Context:
Evaluate \( \mathop{\int\!\!\! \int}_{R_{1}}f(x,y)\,dA \) where \( R_{1} = [0, 5] \times [-4, 4] \), given that \( \mathop{\int\!\!\! \int}_{R_{2}}f(x,y)\,dA = 10 \) for \( R_{2} = [0, 5] \times [0, 4] \), \( \mathop{\int\!\!\! \int}_{R_{3}}f(x,y)\,dA = -24 \) for \( R_{3} = [-5, 0] \times [0, 4] \), and \( f(x,-y) = -f(x,y) \) for all \((x,y)\).
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Context:
Two circles are externally tangent. Lines \(\overline{PAB}\) and \(\overline{PA'B'}\) are common tangents with \(A\) and \(A'\) on the smaller circle and \(B\) and \(B'\) on the larger circle. If \(PA = AB = 4\), then what is the value of \(x\) if the area of the smaller circle is \(x\pi\)?
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Context:
On the New Year's table, there are 4 glasses in a row: the first and third contain orange juice, and the second and fourth are empty. While waiting for guests, Valya absent-mindedly and randomly pours juice from one glass to another. Each time, she can take a full glass and pour all its contents into one of the two empty glasses.

Find the expected number of pourings for the first time when the situation is reversed: the first and third glasses are empty, and the second and fourth glasses are full.
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Context:
## Problem Statement

Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.

$M_{1}(-3 ; 4 ;-7)$

$M_{2}(1 ; 5 ;-4)$

$M_{3}(-5 ;-2 ; 0)$

$M_{0}(-12 ; 7 ;-1)$
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Context:
Given an angle \( \angle BAC = \alpha \) with vertex at point \( A \). A perpendicular line \( CD \) is drawn from point \( C \) on \( AC \) to intersect \( AB \). From point \( D \), another perpendicular \( DE \) is drawn to intersect \( AC \), continuing in this fashion infinitely. What is the sum of the lengths of these perpendiculars if \( AC = b \)?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{b \cot \left(\frac{\alpha}{2} \right)}


Context:
Hannah uses three different AdBlockers with varying effectiveness. AdBlocker A blocks 75% of ads, AdBlocker B blocks 85% of ads, and AdBlocker C blocks 95% of ads. When she tries a combination of these AdBlockers, none of the ads are blocked more than once. Among the ads that don't get blocked by any AdBlocker, 15% are actually interesting. What percentage of ads are not interesting and don't get blocked by any of the AdBlockers?
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Context:
Arrange the numbers $2011, \sqrt{2011}, 2011^{2}$ in increasing order.
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Context:
Let $ABC$ be a triangle with $AB = 15, BC = 17$, $CA = 21$, and incenter $I$. If the circumcircle of triangle $IBC$ intersects side $AC$ again at $P$, find $CP$.
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Context:
In a physics club, the teacher conducted the following experiment. He placed 16 weights of masses $1,2,3, \ldots, 16$ grams on a balance scale so that one of the pans was heavier. Fifteen students took turns leaving the classroom, each taking one weight with them. After each student left, the scales tipped in the opposite direction. Which weight could have remained on the scales?
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Context:
On  $x-y$  plane,  let  $C: y=2006x^{3}-12070102x^{2}+\cdots.$
Find the area of the region surrounded by the tangent line of  $C$  at  $x=2006$  and the curve  $C.$
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Context:
Hooligan Vasya loves to run on the escalator in the metro, and he runs down twice as fast as he runs up. If the escalator is not working, it takes Vasya 6 minutes to run up and down. If the escalator is running downwards, it takes Vasya 13.5 minutes to run up and down. How many seconds will it take Vasya to run up and down the escalator if it is running upwards? (The escalator always moves at a constant speed.)
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Context:
Let $a, b, c, d$ be real numbers. Suppose that all the roots of $z^4+az^3+bz^2+cz+d=0$ are complex numbers lying on a circle in the complex plane centered at $0+0i$ and having radius $1$. Find the sum of the reciprocals of the roots. Express your answer in terms of the coefficients of the polynomial.
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Context:
In how many ways can the number 10000 be factored into three natural factors, none of which is divisible by 10? Factors that differ only in the order of multiplication are considered the same.
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Context:
Find the value of the integral $\frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{1}{M+1} \frac{1 - \cos((M+1)t)}{1 - \cos(t)} dt$ where $M \geq 0$.
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Context:
Given a right triangular prism \(ABC-A_{1}B_{1}C_{1}\) with the base being a right triangle, \(\angle ACB = 90^{\circ}\), \(AC = 6\), \(BC = CC_{1} = \sqrt{2}\), and \(P\) is a moving point on \(BC_{1}\), find the minimum value of \(CP + PA_{1}\).
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Context:
It is given that \( k \) is a positive integer not exceeding 99. There are no natural numbers \( x \) and \( y \) such that \( x^{2} - k y^{2} = 8 \). Find the difference between the maximum and minimum possible values of \( k \).
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Context:
Compute \(\lim_{n\to\infty}\int_{0}^{1}\frac{nx}{1+n^2x^4}dx\).
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Context:
What is the minimum number of sides of a regular polygon that approximates the area of its circumscribed circle with an error of less than 1 per thousand (0.1%)?
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Context:
$In the Cartesian coordinate plane, the minimum value of the perimeter of a convex pentagon with lattice points.
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Context:
On the sides $AB$, $BC$, and $CA$ of $\triangle ABC$, points $D$, $E$, and $F$ are taken respectively such that $AD:DB = BE:CE = CF:FA = 1:n$. What is the ratio of the areas of $\triangle DEF$ to $\triangle ABC$?
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Context:
18. (USA 5) Inside triangle \( A B C \) there are three circles \( k_{1}, k_{2}, k_{3} \) each of which is tangent to two sides of the triangle and to its incircle \( k \). The radii of \( k_{1}, k_{2}, k_{3} \) are 1, 4, and 9. Determine the radius of \( k \).
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Context:
设复数 $z$ 满足 $|3 z-7 i|=3$ ，求 $\left|\frac{z^{2}-2 z+2}{z-1+\mathrm{i}}\right|$ 的最大值。答案的形式为\frac{k}{m}，请计算k+m的值。
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Context:
The increasing sequence $3, 15, 24, 48, \cdots$ consists of those positive multiples of 3 that are one less than a perfect square . What is the remainder when the $1994$th term of the sequence is divided by $1000$?
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Context:
At a reunion, 10 boys meet and each boy shakes hands with each of the others a certain number of times. The total number of handshakes is 45. How many times does each boy shake hands with the others?
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Context:
Find the value of \( \sqrt{i} + \sqrt{-i} \).
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Context:
Given a hyperbola $C:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$, there exist two points $M$ and $N$ symmetric about the origin $O$ on one of the asymptotes of the hyperbola. If the quadrilateral formed by the left and right foci $F_{1}$, $F_{2}$ of the hyperbola and points $M$ and $N$ is a rectangle, and the area of this rectangle is $2\sqrt{6}a^2$, calculate the eccentricity of the hyperbola. Express your answer as a single value.
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Context:
Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be non-zero vectors, where $|\overrightarrow{b}| = 2|\overrightarrow{a}|$. Four pairs of vectors $\overrightarrow{x_{1}}$, $\overrightarrow{x_{2}}$, $\overrightarrow{x_{3}}$, $\overrightarrow{x_{4}}$ and $\overrightarrow{y_{1}}$, $\overrightarrow{y_{2}}$, $\overrightarrow{y_{3}}$, $\overrightarrow{y_{4}}$ are composed of 2 $\overrightarrow{a}$'s and 2 $\overrightarrow{b}$'s. If the minimum possible value of $\overrightarrow{x_{1}} \cdot \overrightarrow{y_{1}} + \overrightarrow{x_{2}} \cdot \overrightarrow{y_{2}} + \overrightarrow{x_{3}} \cdot \overrightarrow{y_{3}} + \overrightarrow{x_{4}} \cdot \overrightarrow{y_{4}}$ is $4|\overrightarrow{a}|^{2}$, then find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ in radians.
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Context:
Let $r_{1}, \ldots, r_{n}$ be the distinct real zeroes of the equation $x^{8}-14 x^{4}-8 x^{3}-x^{2}+1=0$. Evaluate $r_{1}^{2}+\cdots+r_{n}^{2}$
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Context:
With all angles measured in degrees, the product $\prod_{k=1}^{22} \sec^2(4k)^\circ=p^q$, where $p$ and $q$ are integers greater than 1. Find the value of $p+q$.
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Context:
Let \(A B C\) be a triangle with \(\angle A=18^{\circ}, \angle B=36^{\circ}\). Let \(M\) be the midpoint of \(A B, D\) a point on ray \(C M\) such that \(A B=A D ; E\) a point on ray \(B C\) such that \(A B=B E\), and \(F\) a point on ray \(A C\) such that \(A B=A F\). Find \(\angle F D E\).
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Context:
Consider a discrete random walk starting at position $x=0$ with equal probabilities of moving left or right. What is the probability that the particle will never leave the interval $[-n, n]$, where $n$ is a positive integer?
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Context:
A box initially contains a ball marked $1$. At 10:30, the ball marked $1$ is removed, and two balls, labeled $2$ and $3$, are added. At 10:45, the balls labeled $2$ and $3$ are removed, and four balls, marked $4$, $5$, $6$, and $7$, are added. At 10:52:30, the balls labeled $4$, $5$, and $6$ are removed, and eight balls, labeled $8$, $9$, $10$, $11$, $12$, $13$, $14$, and $15$, are added. This pattern continues, with each action occurring halfway between the previous action and 11 o'clock. Each time, one more ball is removed than was removed last time, but twice as many balls are added as were added last time. Determine the number of balls in the box when the clock strikes 11.
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Context:
Let $f(n)$ be a function satisfying the following three conditions for all positive integers $n$:

1. $f(n)$ is a positive integer.
2. $f(n + 1) > f(n)$.
3. $f(f(n)) = 3n$.

Find $f(2001)$.
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Context:
Person A and Person B start simultaneously from points A and B respectively, walking towards each other. Person A starts from point A, and their speed is 4 times that of Person B. The distance between points A and B is \( S \) kilometers, where \( S \) is a positive integer with 8 factors. The first time they meet at point C, the distance \( AC \) is an integer. The second time they meet at point D, the distance \( AD \) is still an integer. After the second meeting, Person B feels too slow, so they borrow a motorbike from a nearby village near point D. By the time Person B returns to point D with the motorbike, Person A has reached point E, with the distance \( AE \) being an integer. Finally, Person B chases Person A with the motorbike, which travels at 14 times the speed of Person A. Both arrive at point A simultaneously. What is the distance between points A and B?

\[ \text{The distance between points A and B is } \qquad \text{kilometers.} \]
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Context:
Calculate the number of different programs that can be arranged with 5 solo segments and 3 chorus segments, given that no two chorus segments are adjacent and none are placed first.
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Context:
Let $ABC$ be an equilateral triangle and let $O$ be its circumcircle. Let $D$ be a point on $\overline{BC}$, and extend $\overline{AD}$ to intersect $O$ at $P$. If $BP = 5$ and $CP = 4$, then what is the value of $DP$?
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Context:
The base of the pyramid \( S A B C D \) is the trapezoid \( A B C D \) with bases \( B C \) and \( A D \). Points \( P_{1}, P_{2}, P_{3} \) lie on the side \( B C \), with the distances satisfying \( B P_{1} < B P_{2} < B P_{3} < B C \). Points \( Q_{1}, Q_{2}, Q_{3} \) lie on the side \( A D \), with the distances satisfying \( A Q_{1} < A Q_{2} < A Q_{3} < A D \). Let the intersection points of \( B Q_{1} \) with \( A P_{1} \), \( P_{2} Q_{1} \) with \( P_{1} Q_{2} \), \( P_{3} Q_{2} \) with \( P_{2} Q_{3} \), and \( C Q_{3} \) with \( P_{3} D \) be denoted as \( R_{1}, R_{2}, R_{3}, \) and \( R_{4} \) respectively. It is known that the sum of the volumes of the pyramids \( S R_{1} P_{1} R_{2} Q_{1} \) and \( S R_{3} P_{3} R_{4} Q_{3} \) is 78. Find the minimum value of

\[
V_{S A B R_{1}}^{2} + V_{S R_{2} P_{2} R_{3} Q_{2}}^{2} + V_{S C D R_{4}}^{2}
\]

Provide the integer nearest to the found value.
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Context:
Suppose that  $a_1 = 1$ , and that for all  $n \ge 2$ ,  $a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1.$  Suppose furthermore that  $b_n = a_1 + a_2 + \ldots + a_n$  for all  $n$ . If  $b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k$  for some  $k$ , find  $k$ .
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Context:
Find the smallest possible $n$ for which there exist integers $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$ such that each integer between $1000$ and $2000$ (inclusive) can be written as the sum (without repetition), of one or more of the integers $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$.
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Context:
The radius of the circle inscribed in triangle \(ABC\) is 4, with \(AC = BC\). On the line \(AB\), point \(D\) is chosen such that the distances from \(D\) to the lines \(AC\) and \(BC\) are 11 and 3 respectively. Find the cosine of the angle \(DBC\).
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Context:
Evaluate the limit \( \lim_{x\to a}\frac{a^x-x^a}{x-a} \) without using L'Hôpital's rule.
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Context:
Before the soccer match between the "North" and "South" teams, five predictions were made:

a) There will be no draw;

b) "South" will concede goals;

c) "North" will win;

d) "North" will not lose;

e) Exactly 3 goals will be scored in the match.

After the match, it was found that exactly three predictions were correct. What was the final score of the match?
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Context:
On a local co-ed softball team, for every group of 3 players, there is 1 more woman than there are men. If there are a total of 20 players on the team, what is the ratio of men to women?
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Context:
Evaluate the limit: $$\lim_{n\to \infty}\sqrt[n^2 +n]{\prod_{i=0}^n \frac{n!}{i!(n-i)!}} $$
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Context:
A list of positive integers is called good if the maximum element of the list appears exactly once. A sublist is a list formed by one or more consecutive elements of a list. For example, the list  $10,34,34,22,30,22$  the sublist  $22,30,22$  is good and  $10,34,34,22$  is not. A list is very good if all its sublists are good. Find the minimum value of  $k$  such that there exists a very good list of length  $2019$  with  $k$  different values on it.
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Context:
The product $20! \cdot 21! \cdot 22! \cdot \cdot \cdot 28!$ can be expressed in the form $m$ $\cdot$ $n^3$, where m and n are positive integers, and m is not divisible by the cube of any prime. Find m.
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Context:
A circle is drawn through the vertex \( M \) of an angle, intersecting the sides of the angle at points \( N \) and \( K \), and the angle bisector at point \( L \). Find the sum of the lengths of segments \( M N \) and \( M K \) if the area of quadrilateral \( M N L K \) is 25 and angle \( L M N \) is \( 30^\circ \).
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Context:
In triangle \( ABC \), lines are drawn through vertices \( A \) and \( C \) perpendicular to the bisector of angle \( ABC \). These lines intersect \( CB \) and \( BA \) at points \( K \) and \( M \) respectively. Given that \( BM = 8 \) and \( KC = 1 \), find the length of \( AB \).
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Context:
There is a four-digit number less than 2000 that has exactly 14 positive divisors (including 1 and itself). One of its prime divisors ends with the digit 1. Find this four-digit number.
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Context:
Determine the precise cardinality of an $\alpha$-generated ($\alpha>\omega$) infinite $\sigma$-algebra $\mathcal{A}$, where $\alpha$-generated means the $\sigma$-algebra is generated by a set of generators of size $\alpha$. Provide your answer in terms of $\alpha$.
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Context:
Determine the limit of the infinite product as $n$ approaches infinity:

$$\lim_{n \to \infty}{\cal I}_{n} \equiv \lim_{n \to \infty} \left(\frac{2}{1}\right)^2 \left(\frac{4}{3}\right)^2 \left(\frac{6}{5}\right)^2 \ldots \left(\frac{2n}{2n - 1}\right)^2 \frac{1}{2n + 1}$$
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Context:
For a set of natural numbers $\{a_1,a_2,\ldots,a_n\}$, the lead element is defined as $l(\{a_1,a_2,\ldots,a_n\})=|a_1-a_2+\cdots+(-1)^{n-1}a_n|$. Given that $X_n$ is the set of natural numbers from $1$ to $n$, find the sum of the lead elements of all non-empty subsets of $X_n$ when $n$ is even.
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Context:
Onur bikes at a speed of 35 km/h and takes a rest every 3rd day, whereas Hanil bikes at a speed of 45 km/h and takes a rest every 4th day. Onur bikes daily for 7 hours, and Hanil bikes for as many hours as it takes to cover 40 kilometers more than Onur's daily biking distance. Determine the total distance covered by both friends in a week (7 days).
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Context:
Let the two foci of the conic section $C$ be $F_1$ and $F_2$, respectively. If there exists a point $P$ on curve $C$ such that the ratio $|PF_1| : |F_1F_2| : |PF_2| = 4 : 3 : 2$, determine the eccentricity of curve $C$.
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Context:
If $f(x) = (a-1)x^2 + ax + 3$ is an even function, then the increasing interval of $f(x)$ is ____.
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Context:
Let $a_0 = 1$, $a_1 = 2$, and $a_n = 4a_{n-1} - a_{n-2}$ for $n \ge 2$. Find an odd prime factor of $a_{2015}$.
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Context:
There are $2022$ grids in a row. Two people, A and B, play a game with these grids. Initially, each odd-numbered grid from the left is marked with A's name, and each even-numbered grid from the left is marked with B's name. The players take turns performing the following action, starting with player A:

- One should select two grids marked with their own name, such that these two grids are not adjacent, and all grids between them are marked with the opponent's name. Then, change the name in all the grids between those two grids to one's own name.

The game continues until one player can no longer make a move. Find the largest positive integer $m$ such that:

- No matter how player B acts, player A can always ensure that there are at least $m$ grids marked with A's name.
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Context:
Solve the equation $$(1+xi)^n+(1-xi)^n=0$$ for $x$, where $n$ is a positive integer and $i$ is the imaginary unit.
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Context:
In a repeatable bet, the odds of winning a dollar are 51% and the odds of losing are 49%. If you start with $100, what is the probability that you will eventually go broke?
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Context:
For which natural integers \( n \) do there exist integers \( a \) and \( b \) such that \( n + a^2 = b^2 \)?
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Context:
Triangle $ABC$ is an equilateral triangle with each side of length 9. Points $D$, $E$, and $F$ are the trisection points of sides $AB$, $BC$, and $CA$ respectively, such that $D$ is closer to $A$, $E$ is closer to $B$, and $F$ is closer to $C$. Point $G$ is the midpoint of segment $DF$ and point $H$ is the midpoint of segment $FE$. What is the ratio of the shaded area to the non-shaded area in triangle $ABC$? Assume the shaded region is the smaller region formed inside the triangle including points $D$, $E$, $F$, $G$, and $H$.
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Context:
The War of 1812 started with a declaration of war on Thursday, June 18, 1812. The peace treaty to end the war was signed 919 days later. Calculate the day of the week on which the treaty was signed. Express your answer as a day of the week (e.g., Monday, Tuesday, etc.).
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Context:
Identify the linear operator in quantum mechanics that corresponds to the continuous spectrum of physical light, given that light consists of traveling photons with energies proportional to their frequencies.
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Context:
2. How many six-digit numbers of the form $\overline{a b c a c b}$ are divisible by 23?
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Context:
Find the maximum possible number of 9-letter palindromes.
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Context:
Let the ellipse with foci \( F_1(-1,0) \) and \( F_2(1,0) \) have an eccentricity \( e \). A parabola with vertex at \( F_1 \) and focus at \( F_2 \) intersects the ellipse at a point \( P \). If \( \frac{|P F_1|}{|P F_2|} = e \), what is the value of \( e \)?
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Context:
Determine the integer $n$ for which the set $\mathbb{A}^n(k) \setminus \{0\}$ is an affine variety.
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Context:
In a regular tetrahedron $ABCD$ with edge length 1, $G$ is the centroid of $\triangle BCD$, and $M$ is the midpoint of segment $AG$. Calculate the surface area of the circumscribed sphere of the triangular prism $M-BCD$. Express your answer in terms of $\pi$.
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Context:
In a 200 meter uphill race with a 5% incline and 3 hurdles spaced evenly along the course, runner A beats runner B by 35 meters in 7 seconds. What is runner A's time to complete the course?
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Context:
Given real numbers $a \geq b \geq c \geq d > 0$ such that $a + b + c + d = 9$ and $a^2 + b^2 + c^2 + d^2 = 21$, determine the minimum value of $a + b$. Express your answer as a single number.
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Context:
Calculate the following integral on the interval $[0,2\pi]$:
$$ I=\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t $$
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Context:
Example 16. Find the function that maps the upper half of the circle $|z|<1$, $\text{Im } z>0$, onto the upper half-plane $\text{Im } w>0$.
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Context:
Two rays emanate from the origin $O$ and form a $45^\circ$ angle in the first quadrant of the Cartesian coordinate plane. For some positive numbers $X$, $Y$, and $S$, the ray with the larger slope passes through point $A = (X, S)$, and the ray with the smaller slope passes through point $B = (S, Y)$. If $6X + 6Y + 5S = 600$, then determine the maximum possible area of $\triangle OAB$.
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Context:
On a standard die one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Using Stokes' theorem, compute the integral \( \int_{\partial S} \cos z \, dx + (xz + \tan y) \, dy + y^2z^3 \, dz \), where \( B = \{(x,y) \mid x^2+y^2 \le 1\} \) is a closed ball and \( S = \{(x,y,z) \mid z = x^2+y^2, (x,y) \in B\} \) is oriented such that the map \( f: B \to S \) defined by \( f(x,y) = (x,y,x^2+y^2) \) is orientation preserving.
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Context:
Find the number of 9-digit numbers in which each digit from 1 to 9 occurs exactly once, the digits 1, 2, 3, 4, 5 are in ascending order, and the digit 6 is placed before the digit 1 (for example, 916238457).
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Context:
## Condition of the problem

Calculate the definite integral:

$$
\int_{2 \operatorname{arctg} \frac{1}{2}}^{\frac{\pi}{2}} \frac{\cos x d x}{(1-\cos x)^{3}}
$$
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Context:
From the numbers $1, 2, \cdots, 1000$, choose $k$ numbers such that any three of the chosen numbers can form the side lengths of a triangle. Find the minimum value of $k$.
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Context:
A certain sports team has members with distinct numbers chosen from the positive integers 1 to 100. If the number of any member is neither the sum of the numbers of two other members nor twice the number of another member, what is the maximum number of members this sports team can have?
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Context:
$\begin{aligned} \int_0^{\frac{\pi}{2}}\cos^{2004}x\cos2004xdx \end{aligned}$
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Context:
已知椭圆 $\frac{x^2}{9}+\frac{y^2}{8}=1$, 焦点为 $F_1, F_2, P$ 为椭圆上任一点 (当不在 $x$ 轴上), $\triangle P F_1 F_2$ 的内心为 $I$, 过 $I$ 作平行于 $x$ 轴的直线交 $P F_1, P F_2$ 于 $A, B$, 则 $\frac{S_{\triangle P A B}}{S_{\triangle P F_1 F_2}}=$ $\qquad$ .
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Context:
Todd borrowed $300 from his brother to start a snow-cone stand. He has to repay him $330 as soon as he can. To set up the stand, Todd spent $120 on renting the equipment, $60 on ingredients for the snow cones, $40 on marketing, and $10 on miscellaneous items. He sells 500 snow-cones for $1.75 each. In addition to the snow cone sales, he sells custom cups at $2 each, with a total sales volume of 250 cups. However, Todd encountered a sudden increase in ingredient costs by 20% after selling 300 snow cones. Taking all these factors into account, how much money does Todd have after he pays back his brother?
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Context:
Define the sequences $a_n$, $b_n$, and $c_n$ as follows:
- $a_0 = k$, $b_0 = 4$, $c_0 = 1$.

For each $n$:
- If $a_n$ is even, then:
  - $a_{n+1} = \frac{a_n}{2}$,
  - $b_{n+1} = 2b_n$,
  - $c_{n+1} = c_n$.
- If $a_n$ is odd, then:
  - $a_{n+1} = a_n - \frac{b_n}{2} - c_n$,
  - $b_{n+1} = b_n$,
  - $c_{n+1} = b_n + c_n$.

Find the number of positive integers $k < 1995$ such that for some $n$, $a_n = 0$.
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Context:
If you roll three fair 6-sided dice, what is the probability that exactly two of them show even numbers and the third one shows an odd number less than 3?
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Context:
A merchant purchased a vintage item and sold it for a profit of 7%. If he had managed to negotiate the purchase price down by 8% and sold it with an additional discount of $3 during a holiday sale, he still would have gained a profit of 12%. What was the cost price of the vintage item?
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Context:
Approximate the integral \( \int_3^4 \frac{x}{\sqrt{(7x-10-x^2)^3}}\,dx \).
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Context:
On the New Year's table, there are 4 glasses arranged in a row: the first and third glasses contain orange juice, while the second and fourth glasses are empty. In anticipation of guests, Valya absentmindedly and randomly pours juice from one glass to another. Each time, she can take a full glass and pour its entire contents into one of the two empty glasses.

Find the expected number of pourings needed to achieve the opposite situation: the first and third glasses are empty, while the second and fourth glasses are full.
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Context:
Calculate the argument of the sum:
\[ e^{5\pi i/36} + e^{11\pi i/36} + e^{17\pi i/36} + e^{23\pi i/36} + e^{29\pi i/36} \]
in the form $r e^{i \theta}$, where $0 \le \theta < 2\pi$.
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Context:
Two milk processing plants must process an equal amount of a certain volume of milk together. The second plant started working $a$ working days later but processed daily $m$ liters more than the first one. After $\frac{5a}{9}$ working days since the start of their joint work, $\frac{1}{3}$ of the entire task remained incomplete. How many working days were required to complete the task if the work was finished simultaneously and each plant processed half of the specified volume of milk?
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Context:
Let $T = (a,b,c)$ be a triangle with sides $a$, $b$, and $c$, and area $\triangle$. Denote by $T' = (a',b',c')$ the triangle whose sides are the altitudes of $T$ (i.e., $a' = h_a$, $b' = h_b$, $c' = h_c$) and denote its area by $\triangle'$. Similarly, let $T'' = (a'',b'',c'')$ be the triangle formed from the altitudes of $T'$, and denote its area by $\triangle''$. Given that $\triangle' = 30$ and $\triangle'' = 20$, find $\triangle$.
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Context:
Given the line \( L: x + y - 9 = 0 \) and the circle \( M: 2x^2 + 2y^2 - 8x - 8y - 1 = 0 \), point \( A \) is on line \( L \), and points \( B \) and \( C \) are on circle \( M \). In triangle \( \triangle ABC \), \(\angle BAC = 45^\circ\) and \( AB \) passes through the center of circle \( M \). Then, the range of the x-coordinate of point \( A \) is \(\quad\)
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Context:
Let $R$ be the set of all rectangles centered at the origin and with perimeter $1$. Let $S$ be a region that contains all of the rectangles in $R$. The minimum possible area of $S$ is $\pi a$. Find $a$.
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Context:
How many $3\times 2$ matrices with entries in $\mathbb{Z}/3\mathbb{Z}$ are in reduced row echelon form?
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Context:
Gavrila throws a ball from behind his head at a speed of $5 \text{ m/s}$ against the wind at an angle to the horizontal. After 1 second, the ball hits his stomach 1 meter below the point of release. Given that the force exerted on the ball by the air is directed horizontally and does not depend on the position and speed, and the acceleration due to gravity, $g$, is $10 \text{ m/s}^2$, determine the maximum distance the shadow of the ball moves away from Gavrila's feet.
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Context:
Determine the number of lines that can be drawn such that they are a distance of \((\sec(\theta))^2 + 2 (\csc(\theta))^2\) from the point \((1,3)\) and intersect the line \(3x+4y=5\).
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Context:
On a circle, there are 2009 nonnegative integers, each not greater than 100. If two adjacent numbers can be increased by 1 simultaneously, what is the minimum number of such operations required to make all the numbers on the circle equal?
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Context:
Suppose I have two orthonormal bases for $\mathbb{C}^d$, denoted $\mathbf{B_1}$ and $\mathbf{B_2}$, such that for any $|e \rangle \in \mathbf{B_1}$ and $|f \rangle \in \mathbf{B_2}$, it is the case that $| \langle e| f \rangle| = 1/\sqrt{d}$ (in other words, the bases are mutually unbiased). For a given vector $| \alpha \rangle \in \mathbb{C}^d$, let $Z_{\alpha} = \{ | v \rangle \in \mathbf{B_1} \cup \mathbf{B_2} | \langle v | \alpha \rangle =0 \}$, be the set of vectors from both bases that are orthogonal to $| \alpha \rangle$. What is the maximum size of $Z_{\alpha}$ for non-zero $| \alpha \rangle$ (i.e. $max_{|\alpha \rangle \in \mathbb{C}^d, |\alpha \rangle \neq 0} |Z_{\alpha}|$)? In other words, for a suitable choice of the vector $| \alpha \rangle$ (that isn't the zero vector), what is the most number of vectors from the two bases that are simultaneously orthogonal to $| \alpha \rangle$?
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Context:
1.52 In coin tossing, if $Z$ represents heads and $F$ represents tails, then the sequence of coin tosses is represented by a sequence of $Z$ and $F$. We can count the occurrences of heads followed by tails $(F Z)$, heads followed by heads $(Z Z)$, etc. For example, the sequence
ZZFFZZZZFZZFFFF

is the result of 15 coin tosses, which includes 5 $Z Z$, 3 $Z F$, 2 $F Z$, and 4 $F F$. How many sequences of 15 coin tosses have exactly 2 $Z Z$, 3 $Z F$, 4 $F Z$, and 5 $F F$?
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Context:
Catherine originally had an equal number of pencils and pens. She gave 12 pens and 4 pencils to her first friend, 15 pens and 7 pencils to her second friend, and 9 pens and 5 pencils to her third friend. If, after distributing these items, she was left with 36 pens and 28 pencils, how many pens and pencils did Catherine originally have?
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Context:
Egor wrote a number on the board and encrypted it according to the rules of letter puzzles (different letters correspond to different digits, identical letters - identical digits). The result was the word "GWATEMALA". How many different numbers could Egor have originally written if his number was divisible by 5?
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Context:
Let \( n \) be a positive integer. Given a real number \( x \), let \( \lfloor x \rfloor \) be the greatest integer less than or equal to \( x \). For example, \( \lfloor 2.4 \rfloor = 2 \), \( \lfloor 3 \rfloor = 3 \), and \( \lfloor \pi \rfloor = 3 \). Define a sequence \( a_1, a_2, a_3, \ldots \) where \( a_1 = n \) and
\[
a_m = \left\lfloor \frac{a_{m-1}}{3} \right\rfloor,
\]
for all integers \( m \geq 2 \). The sequence stops when it reaches zero. The number \( n \) is said to be lucky if 0 is the only number in the sequence that is divisible by 3. For example, 7 is lucky, since \( a_1 = 7, a_2 = 2, a_3 = 0 \), and none of 7, 2 are divisible by 3. But 10 is not lucky, since \( a_1 = 10, a_2 = 3, a_3 = 1, a_4 = 0 \), and \( a_2 = 3 \) is divisible by 3. Determine the number of lucky positive integers less than or equal to 1000.
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Context:
Evaluate the integral \( \int_{0}^{\pi/2} (\cos^{5}x)\sin (7x)\,dx \).
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Context:
Evaluate the limit: $$\lim_{n\to \infty}\sqrt[n^2 +n]{\prod_{i=0}^n \frac{(n+1)!}{(i+1)!(n-i+1)!}}$$
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Context:
In triangle $ABC$, where $AB = AC = 20$ and $BC = 14$, find the minimum value of the sum $BN + MN + MC$ for points $M$ on $\overline{AB}$ and $N$ on $\overline{AC}$. Compute $100x$, where $x$ is this minimum value.
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Context:
Julia played tag with 4 kids on Wednesday. She had played with 6 kids on Monday and with some kids on Tuesday. She played with 2 more kids on Monday than on Wednesday. How many kids did she play with on Tuesday?
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Context:
The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is
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Context:
In how many ways can a \(3 \times 6\) rectangle be divided into L-shaped pieces of three cells?
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Context:
Given the vector $$\overrightarrow {a_{k}} = (\cos \frac {k\pi}{6}, \sin \frac {k\pi}{6} + \cos \frac {k\pi}{6})$$ for k=0, 1, 2, …, 12, find the value of $$\sum\limits_{k=0}^{11} (\overrightarrow {a_{k}} \cdot \overrightarrow {a_{k+1}})$$.
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Context:
Ivan Semenovich leaves for work at the same time every day, travels at the same speed, and arrives exactly at 9:00 AM. One day, he overslept and left 40 minutes later than usual. To avoid being late, Ivan Semenovich increased his speed by 60% and arrived at 8:35 AM. By what percentage should he have increased his usual speed to arrive exactly at 9:00 AM?
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Context:
Calculate the correlation coefficient for the random variables X and Y given the joint density function: \[ f(x, y) = \begin{cases} 4 & 0 < y \le x < 1 \\ 0 & \text{otherwise} \end{cases} \]
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Context:
满足下述条件的最小正实数 k：对任意不小于 k 的 4 个互不相同的实数 a,b,c,d，都存在 a,b,c,d 的一个排列 p, q, r, s，使得方程 (x^2+px+q)(x^2+rx+s)=0 有 4 个互不相同的实根.
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Context:
In the answer obtained by subtracting 101011 from 10000000000, how many times does the digit 9 appear?
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Context:
Consider  $9$  points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of   $\,n\,$  such that whenever exactly  $\,n\,$  edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.
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Context:
Evaluate the limit $$\lim_{x \to 0} \frac{e^{\tan x} - e^x + \ln(\sec x + \tan x) - x}{\tan x - x}$$ using L'Hopital's rule or alternative methods.
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Context:
Evaluate the product
$$
\prod_{n=3}^{\infty} \frac{\left(n^{3}+3 n\right)^{2}}{n^{6}-64}
$$  The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
A function  $f$  satisfies the equation
\[f\left(x\right)+f\left(1-\frac{1}{x}\right)=1+x\]
for every real number  $x$  except for  $x = 0$  and  $x = 1$ . Find a closed formula for  $f$ .
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Context:
Let the even function $f(x) = \log_a |x+b|$ be monotonically increasing on the interval $(0, +\infty)$. Then, compare $f(b-2)$ and $f(a+1)$ (fill in with an equality or inequality sign).
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Context:
Jerry had 7 action figures, but needed a certain number of action figures for a complete collection. If each 1 costs $8, he would need $72 to finish his collection. How many action figures are needed for a complete collection?
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Context:
Five contestants $A, B, C, D, E$ participate in the "Good Voice" competition, and they stand in a line for a group appearance. Each of them has a contestant number badge on their chest. The sum of the five numbers is 35. It is known that the sum of the numbers of the contestants standing to the right of $E$ is 13; the sum of the numbers of the contestants standing to the right of $D$ is 31; the sum of the numbers of the contestants standing to the right of $A$ is 21; and the sum of the numbers of the contestants standing to the right of $C$ is 7. What is the sum of the numbers of the contestants standing at the far left and far right $\qquad$?
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Context:
What is the residue at $0$ of $(1-e^{-z})^{-n}$ for $n\in\mathbb{N}^+$?
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Context:
What is the sum of non-negative integer solutions of $\{(x,y,z)\mid x+y+z=k\}$, $k = 1,\ldots,N$?
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Context:
Determine whether a closed form formula exists for the series \( \sum_{k=0}^{n-1} \binom{n+k}{n-k-1} \).
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Context:
Given a program with a 20% probability of having a bug that causes it to crash, there are two testing procedures that each examine 50% of the program's code. Procedure 1 detects 50% of bugs in its coverage area, while Procedure 2 detects 30%. The overlap between the two procedures is 50% of their respective code areas, meaning they can both detect bugs in this common region, independently. Neither test generates false bug warnings, and bugs are assumed to be equally likely throughout the code (not reflecting real-world software testing). What is the probability that a program is free of major bugs if it passes both tests?
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Context:
Given the number $200\ldots 002$ (100 zeros), determine how many ways two of the zeros can be replaced with non-zero digits such that the resulting number is divisible by 66.
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Context:
Given a function $f(x)=a{e^x}-\frac{1}{2}{x^2}$, where $a\in\mathbb{R}$. If the function has two critical points $x_{1}$ and $x_{2}$, and $\frac{x_{2}}{x_{1}}\geq 2$, then the range of real number $a$ is ____.
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Context:
A circle is tangent to the extensions of two sides \(AB\) and \(AD\) of the square \(ABCD\), and the points of tangency cut off segments of length 4 cm from vertex \(A\). From point \(C\), two tangents are drawn to this circle. Find the side length of the square if the angle between the tangents is \(60^\circ\).
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Context:
The digits $1,2,3,4,5,6$ are randomly chosen (without replacement) to form the three-digit numbers $M=\overline{A B C}$ and $N=\overline{D E F}$. For example, we could have $M=413$ and $N=256$. Find the expected value of $M \cdot N$.
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Context:
Determine the values of \( \alpha \) for which the integral \( \int_E \frac{x^\alpha}{\sqrt{x^4+y^2}} \, dx \, dy \) converges, where \( E \) is the region defined by the area of the circle \( x^2+(y-1)^2=1 \) minus the area of the circle \( x^2+(y-0.5)^2=0.5^2 \).
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Context:
Compute the distribution $x\delta'_0$, where $\delta'_0$ is defined by $\delta'_0(u) = -u'(0)$ for $u \in D\mathbb{R}$.
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Context:
Determine the area of the region \( R \) that is the intersection of all regions \( R(\alpha) \) for \( 0 < \alpha < 1 \), where each \( R(\alpha) \) is the region in \( \mathbb{R}^2 \) bounded by the convex pentagon with vertices \((0,1-\alpha), (\alpha, 0), (1, 0), (1,1), (0, 1)\).
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Context:
9.1. (13 points) The surface of a round table is divided into 9 identical sectors, in which the numbers from 1 to 9 are written sequentially clockwise. Around the table sit 9 players with numbers \(1, 2, \ldots, 9\), going clockwise. The table can rotate around its axis in both directions, while the players remain in place. The players are seated at equal distances from each other, so when the table stops rotating, exactly one player is opposite each sector and receives the number of coins written on that sector. It is known that after 11 rotations of the table, player ㄱo4 received 90 coins, and player № 8 received 35 coins. How many coins did player № 1 receive?
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Context:
Find the maximum value of $y$ given the equation $2(x^3+y)^4+(x^3+y)^2=2x^3+y$.
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Context:
Let $a$, $b$, $c$, $d$ be complex numbers satisfying
\begin{align*}
  5 &= a+b+c+d \\
  125 &= (5-a)^4 + (5-b)^4 + (5-c)^4 + (5-d)^4 \\
  1205 &= (a+b)^4 + (b+c)^4 + (c+d)^4 + (d+a)^4 + (a+c)^4 + (b+d)^4 \\
  25 &= a^4+b^4+c^4+d^4
\end{align*}
Compute $abcd$.

[i]Proposed by Evan Chen[/i]
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Context:
Evaluate the limit: $$\lim_{x\to-1}\frac{\sqrt[3]{26-x}-\sqrt{8-x}}{3x^2+4x+1}$$ without using L'Hôpital's rule.
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Context:
Let $D \subset \mathbb{R}^3$ be the set $D = \{(x, y, z) \mid x^2 + y^2 \leq 1~\text{and}~z = 0\}$, representing a disk in the $xy$-plane. Consider the linear transformation $T : \mathbb{R}^3 \to \mathbb{R}^3$ defined by $T(x, y, z) = (2x − y + z, x − y + 10z, 3x − 7y + 2z)$. Find the area of the image $T(D)$ in $\mathbb{R}^3$. \(\boxed{\text{Area}}\)
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Context:
Determine the values of $x$ for which the series \( \sum _ { n = 1 } ^ \infty \frac { x ^ n } { \sin ( n \theta ) } \) converges when \( \theta = t \pi \), where \( t \) is an irrational number, such as \( t = \sqrt{3} \).
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Context:
A set S consisting of $2019$ (different) positive integers has the following property:
the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements.
What is the maximum number of prime numbers that $S$ can contain?
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Context:
Find the residue of the function \( \sin(z + \frac{1}{z}) \) at \( z = 0 \) using the Laurent series expansion.
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Context:
Evaluate the limit of the sequence as \( n \) approaches infinity: \[ \lim_{{n \to \infty}} \left( \sum_{i={(n+1)}/{2}}^n {n \choose i} \times 0.51^i \times 0.49^{n-i} \right). \]
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Context:
Find the remainder when the binary message 010100011010000 is divided by the polynomial divisor $x^{14} + x^{12} + x^8 + x^7 + x^5$ using the generator polynomial $x^5 + x^4 + x^2 + 1$ in the field $GF(2)$. What is the correct remainder?
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Context:
Let $\mathcal E$ be an ellipse with foci $F_1$ and $F_2$.  Parabola $\mathcal P$, having vertex $F_1$ and focus $F_2$, intersects $\mathcal E$ at two points $X$ and $Y$.  Suppose the tangents to $\mathcal E$ at $X$ and $Y$ intersect on the directrix of $\mathcal P$.  Compute the eccentricity of $\mathcal E$.

(A [i]parabola[/i] $\mathcal P$ is the set of points which are equidistant from a point, called the [i]focus[/i] of $\mathcal P$, and a line, called the [i]directrix[/i] of $\mathcal P$.  An [i]ellipse[/i] $\mathcal E$ is the set of points $P$ such that the sum $PF_1 + PF_2$ is some constant $d$, where $F_1$ and $F_2$ are the [i]foci[/i] of $\mathcal E$.  The [i]eccentricity[/i] of $\mathcal E$ is defined to be the ratio $F_1F_2/d$.)
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Context:
Split the number 10 into the sum of two positive integers. Under the condition that both addends are greater than 2, calculate the probability that both addends are prime numbers. Express your answer as a simplified fraction.
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Context:
Let \(a\) and \(b\) be any positive real numbers. Determine the value of the expression

\[
\frac{\sqrt{\frac{a b}{2}}+\sqrt{8}}{\sqrt{\frac{a b+16}{8}+\sqrt{a b}}}
\]
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Context:
Evaluate the integral $$∫_{ -1 }^{ 1 }(\sqrt {4-x^{2}}+x^{3})dx$$.
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Context:
Solve the system of equations for real numbers:
\[ \begin{cases} \sqrt{x^2-2x+6}\cdot \log_{3}(6-y) =x \\ \sqrt{y^2-2y+6}\cdot \log_{3}(6-z)=y \\ \sqrt{z^2-2z+6}\cdot\log_{3}(6-x)=z \end{cases}. \]
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Context:
8. Given a non-empty set $M$ satisfying
$$
M \subseteq\{0,1, \cdots, n\}\left(n \geqslant 2, n \in \mathbf{Z}_{+}\right) \text {. }
$$

If there exists a non-negative integer $k(k \leqslant n)$, such that for any $a \in M$, we have $2 k-a \in M$, then the set $M$ is said to have property $P$. Let the number of sets $M$ with property $P$ be $f(n)$. Then the value of $f(9)-f(8)$ is $\qquad$
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Context:
Arrange 5 people to be on duty from Monday to Friday, with each person on duty for one day and one person arranged for each day. The conditions are: A and B are not on duty on adjacent days, while B and C are on duty on adjacent days. The number of different arrangements is $\boxed{\text{answer}}$.
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Context:
A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \left\lbrace\frac{1}{z} \ | \ z \in R\right\rbrace$.  Find the area of $S.$
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Context:
The GDP of a certain city last year was 84,300,000 ten thousand yuan. Represent this number in scientific notation.
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Context:
Find the least possible number of elements which can be deleted from the set  $\{1,2,...,20\}$  so that the sum of no two different remaining numbers is not a perfect square.

N. Sedrakian , I.Voronovich
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Context:
A little rascal ran a race with a horse mounted on a mechanical carousel. After \(a\) seconds, she found that the horse, having made a circle, had caught up to her. Instantly turning around, the little rascal ran towards the horse at the same speed and met it after \(\frac{a}{2}\) seconds. Determine how long it takes for the carousel to make a full rotation, assuming all movements are uniform.
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Context:
Suppose we have a convex polygon in which all interior angles are integers when measured in degrees, and the interior angles at every two consecutive vertices differ by exactly \(1^{\circ}\). If the greatest and least interior angles in the polygon are \(M^{\circ}\) and \(m^{\circ}\), what is the maximum possible value of \(M - m\)?
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Context:
1. On the cube $A B C D A_{1} B_{1} C_{1} D_{1}$, points $P$ and $Q$ are chosen, where $P$ is the midpoint of edge $B C$, and $Q$ is the intersection of the diagonals in the square $C C_{1} D D_{1}$. The plane $A P Q$ divides the cube into two parts. Find the ratio of the volumes of the resulting parts of the cube?
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Context:
Determine the minimum value of the expression $2x^4 - 2x^2y^2 + y^4 - 8x^2 + 18$ where $x, y \in \mathbb{R}$.
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Context:
Let \(x\) and \(y\) be independent, uniformly distributed random variables over \([0,1]\). Let \(a\) and \(R\) be positive real numbers where \(R>a\). How can we find the probability of
$$P(x>\frac{ya}{R-a})?$$
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Context:
Calculate the sum of the largest and smallest angles in a triangle with side lengths of 5, 7, and 8.
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Context:
A machine has a present value of $1200. Its value depletes based on a variable depreciation rate depending on the age of the machine (in years): 10% for the first year, 12% for the second year, and so on, increasing by 2% for each subsequent year. Additionally, the machine undergoes annual repairs and maintenance which add 3% of the previous year's value back to the machine. However, the machine requires major overhauls at irregular intervals, with the next major overhaul scheduled at the end of the second year, which will decrease its value by an additional 15%. Calculate the machine's value after the major overhaul at the end of the 2nd year, considering all depreciation, repairs, and the major overhaul costs.
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Context:
Let $\Omega$ be a circle with radius $18$ and let $\mathcal{S}$ be the region inside $\Omega$ that the centroid of $\triangle XYZ$ sweeps through as $X$ varies along all possible points lying outside of $\Omega$, $Y$ varies along all possible points lying on $\Omega$ and $XZ$ is tangent to the circle. Compute the greatest integer less than or equal to the area of $\mathcal{S}$.

[I]Proposed by [b]FedeX333X[/b][/I]
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Context:
In the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($0 < b < a$), a line passing through the center intersects the ellipse at points $A$ and $B$, and the right focus is $F_2(c,0)$. Calculate the maximum area of $\triangle ABF_2$ and express your answer as a product of two variables.
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Context:
Evaluate the expression: \[ \lg_{2} \left( \prod\limits_{a=1}^{2015} \prod\limits_{b=1}^{2015} (1 + e^{\frac{2\pi iab}{2015}}) \right) \]
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Context:
In Plutarch Enterprises, 70% of the employees are marketers, some percentage are engineers, and the rest are managers. Marketers make an average salary of $50,000 a year, engineers make an average of $80,000, and managers make an average of $370,000. If the average for all employees is also $80,000, what percentage of the employees are engineers?
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Context:
On a chess board ( $8*8$ ) there are written the numbers  $1$  to  $64$ : on the first line, from left to right, there are the numbers  $1, 2, 3, ... , 8$ ; on the second line, from left to right, there are the numbers  $9, 10, 11, ... , 16$ ;etc. The  $\"+\"$  and  $\"-\"$  signs are put to each number such that, in each line and in each column, there are  $4$   $\"+\"$  signs and  $4$   $\"-\"$  signs. Then, the  $64$  numbers are added. Find all the possible values of this sum.
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Context:
In triangle \( ABC \), the angle bisectors intersect at point \( O \). The line \( AO \) intersects the circumcircle of triangle \( OBC \) at points \( O \) and \( M \). Find \( OM \), given that \( BC = 2 \) and angle \( A \) is \( 30^{\circ} \).
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Context:
Daniel has a (mostly) standard deck of 54 cards, consisting of 4 suits each containing the ranks 1 to 13 as well as 2 jokers. Daniel plays the following game: He shuffles the deck uniformly randomly and then takes all of the cards that end up strictly between the two jokers. He then sums up the ranks of all the cards he has taken and calls that his score.

Let $p$ be the probability that his score is a multiple of 13. There exists relatively prime positive integers $a$ and $b,$ with $b$ as small as possible, such that $|p - a/b| < 10^{-10}.$ What is $a/b?$

[i]Proposed by Dilhan Salgado, Daniel Li[/i]
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Context:
In the convex quadrilateral $ABCD$, we have the following side lengths: $AB = 15$, $BC = 16$, $CD = 12$, $DA = 25$, and the diagonal $BD = 20$. Let $M$ be the circumcenter and $\gamma$ the circumcircle of $\triangle ABD$. Line $CB$ intersects $\gamma$ again at point $F$. Line $AF$ intersects $MC$ at point $G$, and line $GD$ intersects $\gamma$ again at point $E$. Determine the area of the pentagon $ABCDE$.
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Context:
Captain Billy the Pirate looted 1010 gold doubloons and set sail on his ship to a deserted island to bury his treasure. Each evening of their voyage, he paid each of his pirates one doubloon. On the eighth day of sailing, the pirates plundered a Spanish caravel, doubling Billy's treasure and halving the number of pirates. On the 48th day of sailing, the pirates reached the deserted island, and Billy buried all his treasure at the marked spot—exactly 1000 doubloons. How many pirates set off with Billy to the deserted island?
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Context:
设实数 $a$ 满足 $a<9 a^{3}-11 a<|a|$, 则 $a$ 的取值范围是一个区间，区间的两端点和形如\frac{a\sqrt{b}+c\sqrt{d}}{e}，请给出a+b+c+d+e
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Context:
Calculate the nearest integer to $(3+\sqrt{5})^6$.
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Context:
Determine the number of permutations \( a_1, a_2, \ldots, a_{10} \) of the numbers \( 1, 2, \ldots, 10 \) such that \( a_i > a_{2i} \) for \( 1 \le i \le 5 \) and \( a_i > a_{2i+1} \) for \( 1 \le i \le 4 \).
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Context:
Given that $x,y,z$ are reals such that $16 \sin^4(x+y) + 49 \cos^4(x+y) = \sin^2(2x + 2y)(8\sin^2(x+z) + 6 \cos^2(y+z))$, the largest possible value of $3\cos^2(x+y+z)$ can be expressed in the form $\frac{a+b\sqrt{c}}{d}$, where $a$ and $b$ are integers, $c$ is a positive integer not divisible by the square of any prime, and $d$ is a positive integer such that $\gcd(a,b,d)=1$. Find $a+b+c+d$.
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Context:
Jackson starts at position 1 on a number line. At each step, he stays in place with a probability of 85% and moves one position to the right with a probability of 15%. Let \(d_n\) represent his position after \(n\) steps, and let \(E_n\) be the expected value of \(\frac{1}{d_n}\). Determine the smallest \(n\) such that \(\frac{1}{E_n} > 2017\).
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Context:
Determine the asymptotic behavior \( \Theta(S_n) \) of the summation \( S_n = \sum_{k=1}^{n}\frac{n-k+1}{k} \), where \( n \in \mathbb{Z}^{+} \).
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Context:
Determine the number of $2 \times 2$ matrices $A$ with entries in $\mathbb{Z}/3\mathbb{Z}$ such that $A^{50} = I$, where $I$ is the identity matrix.
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Context:
Evaluate the outward flux of the vector field \( V = x(x^2 + y^2 + z^2)^{-3/2}\mathbf{i} + y(x^2 + y^2 + z^2)^{-3/2}\mathbf{j} + z(x^2 + y^2 + z^2)^{-3/2}\mathbf{k} \) across the boundary of the rectangular box defined by the points \((-1, -1, -1)\) to \((1, 1, 1)\). Note that \( V \) is not defined at the origin.
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Context:
Given four points on a plane and their displacement vectors after a rotation, determine the center of rotation.
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Context:
Simon expands factored polynomials with his favorite AI, ChatSFFT. However, he has not paid for a premium ChatSFFT account, so when he goes to expand $(m - a)(n - b),$ where $a, b, m, n$ are integers, ChatSFFT returns the sum of the two factors instead of the product. However, when Simon plugs in certain pairs of integer values for $m$ and $n,$ he realizes that the value of ChatSFFTs result is the same as the real result in terms of $a$ and $b$. How many such pairs are there?
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Context:
Consider natural numbers \(a\), \(b\), and \(c\) such that the fraction

\[
k = \frac{ab + c^2}{a + b}
\]

is a natural number less than \(a\) and \(b\). What is the smallest number of natural divisors that \(a + b\) can have?
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Context:
Let $X_1,X_2,..$ be independent random variables with the same distribution, and let $S_n=X_1+X_2+...+X_n, n=1,2,...$. For what real numbers $c$ is the following statement true:
$$P\left(\left| \frac{S_{2n}}{2n}- c \right| \leqslant \left| \frac{S_n}{n}-c\right| \right)\geqslant \frac{1}{2}$$
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Context:
Evaluate the integral $\int_0^{2\pi}e^{-\cos\theta} \cos(n\theta+\sin\theta)~d\theta$ using contour integration for $n\in\mathbb{Z}$.
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Context:
Find the value of the following integral:

$$\int _0^{\infty }\ln \left(\frac{e^x+1}{e^x-1}\right)dx$$
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Context:
There are 2 sections, A and B, in a class, consisting of 26 and 34 students respectively. The average weight of section A is some kg and that of section B is 30 kg. The average weight of the whole class is 38.67 kg. What is the average weight of section A?
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Context:
A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square?
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Context:
2. Let $A B C D$ be a square of side 4. Determine the largest positive integer $k$ with the following property: For an arbitrary arrangement of $k$ points strictly inside square $A B C D$, one can always find a square of side 1 , entirely contained in square $A B C D$ (with sides not necessarily parallel to the sides of square $A B C D$ ), whose strict interior contains none of the $k$ given points.

(Bojan Bašić)
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Context:
For certain pairs $(m,n)$ of positive integers with $m\geq n$ there are exactly $50$ distinct positive integers $k$ such that $|\log m - \log k| < \log n$. Find the sum of all possible values of the product $mn$.
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Context:
Given that $A, B, C, D$ are points along a circle in that order, and $AC$ intersects $BD$ at point $X$. If $BC = 6$, $BX = 4$, $XD = 5$, and $AC = 11$, find the length of $AB$.
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Context:
Evaluate the limit:
$$\lim_{x\to 0}\frac{x^{(\sin x)^{\textstyle x}}-(\textstyle \sin x)^{\textstyle x^{\textstyle \sin x}}}{\textstyle x^3}$$
without using Taylor's formula.
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Context:
Given the sample data set $3$, $3$, $4$, $4$, $5$, $6$, $6$, $7$, $7$, calculate the standard deviation of the data set.
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Context:
In tetrahedron $V-ABC$, it is known that $VA=VB=VC=\sqrt{3}$, $AB=\sqrt{2}$, and $∠ACB=\frac{π}{4}$. Find the surface area of the circumscribed sphere of this tetrahedron.
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Context:
How many $r$-class variations can be formed from $n$ elements in which the first $s$ elements occur?
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Context:
9548 plus a certain number equals 3362 plus 13500. What is the second number?
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Context:
A cyclist is walking at a constant rate of 15 miles per hour along a path with a flat surface when she is passed by a car traveling in the same direction at 60 miles per hour. After passing the cyclist, the car continues for another mile before stopping to wait for her. Along this mile, there is a continuous uphill slope that starts right after the cyclist is passed by the car. As the cyclist enters the uphill slope, her speed decreases linearly from 15 miles per hour to 10 miles per hour by the time she reaches its end (the point where the car is waiting). Find how many minutes the car must wait for the cyclist to catch up.
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Context:
a number when divided by 5 gives a number which is 8 more than the remainder obtained on dividing the same number by 34 . such a least possible number x is
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Context:
Given the base of a triangle is $24$ inches, two lines are drawn parallel to the base, with one of the lines dividing the triangle exactly in half by area, and the other line dividing one of the resulting triangles further into equal areas. If the total number of areas the triangle is divided into is four, find the length of the parallel line closer to the base.
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Context:
The graph of the function y=sin(2x+φ) is shifted to the left by π/6 units along the x-axis, resulting in an even function graph. Determine the value of φ such that the equation 2(x + π/6) + φ = -x + 2πk is satisfied for some integer k.
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Context:
A vendor sells 80 percent of the pears he had and throws away 50 percent of the remainder. The next day, the vendor sells 80 percent of the remaining pears and throws away the rest. In total, what percent of his pears does the vendor throw away?
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Context:
How many 11 step paths are there from point A to point D, which pass through points B and C in that order? Assume the grid layout permits only right and down steps, where B is 2 right and 2 down steps from A, and C is 1 right and 3 down steps from B, and finally, D is 3 right and 1 down step from C.
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Context:
Using the digits 1, 2, 3, 4 only once to form a 4-digit number, how many of them are divisible by 11?
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Context:
10. There is a sequence of numbers +1 and -1 of length $n$. It is known that the sum of every 10 neighbouring numbers in the sequence is 0 and that the sum of every 12 neighbouring numbers in the sequence is not zero.
What is the maximal value of $n$ ?
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Context:
Yatsenko I.V.

In the elections to the 100-seat parliament, 12 parties participated. Parties that received strictly more than $5\%$ of the voters' votes enter the parliament. Among the parties that entered the parliament, seats are distributed proportionally to the number of votes they received. After the elections, it turned out that each voter voted for exactly one party (there were no invalid ballots, votes "against all" and the like) and each party received an integer number of seats. At the same time, the Party of Mathematics Enthusiasts received 25% of the votes. What is the maximum number of seats in the parliament that it could have received?
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Context:
Casey wants to decorate her toenails, fingernails, and her friend's toenails and fingernails with a more intricate design. First, she wants to do a base coat on each nail, then a coat of paint, followed by a coat of glitter on every nail, and finally adding a pattern which takes additional time. It takes 20 minutes to apply each coat, 20 minutes for each coat to dry before applying the next one, and an extra 40 minutes to meticulously create the pattern on all nails. Assuming Casey's nails and her friend's nails will be done when all coats have been applied and dried, and the pattern is completed, how many minutes will it take her to finish decorating her and her friend's fingernails and toenails?
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Context:
How many four-digit integers $abcd$, with $a \neq 0$, have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$, where $a=4$, $b=6$, $c=9$, and $d=2$.
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Context:
On a circle, points $A, B, C, D, E, F, G$ are arranged clockwise, as shown in the diagram. It is known that $AE$ is the diameter of the circle. Additionally, it is known that $\angle ABF = 81^\circ$ and $\angle EDG = 76^\circ$. How many degrees is $\angle FCG$?
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Context:
Alice has 8 keys strung on a ring. The keys are indistinguishable at a glance, and both sides of each key are identical. In order to differentiate the keys, Alice places a colored cap on each one. How many colors does she need?
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Context:
Let $O$ be the center of the base $\triangle ABC$ of a regular triangular pyramid $P-ABC$. A moving plane passing through $O$ intersects the three lateral edges or their extensions of $P-ABC$ at points $Q, R, S$ respectively. Calculate the sum $\frac{1}{PQ}+\frac{1}{PR}+\frac{1}{PS}$. Express your answer as a single numerical value.
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Context:
Determine the smallest real number $m$ such that for any positive integers $a, b, c$ satisfying $a + b + c = 1$, the inequality $m(a^3 + b^3 + c^3) \geq 6(a^2 + b^2 + c^2) + 1$ holds.
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Context:
Let the natural number \( n \) have the following property: If you choose 50 different numbers from the natural numbers \( 1, 2, \cdots, n \), there must be two numbers whose difference is equal to 7. Find the maximum value of \( n \).
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Context:
Point  $A,B$  are marked on the right branch of the hyperbola   $y=\frac{1}{x},x>0$ . The straight line  $l$  passing through the origin  $O$  is perpendicular to  $AB$  and meets   $AB$  and given branch of the hyperbola at points  $D$  and  $C$  respectively. The circle through  $A,B,C$  meets  $l$  at  $F$ .
Find  $OD:CF$
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Context:
Let \( \triangle ABC \) be an acute triangle with \( BC = 5 \). \( E \) is a point on \( AC \) such that \( BE \perp AC \). \( F \) is a point on \( AB \) such that \( AF = BF \). Moreover, \( BE = CF = 4 \). Find the area of the triangle.
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Context:
In triangle $ABC$, the angle $C$ is obtuse, $m(\angle A) = 2m(\angle B)$, and the side lengths are integers. Find the smallest possible perimeter of this triangle.
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Context:
Lyndy conducted a survey of the 300 students in her school. She found that 60 students own dogs, while only 30 students own cats. What percent of the students own cats?
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Context:
【Question 14】Select 5 digits from $1 \sim 9$, to form a five-digit number, such that this five-digit number can be evenly divided by any of the 5 selected digits, but cannot be evenly divided by any of the 4 unselected digits. Then, the smallest value of this five-digit number is $\qquad$.

---

Note: The blank at the end of the question is intended for the answer to be filled in.
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Context:
Let $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of $M, 32, 33, 34, 35, 36, 37, 38, 39,$ and $40.$ Find the value of $\frac{N}{M}.$
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Context:
The line $x+2ay-1=0$ is parallel to $\left(a-1\right)x+ay+1=0$, find the value of $a$.
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Context:
Calculate the curl of the cross product $v(x,y,z) = (x,y,z)^T \times \omega$, where $\omega \in \mathbb{R}^3$ and $v(x,y,z)$ is a vector function.
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Context:
(10 Given that the three lateral edges of a triangular pyramid are pairwise perpendicular, and the base area is 1, find the maximum volume of this triangular pyramid.

保留源文本的换行和格式，翻译结果如下：

(10 Given that the three lateral edges of a triangular pyramid are pairwise perpendicular, and the base area is 1, find the maximum volume of this triangular pyramid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCDE$ be a convex pentagon, and let $H_A,$ $H_B,$ $H_C,$ $H_D$ denote the centroids of triangles $BCD,$ $ACE,$ $ABD,$ and $ABC,$ respectively. Determine the ratio $\frac{[H_A H_B H_C H_D]}{[ABCDE]}.$
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Context:
A trapezoid is divided into three smaller trapezoids by lines parallel to the bases. It is known that each of the three resulting trapezoids can have an inscribed circle. Find the radius of the circle inscribed in the middle trapezoid if the radii of the circles inscribed in the other two trapezoids are $R$ and $r$.
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Context:
In the diagram, $D$ is on side $A C$ of $\triangle A B C$ so that $B D$ is perpendicular to $A C$. Also, $\angle B A C=60^{\circ}$ and $\angle B C A=45^{\circ}$. If the area of $\triangle A B C$ is $72+72 \sqrt{3}$, what is the length of $B D$?
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Context:
What is the cardinality of the set of all subsets of the real numbers $\mathbb{R}$ that are neither open nor closed and have no intersection with the set of integers $\mathbb{Z}$ (i.e., if $A$ is such a set, then $A \cap \mathbb{Z} = \emptyset$)?
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Context:
For positive integers $n$, let $L(n)$ be the largest factor of $n$ other than $n$ itself. Determine the number of ordered pairs of composite positive integers $(m, n)$ for which $L(m) L(n)=80$.
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Context:
Two different digits from 1 to 9 are chosen. One digit is placed in each box to complete the two 2-digit numbers shown. The result of subtracting the bottom number from the top number is calculated. How many of the possible results are positive?
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Context:
How many positive integers which divide  $5n^{11}-2n^5-3n$  for all positive integers  $n$  are there?
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Context:
If k^4 is divisible by 1680, and the prime factorization of k is in the form 2^a * 3^b * 5^c, where a, b, and c are positive integers, what is the least possible value of integer k?
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Context:
For  $i=1,2,\ldots,7$ , Zadam Heng chooses a positive integer  $m_i$  at random such that each positive integer  $k$  is chosen with probability  $\frac{2^i-1}{2^{ik}}$ . If  $m_1+2m_2+\ldots+7m_7\neq35$ , Zadam keeps rechoosing the  $m_i$  until this equality holds. Given that he eventually stops, what is the probability that  $m_4=1$  when Zadam stops?
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Context:
The least common multiple of three consecutive natural numbers is 660. Find these three numbers.
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Context:
11.2. What is the minimum number of factors that need to be crossed out from the number $99!=1 \cdot 2 \cdot \ldots \cdot 99$ so that the product of the remaining factors ends in $2?$
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Context:
8. There is an unlimited number of test tubes of three types - A, B, and C. Each test tube contains one gram of a solution of the same substance. Test tubes of type A contain a $10\%$ solution of this substance, type B $-20\%$ solution, and type C $-90\%$ solution. Sequentially, one after another, the contents of the test tubes are poured into a certain container. In this process, two consecutive pourings cannot use test tubes of the same type. It is known that a $20.17\%$ solution was obtained in the container, performing the minimum number of pourings. What is the maximum number of test tubes of type C that can be used in this process?
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Context:
In a table \( A \) of size \( 10 \times 10 \), some numbers are written. Let the sum of all numbers in the first row be denoted as \( s_{1} \), in the second row as \( s_{2} \), and so on. Similarly, the sum of the numbers in the first column is denoted as \( t_{1} \), in the second column as \( t_{2} \), and so on.

A new table \( B \) of size \( 10 \times 10 \) is constructed such that in the first cell of the first row is written the smallest of the numbers \( s_{1} \) and \( t_{1} \), in the third cell of the fifth row is written the smallest of the numbers \( s_{5} \) and \( t_{3} \), and so on for the whole table. It turns out that it is possible to number the cells of table \( B \) with numbers from 1 to 100 such that in the cell with number \( k \) there will be a number less than or equal to \( k \). What is the maximum value that the sum of all the numbers in table \( A \) can take under these conditions?
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Context:
Given integers \( a, b, c \) from the set \( \{1, 2, 3, \ldots, 9\} \), where \( \overline{ab} = b^2 \) and \( \overline{acbc} = (\overline{ba})^2 \), find the value of \( \overline{abc} \).
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Context:
Find the number(s) that satisfy the equation x = 1/x, where x is the original number. Express your answer using the ± symbol if there are multiple solutions.
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Context:
Find an integer $1 < a < 9991$ such that:
\[
a^{4995} \not\equiv 1 \pmod{9991}, \\
a^{4995} \not\equiv -1 \pmod{9991}, \\
a^{2 \cdot 4995} \not\equiv -1 \pmod{9991}.
\]
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Context:
Determine a function that, when applied to a triangular number \( M \), always results in an odd square number.
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Context:
Determine the asymptotic growth rate, in big O notation, of the divisor function \( \sigma_{-1}(n) \).
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Context:
What is the maximum area of a triangle that can be covered by a square if each vertex of the square is on one of the triangle's sides?
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Context:
If a sphere $O$ with a radius of $2$ has a point $A$ on its surface, and $A$ creates a section of the sphere $O$, and the angle formed by $OA$ and this section is $60^{\circ}$, then the area of this section is
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Context:
Compute the measure of the set \( A = \{ (x, x): x \in [0, 1] \} \) under the product measure \( \mu_1 \times \mu_2 \), where \( \mu_1 \) is the Lebesgue measure and \( \mu_2 \) is the counting measure on \([0, 1]\). Express \( A \) as a Cartesian product to facilitate the computation.
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Context:
Consider a homomorphism $f: D_{18} \to GL(2,\Bbb R)$, where $D_{18}$ is the dihedral group with 18 elements and $f(r) = R := \begin{pmatrix}1& 1 \\ -1& 0\end{pmatrix}$. Given that the reflection $s$ maps to $f(s) = S := \begin{pmatrix}a &b \\ b-a &-a\end{pmatrix}$ with $a^2 -ab + b^2 = 1$, determine the order of $\ker(f)$, denoted as $\lvert \ker(f) \rvert$. What is the order of the kernel?
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Context:
In the convex pentagon $ABCDE$, $\angle A = \angle B = 120^{\circ}$, $EA = AB = BC = 2$, and $CD = DE = 4$. The area of $ABCDE$ is
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Context:
In triangle \(ABC\), median \(BM\) and height \(AH\) are drawn. It is known that \(BM = AH\). Find the angle \(\angle MBC\).
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Context:
Given that \( x_{1}, \cdots, x_{n} \) are all real numbers within the interval \(\left(\frac{1}{4}, 1\right)\), find the minimum value of the sum \( \sum_{k=1}^{n} \log _{x_{4}}\left(x_{k+1} - \frac{1}{4}\right) \), where \( x_{n+1} = x_{1} \).
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Context:
Simplify the expression: \(\frac{a^{1 / 2}+a b^{-1}}{a^{-1 / 3}-a^{-1 / 6} b^{-1 / 3}+b^{-2 / 3}}-\frac{a}{\sqrt[3]{b}}\).
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Context:
Tokorev S.i.

In each cell of a $4 \times 4$ table, there is a sign "+" or "-". It is allowed to simultaneously change the signs to the opposite in any cell and in all cells that share a side with it. How many different tables can be obtained by repeatedly applying such operations?
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Context:
Calculate the degree of the field extension \( \mathbb{Q}(\sqrt[80]{43},\sqrt[84]{43})/\mathbb{Q} \).
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Context:
Given the function $f(x) = A \sin(\omega x + \varphi)$, where $A$, $\omega$, and $\varphi$ are constants with $A > 0$ and $\omega > 0$. If $f(x)$ is monotonic in the interval $[\frac{\pi}{6}, \frac{\pi}{2}]$ and $f(\frac{\pi}{2}) = f(\frac{2\pi}{3}) = -f(\frac{\pi}{6})$, determine the smallest positive period of $f(x)$.
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Context:
Calculate the volume of the region enclosed by the cone $z^2 = x^2 + y^2$ and the sphere $x^2 + y^2 + z^2 = z$ using cylindrical coordinates.
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Context:
On June 1, a bicycle dealer noted that the number of bicycles in stock had decreased by 4 for each of the past few months. If the stock continues to decrease at the same rate for the rest of the year, there will be 36 fewer bicycles in stock on October 1 than were in stock on January 1. For how many months has the stock been decreasing?
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Context:
Find the norm of the linear functional \( \phi(x_n) = \sum_{n\ge 1} (-2)^{1-n}x_n \), where \( \phi: c_0 \to \mathbb{R} \) and \( c_0 \) is the set of sequences converging to 0.
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Context:
Find the expected value of the function \( \phi(x) = x + \frac{F(x) - 1}{f(x)} \), where \( x \in [\underline{x}, \bar{x}] \) and \( f(x) \) is a density function.
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Context:
A, B, and C are standing on a staircase with 7 steps, where each step can accommodate up to 2 people. If the positions of people on the same step are not distinguished, calculate the number of different ways they can stand.
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Context:
The area of the figure enclosed by the curve $y= \sqrt{x}$, the line $y=2-x$, and the x-axis is \_\_\_\_\_\_.
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Context:
Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between $12^2$ and $13^2$?

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Context:
In a directed graph, there are two cycles: one cycle consists of 2 vertices, and the other consists of 3 vertices. Determine the number of 12-step paths from a vertex 'w' back to itself, denoted as \( f(12) \), using the edges of these two cycles.
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Context:
The largest prime factor of the number $520$, $302$, $325$ has $5$ digits. What is this prime factor?
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Context:
In the figure, \(O\) is the center of the circle and \(AB = 5 \text{ cm}\). What is the diameter of this circle?
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Context:
Let $X_1, X_2, \ldots$ be an i.i.d. sequence of random variables with a distribution that has a continuous density function on $[0,1]$. Determine whether the following limit holds almost surely:

$$\lim_{n\to \infty}\frac{\sum_{1\le i<j \le n}{\bf{1}}_{(-1/n,1/n)}{(X_i-X_j)}}{n}=1$$
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Context:
李阿姨配制一种洗洁水，由洗洁液和水按10mL：1L的浓度配成．现有30mL的洗洁液，可以配制多少升的洗洁水？
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Context:
A disphenoid is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a
 disphenoid whose faces are scalene triangles with integer side lengths?

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Context:
Let $z_1,$ $z_2,$ $z_3$ be complex numbers such that $|z_1| = 2,$ $|z_2| = 3,$ and $|z_3| = 4.$  Find the largest possible value of
\[|z_1 - z_2|^2 + |z_1 - z_3|^2 + |z_2 - z_3|^2.\]
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Context:
Consider the set \( M = \{x \in \mathbb{R}^n : x^{⊤} A x = r\} \subset \mathbb{R}^n \), where \( r > 0 \) and \( A \) is a symmetric, positive semidefinite matrix \( A \in \mathbb{R}^{n \times n} \). Determine if \( M \) is a \( C^l \)-submanifold of \( \mathbb{R}^n \). If it is, find its dimension.
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Context:
Evaluate the limit of the integral \( I(r) = \int_{\gamma_r} \frac{e^{iz}}{z^2} \, dz \) as \( r \to \infty \), where \( \gamma_r \) is a specified contour.
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Context:
Given a Poisson random variable $Z$ with parameter $\lambda = \mathbb{E}[Z]$, and another random variable $G$ that converges in distribution to $Z$ with a total variation distance bound $d_{TV}(G,Z) \leq c$, where $c$ is a constant, find an upper bound for the probability $P(|G-Z| > t)$ for a fixed real number $t > 0$. Use the convergence in distribution to derive the bound.
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Context:
Let \( g(x) \) be a continuous and bijective function on \( \mathbb{R} \). Suppose that for every \( t \in \mathbb{R} \), the area bounded by \( y=g(x) \), \( x=b-t \), \( x=b \), and the \( x \)-axis is equal to the area bounded by \( y=g(x) \), \( x=b+t \), \( x=b \), and the \( x \)-axis. If \( g(b)=0 \) and \( \int_{\alpha}^{\beta}g^{-1}(x)~\mathrm{d}x = (\beta-\alpha) mb \), find the value of \( m \).
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Context:
Evaluate the integral \( \int_{-\infty}^\infty \frac{e^{ix}}{x^2} \, dx \).
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Context:
Determine the values of \( p \in (1, +\infty) \) for which the Banach space \( l_p \) is a Fréchet differentiable space (FD-space).
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Context:
Determine the range of values for $a$ such that the function \( f(x) = ax + \cos(2x) + \sin(x) + \cos(x) \) is strictly increasing for all real numbers $x$.
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\boxed{\left( \dfrac{17}{8}, \infty \right)}


Context:
Determine whether the infinite product \( \prod_{k=1}^n \cos(bk) \) converges to zero as \( n \to \infty \), where \( b \) is an irrational number.
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Context:
In a measure space $(X, \mathcal F, \mu)$ with $\mu(X)=1$, there are $n$ measurable sets $F_1, F_2, \ldots, F_n$ such that $\mu(F_j) \geq 1/2$ for all $j$. Determine the smallest integer $k$ such that there exists a subset of indices $\{i_1, i_2, \ldots, i_k\}$ for which $F_{i_1} \cap F_{i_2} \cap \cdots \cap F_{i_k} \neq \varnothing$. What is the smallest possible value of $k$?
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Context:
Evaluate the limit of the integral \( \int_{0}^{\pi}\frac {\sin x}{1+\cos^2(nx)}\mathrm{d}x \) as \( n \to +\infty \).
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Context:
Determine the minimum number of terms required in the Maclaurin series to approximate the square root of $e$ with an error less than $10^{-3}$. Use the inequality \( \left| \frac{e^c}{(n+1)!} \cdot 0.5^{(n+1)} \right| \leq 10^{-3} \), where $0 < c < 0.5$, to find your answer.
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Context:
Calculate the integral \( \int_{-2}^2 \left(x^3\cos\frac{x}{2}+\frac{1}{2}\right)\sqrt{4-x^2}\ dx \) and provide the first 10 digits of the result.
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Context:
Consider a bin containing 1047 distinct numbers. You extract 10 elements without replacement, note them, and then reinsert them into the bin. This process is repeated 50 times. What is the expected number of different elements you will have seen after these 50 extractions?
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Context:
某船上的人开始看见灯塔在南偏东30°方向，后来船沿南偏东60°方向航行45n mile后，看见灯塔在正西方向，则这时船与灯塔的距离是多少？
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Context:
Let \( V \) be an \( n \)-dimensional real inner product space. Define the operator \( M: L(V) \to L(V) \) by \( M(S) = S + S^* \), where \( S^* \) is the adjoint of \( S \). Determine the trace of the operator \( M \).
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Context:
In a shop, two employees are each serving a customer. The service time for each customer is an independent random variable following an exponential distribution with parameter \( \lambda \). After finishing with their current customer, each employee will serve you. What is the probability that you complete your transaction and leave the shop before both of the initial customers finish their transactions?
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Context:
In a vector space of dimension $m$, determine the maximum total number of dimensions that two non-intersecting affine subspaces can have.
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Context:
Let $ \triangle ABC $ be a non-degenerate triangle with perimeter 4 such that $ a = bc \sin^2 A $. If $ M $ is the maximum possible area of $ \triangle ABC $, and $ m $ is the minimum possible area of $ \triangle ABC $, then $ M^2 + m^2 $ can be expressed in the form $ \frac{a}{b} $ for relatively prime positive integers $ a $ and $ b $. Compute $ a + b $.

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Context:
The non-terminating periodic decimal $0.124124 \ldots = 0.\overline{124}$ has period three and is abbreviated by placing a bar over the shortest repeating block.

 If all digits $0$ through $9$ are allowed, how many distinct periodic decimals $0.\overline{d_1d_2 \ldots d_6}$ have period exactly six? Do not include patterns like $0.323$ and $0.17$ that have shorter periods.

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Context:
Determine whether the equation \( x^{2}+y^{2}+z^{2}=2 x y z \) has a solution in non-negative integers.
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Context:
The distance from the point $P$ to two vertices $A$ and $B$ of an equilateral triangle are $|P A| = 2$ and $|P B| = 3$. Find the greatest possible value of $|P C|$.

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Context:
Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of (positive) integers such that $k$ divides $\gcd(a_{k-1},a_k)$ for all $k\geq 2$.  Compute the smallest possible value of $a_1+a_2+\cdots+a_{10}$.
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Context:
A polygon $A_1A_2A_3\dots A_n$ is called beautiful if there exist indices $i$, $j$, and $k$ such that $\measuredangle A_iA_jA_k = 144^\circ$. Compute the number of integers $3 \le n \le 2012$ for which a regular $n$-gon is beautiful.

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Context:
Evaluate the infinite series: $$\sum_{k=0}^{\infty} \sum_{j=0, j\neq k}^{\infty} \sum_{i=0, i\neq j, i\neq k}^{\infty} \frac{1}{3^i3^j3^k}$$
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Context:
In triangle $\triangle ABC$, $\angle C = 120^\circ$. Point $M$ is located on side $AB$ such that $\angle CMB = 60^\circ$ and the ratio $BM:AB = 1:3$. Determine the measure of $\angle B$.
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Context:
In the expansion of \((2x+3y-4z+w)^9\), determine the coefficient of the term \(x^3y^2z^3\).
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Context:
Evaluate the integral \( \int_{-\pi}^{\pi} \frac{d\theta}{1+\sin^2(\theta)} \) using residue theory.
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Context:
Calculate the limit \( \lim_{n \to \infty} x_n \) for the sequence defined by:
\[ x_n = \frac{n}{2^{n+1}} \sum_{i=1}^{n} \frac{2^i}{i} \]
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Context:
Identify the second nonabelian simple group and determine its order.
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Context:
Determine whether the following limit diverges:
\[ \lim_{n \to +\infty} \frac{n+o(n)}{\sqrt{n+o(n)} - \sqrt{n+o(n)}} \cdot \cos(1) \]
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Context:
Determine the complexity class for subsumption in the description logic \(\mathcal{AL}(\circ, ^{-})\). Is it in \(\mathrm{P}\) or does it belong to a different complexity class?
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Context:
Let \( n \in \mathbb{N}_{\geq 2} \). For any real numbers \( a_1, a_2, \ldots, a_n \), denote \( S_0 = 1 \) and for \( 1 \leq k \leq n \), denote \[ S_k = \sum_{1 \leq i_1 < i_2 < \ldots < i_k \leq n} a_{i_1} a_{i_2} \ldots a_{i_k}. \] Find the number of \( n \)-tuples \((a_1, a_2, \ldots, a_n)\) such that \[ (S_n - S_{n-2} + S_{n-4} - \ldots)^2 + (S_{n-1} - S_{n-3} + S_{n-5} - \ldots)^2 = 2^n S_n. \]
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Context:
 The deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation.
Find the smallest possible number of independent deputies after the vacation.

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Context:
Determine whether the limit of the summation diverges: $$\lim\limits_{n \to +\infty} \sum_{k=1}^n \frac{n\cdot \ln (k)}{n^2+k^2}$$
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Context:
In the given geometric figure, calculate the ratio \( \frac{\text{area}(\triangle CDF)}{\text{area}(\triangle CEF)} \).
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