Brenda now ahead?
Please reason step by step, and put your final answer within \boxed{}.


Context:
60. Given $\triangle A B C$. On the ray $B A$, take a point $D$ such that $|B D|=|B A|+|A C|$. Let $K$ and $M$ be two points on the rays $B A$ and $B C$ respectively such that the area of $\triangle B D M$ is equal to the area of $\triangle B C K$. Find $\widehat{B K M}$, if $\widehat{B A C}=\alpha$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find $a$ such that $83359654581036155008716649031639683153293510843035531 ^ a \equiv 1 \pmod{32003240081}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all such positive integers \( k \) that the number

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

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


Context:
Trapezoid $ABCD$ has $AD||BC$, $BD = 1$, $\angle DBA = 23^{\circ}$, and $\angle BDC = 46^{\circ}$.  The ratio $BC: AD$ is $9: 5$.  What is $CD$?

$
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知 $f(x)=2 m x^{2}-2 m x-8 x+9, g(x)=m x-m$ ，对任意 $x \in \mathbf{R}, f(x)$ 与 $g(x)$ 至少有一个为正数，则实数 m 的取值范围是. 请给出该区间的两个端点之和.
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Context:
Compute the limit of the function:

\[
\lim _{x \rightarrow 0}\left(\frac{1+\sin x \cdot \cos \alpha x}{1+\sin x \cdot \cos \beta x}\right)^{\operatorname{ctg}^{3} x}
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Grass on an entire meadow grows uniformly dense and fast. It is known that 70 cows would eat all the grass in 24 days, and 30 cows would eat it in 60 days. How many cows would eat all the grass in 96 days?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Michael scored 260 points during his freshman year on the school basketball team. In his sophomore year, he improved by 15% and scored P% more points during his junior year than he did in his sophomore year. Finally, in his senior year, he scored 12% more points than in his junior year, plus an extra 17 points from successful free throws. If the total number of points scored during all four years is 1,450, find the value of P.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a caravan, in addition to 50 hens, there are 45 goats, 8 camels, 12 spiders, and 6 octopuses with some keepers. 3 of the keepers have a prosthetic leg, while 2 of the keepers have a missing leg. If the total number of feet is 372 more than the number of heads in the caravan, the number of keepers is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the fifth-largest divisor of 2,500,000,000.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of values for the real number \(\lambda\) such that the inequality \(\sqrt{x(x^2 + 8)(8-x)} < \lambda(x+1)\) holds for all real numbers \(x\) in the interval \((0, 2)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the marginal probability \( P(X_1=27, X_2=20) \) given the joint probability distribution:
\[
f(X_1, X_2 \mid p_1, p_2) = p_1^{x_1} (1 - p_1)^{(n_1-x_1)}p_2^{x_2}(1-p_2)^{(n_2-x_2)},
\]
where \( p_1 \sim \text{Unif}(0,1) \) independently \( p_2 \sim \text{Unif}(0,1) \), \( n_1=34 \), and \( n_2 = 56 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jonsey is awake for 2/3 of the day and spends 1/2 her time awake playing outside and the rest inside. Her brother, Riley, is awake for 3/4 of the day and spends some fraction of his day outside and the rest inside. On average, they spend 10 hours inside. What fraction of his day does Riley spend outside?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify the tensor product $H \otimes_{\mathbb R} \mathbb C$ in familiar terms, where $H$ is the Hamiltonian quaternions, $\mathbb C$ is the complex numbers, and $\mathbb R$ is the real numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A grocer creates a display of cans where the top row contains two cans and each subsequent lower row has three more cans than the row preceding it. If the total number of cans used in the display is 120, how many rows are there in the display?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Rapidly spinning a coin that is standing upright (assuming the axis of rotation remains stationary), the three-dimensional shape formed by the rotation is ___.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For 8-bit symbols, Reed-Solomon codes are of the form \((n,k) = (255-z,255-2t-z)\), where:

- \(n\) is the number of symbols in a codeword.
- \(k\) is the number of data symbols that can be encoded.
- \(t\) is the number of errors that can be corrected.
- \(z\) is the number of implicit zeros that can be used if the code is to be shortened.

Determine the maximum value of \(t\) for these codes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $r$ for which there exists a finite $r$-dense subset in the unit ball of $C([0,1])$ with respect to the supremum norm $d_{	ext{inf}}$. A subset $Y \subset (X, d)$ is $r$-dense if for every $x \in X$, there exists a $y \in Y$ such that $d(x, y) < r$. Provide the set of values for $r$ where such a finite subset exists.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Equilateral $\triangle DEF$ has side length $300$. Points $R$ and $S$ lie outside the plane of $\triangle DEF$ and are on opposite sides of the plane. Furthermore, $RA=RB=RC$, and $SA=SB=SC$, and the planes containing $\triangle RDE$ and $\triangle SDE$ form a $150^{\circ}$ dihedral angle. There is a point $M$ whose distance from each of $D$, $E$, $F$, $R$, and $S$ is $k$. Find $k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $A B C, \angle B A C=60^{\circ}$. Let \omega be a circle tangent to segment $A B$ at point $D$ and segment $A C$ at point $E$. Suppose \omega intersects segment $B C$ at points $F$ and $G$ such that $F$ lies in between $B$ and $G$. Given that $A D=F G=4$ and $B F=\frac{1}{2}$, find the length of $C G$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $\alpha\geq 1$  be a real number. Define the set  $$ A(\alpha)=\{\lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor, \lfloor 3\alpha\rfloor,\dots\} $$  Suppose that all the positive integers that **does not belong** to the  $A(\alpha)$  are exactly the positive integers that have the same remainder  $r$  in the division by  $2021$  with  $0\leq r<2021$ . Determine all the possible values of  $\alpha$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum value of  $$ \int^1_0|f'(x)|^2|f(x)|\frac1{\sqrt x}dx $$  over all continuously differentiable functions  $f:[0,1]\to\mathbb R$  with  $f(0)=0$  and  $$ \int^1_0|f'(x)|^2dx\le1. $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f(x)$ be a strictly increasing function defined for all $x > 0$ such that $f(x) > -\frac{1}{x}$ for all $x > 0$, and
\[f(x) f \left( f(x) + \frac{1}{x} \right) = 1\]for all $x > 0$.  Find $f(1)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x) = \log_{3}(x^{2} + ax + a + 5)$, if $f(x)$ is a decreasing function in the interval $(-\infty, 1)$, then the range of the real number $a$ is:
_______ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Unit size cubes are stacked inside a large rectangular box with dimensions corresponding to four consecutive multiples of a prime number p. Choose the exact number of cubes that can completely fill the box.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Among the numbers $11n + 10^{10}$, where $1 \le n \le 10^{10}$ is an integer, how many are squares?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a horizontal plane, there are 4 touching spheres with a radius of \( 20 - 10 \sqrt{2} \), and their centers form a square. Above the center of the square, a fifth sphere of the same radius is placed on top of the four spheres. Find the distance from its highest point to the plane.
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Context:
Alex is stuck on a platform floating over an abyss at $1 \mathrm{ft} / \mathrm{s}$. An evil physicist has arranged for the platform to fall in (taking Alex with it) after traveling 100ft. One minute after the platform was launched, Edward arrives with a second platform capable of floating all the way across the abyss. He calculates for 5 seconds, then launches the second platform in such a way as to maximize the time that one end of Alex's platform is between the two ends of the new platform, thus giving Alex as much time as possible to switch. If both platforms are 5 ft long and move with constant velocity once launched, what is the speed of the second platform (in $\mathrm{ft} / \mathrm{s}$)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
the original price of a suit is $300. The price increased by 20%, and after this increase, the store published a 30% off coupon for a one-day sale. However, the day after the sale, the price was reduced by an additional 10%. Given that the consumers who used the coupon on the sale day were getting 30% off the increased price, how much did those who bought the suit the day after the sale pay for the suit?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the hyperbola $my^{2}-x^{2}=1$ whose one vertex lies on the axis of the parabola $y= \frac {1}{2}x^{2}$, determine the eccentricity of the hyperbola. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rectangular grass field is 75 m * 55 m, it has a path of 2.5 m wide all round it on the outside. The area of the path is 6750 sq m. If the cost of constructing the path is Rs. 10 per sq m, what is the total cost of constructing the path?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $M$ be a subset of the set of $2021$ integers $\{1, 2, 3, ..., 2021\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$ we have $|a + b - c | > 10$.
Determine the largest possible number of elements of $M$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In rectangle $ABCD$ with center $O$, where $AB=10$ and $BC=8$, circle $\gamma$ is centered at $O$ and is tangent to both $\overline{AB}$ and $\overline{CD}$. Points $M$ and $N$ are located on $\overline{AD}$ and $\overline{BC}$, respectively. The segment $MN$ intersects $\gamma$ at two distinct points $P$ and $Q$, with $P$ between $M$ and $Q$. Given that the ratio $MP : PQ : QN = 3 : 5 : 2$, the length $MN$ can be expressed as $\sqrt{a} - \sqrt{b}$, where $a$ and $b$ are positive integers. Determine the value of $100a + b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the universal set U = A ∪ B = {x ∈ ℤ | 0 ≤ x ≤ 6} and A ∩ (∁_U B) = {1, 3, 5}, deduce the set B. Express your answer as a set of integers, including all elements of the set.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can 1 apple, 2 peaches, 3 cucumbers, and 2 plums be distributed among Annus, Bözsi, Cili, and Dóra so that each girl gets 2 whole fruits? (Fruits of the same type cannot be distinguished from each other.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Juvy has a garden that is divided into 60 equally spaced rows with 18 plants in each row. She plants parsley in 3/5 of the rows, rosemary in 1/4 of the rows, mint in 1/6 of the remaining rows, thyme in 7/12 of the remaining rows after mint, and basil in 2/3 of the remaining rows after thyme. The rest of the rows will be planted with chives. How many chives will Juvy plant?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知两圆$$x^{2}+y^{2}=9$$和$$(x-3)^{2}+y^{2}=27$$，求大圆被小圆截得劣弧的长度.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral:

$$\int_{0}^{1} (1-x^7)^{\frac{1}{3}}-(1-x^3)^{\frac{1}{7}} \: dx$$
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Context:
Each face of a cube is painted either red or blue, each with probability $1/2$. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color? The original answer is in \frac{k}{m} format, please give the value of k + m.
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Context:
Find the angle between the line of intersection of the planes $2 x-y-3 z+5=0$ and $x+y-2=0$ and the plane passing through the points $M(-2 ; 0 ; 3), N(0 ; 2 ; 2)$ and $K(3 ;-3 ; 1)$.

#
Please reason step by step, and put your final answer within \boxed{}.


Context:
A bagel is cut into sectors. A total of 10 cuts were made. How many pieces were formed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( k \) be a positive constant. The sequence \( x_i \) of positive real numbers has a sum equal to \( k \). What are the possible values for the sum of \( x_i^2 \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Karina took several matches from a box and assembled from them a $3 \times 7$ grid of squares with sides equal to one match, as shown in the picture below.

What is the minimum number of matches she needs to take from the box so that with all the matches she can assemble a square grid? (The squares in the grid must again have a side equal to one match. There should not be any extra matches left.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four points $B,$ $A,$ $E,$ and $L$ are on a straight line, as shown. The point $G$ is off the line so that $\angle BAG = 120^\circ$ and $\angle GEL = 80^\circ.$ If the reflex angle at $G$ is $x^\circ,$ then what does $x$ equal?

[asy]
draw((0,0)--(30,0),black+linewidth(1));
draw((10,0)--(17,20)--(15,0),black+linewidth(1));
draw((17,16)..(21,20)..(17,24)..(13,20)..(14.668,16.75),black+linewidth(1));
draw((17,16)..(21,20)..(17,24)..(13,20)..(14.668,16.75),Arrows);
label("$B$",(0,0),S);
label("$A$",(10,0),S);
label("$E$",(15,0),S);
label("$L$",(30,0),S);
label("$G$",(17,20),N);
label("$120^\circ$",(10,0),NW);
label("$80^\circ$",(15,0),NE);
label("$x^\circ$",(21,20),E);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Konsistent Karl is taking this contest. He can solve the first five problems in one minute each, the next five in two minutes each, and the last five in three minutes each. What is the maximum possible score Karl can earn? Recall that this contest is $15$ minutes long, there are $15$ problems, and the $n$th problem is worth $n$ points. Assume that entering answers and moving between or skipping problems takes no time.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For $n \geq 1$, let $a_n$ be the number beginning with $n$ 9's followed by 744; for example, $a_4 = 9999744$. Define \[ f(n) = \max \{ m \in \mathbb{N} \mid 2^m \text{ divides } a_n \}, \] for $n \geq 1$. Find $f(1) + f(2) + f(3) + \cdots + f(10)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a football game, wristbands were given to every spectator for both their hands. There were two types of wristbands - type A and type B. Type A wristbands were given to 3 spectators for each hand, while Type B wristbands were given to 2 spectators for each hand. In total, 460 wristbands were distributed. How many spectators received both type A wristbands for both hands?
Please reason step by step, and put your final answer within \boxed{}.


Context:
At a crossroads, if vehicles are not allowed to turn back, calculate the total number of possible driving routes.
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Context:
Two non-intersecting circles with centers at points \(A\) and \(B\) are inscribed in an angle measuring \(\alpha\) \((0 < \alpha < 180^\circ)\). The distance \(A B\) is \(n\). A segment \(M N\) with its endpoints on the sides of the angle touches both circles. Find the difference in the areas of triangles \(B M N\) and \(A M N\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these 4 numbers?
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Context:
In a faraway kingdom, there are coins of three types in circulation: bronze coins worth 1 ruble, silver coins worth 9 rubles, and gold coins worth 81 rubles. From a treasury that has an unlimited supply of each type of coin, a certain amount of money less than 700 rubles was issued using 23 coins. Find this amount, given that it is not possible to issue this amount with fewer coins.
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Context:
Let $A, B, C, D, E, F$ be 6 points on a circle in that order. Let $X$ be the intersection of $AD$ and $BE$, $Y$ is the intersection of $AD$ and $CF$, and $Z$ is the intersection of $CF$ and $BE$. $X$ lies on segments $BZ$ and $AY$ and $Y$ lies on segment $CZ$. Given that $AX=3, BX=2, CY=4, DY=10, EZ=16$, and $FZ=12$, find the perimeter of triangle $XYZ$.
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Context:
During what year after 1994 did sales increase the most number of dollars?
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Context:
For a positive integer $n$, let $p(n)$ denote the smallest prime divisor of $n$. Find the maximum number of divisors $m$ can have if $p(m)^4 > m$.
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Context:
Consider triangle $A B C$ with side lengths $A B=4, B C=7$, and $A C=8$. Let $M$ be the midpoint of segment $A B$, and let $N$ be the point on the interior of segment $A C$ that also lies on the circumcircle of triangle $M B C$. Compute $B N$.
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Context:
Triangle $A B C$ has incircle $\omega$ which touches $A B$ at $C_{1}, B C$ at $A_{1}$, and $C A$ at $B_{1}$. Let $A_{2}$ be the reflection of $A_{1}$ over the midpoint of $B C$, and define $B_{2}$ and $C_{2}$ similarly. Let $A_{3}$ be the intersection of $A A_{2}$ with $\omega$ that is closer to $A$, and define $B_{3}$ and $C_{3}$ similarly. If $A B=9, B C=10$, and $C A=13$, find \left[A_{3} B_{3} C_{3}\right] /[A B C].
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Context:
In the given diagram, a circle of radius 3 has two tangents that intersect at a point, forming an angle of 60 degrees. Calculate the shaded area, which consists of the interior of the circle and the area between the circle and the two tangents. Express your answer in terms of pi and the square root of 3.
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Context:
Given the quadratic function y = x^2 - (m-2)x + 4, find the value of m that does not satisfy the condition that the vertex lies on the coordinate axis.
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Context:
Given $f(x) = x^2 - ax$, the inequality $|f(f(x))| \leqslant 2$ holds for all $x$ in the interval $[1, 2]$. Determine the maximum possible value of the real number $a$.
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Context:
Five members of the Lexington Math Team are sitting around a table. Each flips a fair coin. Given that the probability that three consecutive members flip heads is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
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Context:
Mrs. Delta's language class has 52 students, each with unique initials, and no two students have initials that are alphabetically consecutive (e.g., AB cannot follow AC directly). Assuming Y is considered a consonant, what is the probability of randomly picking a student whose initials (each first and last name starts with the same letter, like AA, BB) are both vowels? Express your answer as a common fraction.
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Context:
In the expansion of \((x + y + z)^8\), determine the sum of the coefficients of all terms of the form \(x^2 y^a z^b\) (\(a, b \in \mathbf{N}\)).
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Context:
Given the equation $\dfrac{x-1}{x-4}=\dfrac{m}{x-4}$, find the value of $m$ that makes the equation have a root. Express your answer as a single integer.
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Context:
Calculate the degree of the field extension \([F_{p^2}(t)(t^{1/p}):F_p(t)]\) using the tower of fields \(F_p \subseteq F_{p^2} \subseteq F_{p^2}(t) \subseteq F_{p^2}(t)(t^{1/p})\).
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Context:
Calculate $T(10, 4) \pmod{7}$, where $T(n, k) = \sum_{j=1}^{\min(k,n)} j T(n - j, j)$ for a positive integer $n$.
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Context:
5. On a sheet of paper, 14 parallel lines $L$ and 15 lines $P$ perpendicular to them are drawn. The distances between adjacent lines from $L$ from the first to the last are given: 2;4;6;2;4;6;2;4;6;2;4;6;2. The distances between adjacent lines from $P$ are also known: 3;1;2;6;3;1;2;6;3;1;2;6;3;1. Find the greatest length of the side of a square whose boundaries lie on the lines $L$ and $P$.
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Context:
Raymonde's financial institution publishes a list of service charges as follows: Autodebit costs $0.60 per transaction, cheque costs $0.50 per transaction, and cash withdrawal costs $0.45 per transaction. For her first twenty-five transactions, she uses Autodebit three times as often as she writes checks, and she also writes as many checks as she makes cash withdrawals. After her twenty-fifth transaction, she begins to make single transactions. Calculate the smallest number of transactions she needs to make so that her monthly service charges will exceed the $15.95 "all-in-one" fee. Express your answer as a whole number.
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Context:
Evaluate the contour integral $\oint_{|z|=1} z^{-1+i} \,dz$ using the principal branch of the logarithm, defined on the set $\{z: 0 \leq \arg(z) < 2\pi\}$.
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Context:
The circles $k_{1}$ and $k_{2}$, both with unit radius, touch each other at point $P$. One of their common tangents that does not pass through $P$ is the line $e$. For $i>2$, let $k_{i}$ be the circle different from $k_{i-2}$ that touches $k_{1}$, $k_{i-1}$, and $e$. Determine the radius of $k_{1999}$.
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Context:
Given a positive integer \( n \), find the number of positive integer solutions \((x, y)\) for the equation \(\frac{1}{n} = \frac{1}{x} + \frac{1}{y}\) where \( x \neq y \).
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Context:
Geli is committed to her workout routine. On her first day, she started doing 10 push-ups. Her fitness trainer encouraged her to add 5 more push-ups each day. Geli did a total of 45 push-ups throughout her first week. How many times a week does Geli workout?
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Context:
The GooGoo brand of clothing manufactures four types of shirts: Type A with 3 buttons, Type B with 5 buttons, Type C with 6 buttons, and Type D with 4 buttons. A large department store ordered GooGoo to manufacture 200 of Type A, 300 of Type B, 150 of Type C, and 250 of Type D shirts. Additionally, GooGoo provides a discount on button usage for bulk orders, reducing button usage by 10% for orders over 200 shirts. Calculate the total number of buttons GooGoo brand will use to manufacture all the shirts for this order, taking the discount into account.
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Context:
已知$( 1 - \sin \alpha ) ( 1 - \cos \alpha ) = \frac { 2 } { 2 5 }$，设$( 1 + \sin \alpha ) ( 1 + \cos \alpha ) = a + \frac { b } { c }$，其中α是锐角，a、b、c均为正整数，且b，c(b<c)互质，则$\sqrt { a + b + c } =$
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Context:
$I=\int_{50}^{+\infty} e^{-x^{2}} \mathrm{~d} x$, 则 $[-\ln I]=$ $\qquad$
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Context:
将正奇数划分成下列各组：（1），（3，5，7），（9，11，13，15，17），（19，21，23，25，27，29，31}，……则前$\textit{n}$组各数的和是_____．
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Context:
10. Given a sequence $\left\{a_{n}\right\}$ whose terms are all non-negative real numbers, and satisfies: for any integer $n \geqslant 2$, we have $a_{n+1}=$ $a_{n}-a_{n-1}+n$. If $a_{2} a_{2022}=1$, find the maximum possible value of $a_{1}$.
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Context:
Triangle $\triangle ABC$ has $\angle{A}=90^\circ$ with $BC=12$. Square $BCDE$ is drawn such that $A$ is in its interior. The line through $A$ tangent to the circumcircle of $\triangle ABC$ intersects $CD$ and $BE$ at $P$ and $Q$, respectively. If $PA=4\cdot QA$, and the area of $\triangle ABC$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$.
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Context:
Mr. Thompson, a parent who believes that teachers should focus solely on academic subjects, is organizing a math-focused weekend event for students. He decides to create a workbook with a series of math exercises. For every 5 math problems he writes, he takes a 10-minute break. If he spends a total of 3 hours working on the workbook, how many math problems does Mr. Thompson write, assuming he only takes breaks during this time?
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Context:
将一个色子连续投掷五次，则事件 "五次出现的点数既不全相同也不两两互异，且从第二次起每一次的点数都不小于前一次的点数"的概率是 $\qquad$。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:
How many of the coefficients of $(x+1)^{65}$ cannot be divisible by $65$? Find the number of such coefficients.
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Context:
A spider has 8 identical socks and 8 identical shoes. In how many different orders can the spider dress, knowing that, obviously, on each leg, she must put the shoe on after the sock?
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Context:
The height of a triangle, equal to 2, divides the angle of the triangle in the ratio 2:1, and the base of the triangle into parts, the smaller of which is equal to 1. Find the area of the triangle.
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Context:
a's speed is some times that of b. If a and b run a race, a should give b a head start of 0.09523809523809523 of the length of the race, so that the race ends in a dead heat. What is the ratio of a's speed to b's speed?
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Context:
What is the exact closed form of the following expression? $(2^0 + \dots + 2^n) + (2^1 + \dots + 2^{n+1}) + \dots + (2^n + \dots + 2^{2n})$
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Context:
The cost of purchasing a car is 150,000 yuan, and the annual expenses for insurance, tolls, and gasoline are about 15,000 yuan. The maintenance cost for the first year is 3,000 yuan, which increases by 3,000 yuan each year thereafter. Determine the best scrap year limit for this car.
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Context:
A special fair die has 10 faces numbered from 1 to 10. Let \( r_i \) be the result of the \( i \)-th throw and \( a \) be the imaginary fifth root of unity. Calculate the probability that \( \sum_{i=1}^5 a^{r_i} = 0 \).
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Context:
Let $P(A)$ be the arithmetic mean of all elements of the set $A = \{ a_1, a_2, \ldots, a_n \}$, defined as $P(A) = \frac{1}{n} \sum^{n}_{i=1} a_i$. A subset $B$ of $A$ is called a "balanced subset" if $B$ is non-empty and $P(B) = P(A)$. Given the set $M = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$, find the number of all "balanced subsets" of $M$.
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Context:
Calculate the number of 10-letter words that do not contain all five different vowels (A, E, I, O, U).
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Context:
Consider a hyperbola given by $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$). Construct a tangent line to the circle $x^2 + y^2 = a^2$ from the hyperbola's left focus $F(-c,0)$, with the point of tangency being $E$. Extend $FE$ to intersect the parabola $y^2 = 4cx$ at the point $P$. If $E$ is the midpoint of the line segment $FP$, find the eccentricity of the hyperbola.
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Context:
Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \), \( f(1) = 1 \), and \( |f'(x)| \leq 2 \) for all real numbers \( x \). If \( a \) and \( b \) are real numbers such that the set of possible values of \( \int_{0}^{1} f(x) \, dx \) is the open interval \( (a, b) \), determine \( b - a \).
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Context:
Given the functions f(x) = log2x and g(x) = sqrt(x) + sqrt(a - x) (a > 0), if for all x1 in the domain of g(x), there exists an x2 in the interval [4, 16] such that g(x1) = f(x2), then the range of values for the real number a is _______.
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Context:
Given the parabola $y^{2}=2 m x$, what is the length of the shortest chord that is normal to the curve at one end?Write down the value when \(m = \sqrt{3}\).
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Context:
1, 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 1, 2, 3, 4, 5, 4, 2, 1, 2, 3, 4, 5, 6, 5, 3, 1, 2, 3, 4, 5, 6, 7, 6, 4, 2, 1, 2, 3, 4, 5, 6, 7, 8, 7, 5, 3, 1, ___?
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Context:
Given the function $f(x) = e^x + 2x - a$, where $a \in \mathbb{R}$, if there exists a point $(x_0, y_0)$ on the curve $y = \sin x$ such that $f(f(y_0)) = y_0$, then the range of values for the real number $a$ is ______.
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Context:
A square with a side length of one unit has one of its vertices separated from the other three by a line \( e \). The products of the distances of opposite vertex pairs from \( e \) are equal. What is the distance of the center of the square from \( e \)?
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Context:
A wooden cube has edges of length $3$ meters. Square holes, with a side of $1$ meter, are centered in each face and cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. What is the total surface area, including the inside, in square meters?
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Context:
You have six blocks in a row, labeled 1 through 6, each with weight 1. Call two blocks $x \leq y$ connected when, for all $x \leq z \leq y$, block $z$ has not been removed. While there is still at least one block remaining, you choose a remaining block uniformly at random and remove it. The cost of this operation is the sum of the weights of the blocks that are connected to the block being removed, including itself. Compute the expected total cost of removing all the blocks.
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Context:
The "Changfei Class" at Hua Shi No.1 consists of $m$ students, and the school specially arranges $n$ teachers as mentors. In one activity of the class, each pair of students asks each other one question, each student asks each mentor one question, and each mentor asks the whole class one question. All the questions asked are different, and there are a total of 51 questions. Find the sum of $m$ and $n$.
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Context:
8. (10 points) Three households, A, B, and C, plan to subscribe to newspapers. There are 5 different newspapers available. It is known that each household subscribes to two different newspapers, and any two households have exactly one newspaper in common. How many different subscription methods are there for the three households?
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Context:
A pentagon \(ABCD E\) is inscribed in a circle of radius 10 such that \(BC\) is parallel to \(AD\) and \(AD\) intersects \(CE\) at \(M\). The tangents to this circle at \(B\) and \(E\) meet the extension of \(DA\) at a common point \(P\). Suppose \(PB = PE = 24\) and \(\angle BPD = 30^{\circ}\). Find \(BM\).
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Context:
Find the number of distinct monic quadratic trinomials (i.e., with the leading coefficient equal to 1) with integer coefficients such that they have at least one root, all of their roots are powers of 11 with non-negative integer exponents, and their coefficients do not exceed \(1331^{38}\) in absolute value.
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Context:
A class composed of 12 girls and 10 boys was sent to the library for their reading class. Their teacher found out that only 5/6 of the girls and 4/5 of the boys are reading. In addition, the girls read at a rate of 3 books per week on average, while the boys read 2 books per week on average. Furthermore, among the students who are reading, 20% of the girls read at a faster rate of 5 books per week, while 10% of the boys read at a slower rate of 1 book per week. Calculate the total number of books read by the whole class per week, given the varying reading rates and considering the students who are not reading.
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Context:
Marcy is the lunch monitor in an elementary school cafeteria. She gives 5 time-outs for running, 2 more than three times that number of time-outs for throwing food, the square root of triple that food-throwing number for swearing, and twice the number of running time-outs for talking loudly. If each time-out is 5 minutes, how much total time will the students spend in time-out?
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Context:
Let complex number \( z = \cos \frac{4\pi}{7} + i \sin \frac{4\pi}{7} \). Determine the value of \( \left| \frac{z}{1+z^2} + \frac{z^2}{1+z^4} + \frac{z^3}{1+z^6} \right| \).
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Context:
Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(g(x) - y) = g(x) + g(g(y) - g(-x)) + 2x \] for all real numbers \( x \) and \( y \).

Let \( m \) be the number of possible values of \( g(4) \), and let \( t \) be the sum of all possible values of \( g(4) \). Find \( m \times t \).
Please reason step by step, and put your final answer within \boxed{}.