Context:
Evaluate the surface integral \( \int\int_S z \, dx \, dy \) where \( S \) is the complete surface of the cylinder with radius \( a \) and axis \( OZ \) bounded by the planes \( z = 0 \) and \( z = 2a \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
在正四面体 ABCD 中，棱 AD 的中点和面 BCD 的中心的连线为 MN，棱 CD 的中点和面 ABC 的中心的连线为 PQ。求 MN 与 PQ 所成角的余弦值。答案应为\(\frac{k}{m}\)的形式，请给出k+m的值。
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Context:
Find the particular solution to the partial differential equation \( u_{xx}+2u_{xy}-3u_{yy}=0 \) given the general solution \( u(x,y)=F(x+y)+G(x-\frac{1}{3}y) \) and the conditions \( u(x=0,y)=y \) and \( u_x(x=0,y)=y \).
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Context:
Find the smallest positive integer $n$ so that for any integers $a_1, a_2,..., a_{527}$,the number
$$
\left( \prod^{527}_{j=1} a_j\right) \cdot\left( \sum^{527}_{j=1} a^n_j\right)
$$
is divisible by $527$.
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Context:
Find the centroid of the part of a spherical shell with radius 30, centered at the origin, that lies in the first octant.
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Context:
a group of people participate in some curriculum , 35 of them practice yoga , 20 study cooking , 15 study weaving , 7 of them study cooking only , 5 of them study both the cooking and yoga , 3 of them participate all curriculums . how many people study both cooking and weaving ?
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Context:
For every integer $n \ge 1$, the function $f_n : \{ 0, 1, \cdots, n \} \to \mathbb{R}$ is defined recursively by $f_n(0) = 0$, $f_n(1) = 1$, and
\[
(n-k) f_n(k-1) + kf_n(k+1) = nf_n(k)
\]
for each $1 \le k < n$.  Let
\[
S_N = f_{N+1}(1) + f_{N+2}(2) + \cdots + f_{2N}(N)
\]
Find the remainder when $\left\lfloor S_{2013} \right\rfloor$ is divided by $2011$.  (Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$.)
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Context:
Evaluate the sum: 1 - 2 + 3 - 4 + $\cdots$ + 100 - 101
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Context:
Quadrilateral $ABCD$ satisfies $AB = 8$, $BC = 5$, $CD = 17$, and $DA = 10$. Let $E$ be the intersection of $AC$ and $BD$. Suppose the ratio $BE : ED = 1 : 2$. Find the area of quadrilateral $ABCD$.
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Context:
11个集合$M _ { 1 } , M _ { 2 } , \cdots , M _ { 1 1 }$，每个集合有5个元素，并且任意两个集合的交非空.求具有同一公共元素的集合数目的最大值的最小可能值.
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Context:
有 10 个乒兵球选手进行单循环赛, 比赛结果显示, 没有和局, 且任意 5 人中既有 1人胜其余 4 人, 又有 1 人负其余 4 人, 则恰好胜了两场的人数为 $\qquad$个.
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Context:
It is known that the distance between any two of the given $n(n=2,3,4,5)$ points in the plane is at least 1. What is the minimum value that the diameter of this system of points can have?
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Context:
The water level in a rectangular swimming pool measuring 40 feet by 25 feet is to be lowered by some inches. To do this, 3750 gallons of water must be removed. By how many inches should the water level be lowered?
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Context:
In \(\triangle ABC\), \(BC = a\), \(CA = b\), \(AB = c\). If \(2a^{2} + b^{2} + c^{2} = 4\), then the maximum area of \(\triangle ABC\) is ______.
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Context:
Let an integer  $ n > 3$ . Denote the set  $ T\equal{}\{1,2, \ldots,n\}.$  A subset S of T is called *wanting set* if S has the property: There exists a positive integer  $ c$  which is not greater than  $ \frac {n}{2}$  such that  $ |s_1 \minus{} s_2|\ne c$  for every pairs of arbitrary elements  $ s_1,s_2\in S$ . How many does a *wanting set* have at most are there ?
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Context:
Given two sets A and B where A = {x^2, 2x - 1, -4} and B = {x - 5, 1 - x, 9}, and the intersection of B and A is {9}, find the union of sets A and B.
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Context:
Calculate the integral:
\[ I = \int_0^{\infty} \frac{\log(x)}{(x-1)\sqrt{x}} \, dx \]
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Context:
在满足$y<x\leq 100$的有序正整数对（x，y）中，有多少对使得$\frac { x } { y }$与$\frac { x + 1 } { y + 1 }$都是整数？
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Context:
A square piece of paper is folded along its vertical centerline, and then the folded paper is cut along a dotted line, resulting in three rectangular pieces of paper: one large and two small. Calculate the ratio of the perimeter of each small rectangle to the perimeter of the large rectangle. Express your answer as a simplified fraction.
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Context:
Last year, the electricity price in a certain city was 0.80 yuan per kilowatt-hour, with an annual electricity consumption of $a$ kilowatt-hours. This year, it is planned to reduce the electricity price to between 0.55 yuan and 0.75 yuan per kilowatt-hour, while residential users hope for an electricity price of 0.40 yuan per kilowatt-hour (the city's cost price for electricity is 0.30 yuan per kilowatt-hour). It has been calculated that after reducing the electricity price, the increase in electricity consumption in the city is inversely proportional to the difference between the actual electricity price and the user's expected price, with a proportionality constant of $0.2a$. The question is, what is the lowest electricity price that can ensure the revenue of the power department increases by at least 20% compared to last year?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{0.6 \text{ yuan per kilowatt-hour}}


Context:
Let \( N \) be the number of ways to distribute 52 identical balls into 4 distinguishable boxes such that no box is empty and the difference between the number of balls in any two boxes is not a multiple of 6. If \( N = 100a + b \), where \( a \) and \( b \) are positive integers less than 100, find \( a + b \).
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Context:
Three cones with a common vertex \( A \) touch each other externally, with the angle at the vertex of the first two cones being \( \frac{\pi}{3} \). All the cones also touch a plane passing through the point \( A \) and lie on one side of it. Determine the angle at the vertex of the third cone.
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Context:
Evaluate the complex line integral \( \int_\gamma \frac{z^5}{z^7+3z-10}\,dz \), where \( \gamma \) is the boundary of \( D(0,2) \) oriented counterclockwise.
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Context:
The number \( a \) is a root of the equation \( x^{11} + x^{7} + x^{3} = 1 \). Specify all natural values of \( n \) for which the equality \( a^{4} + a^{3} = a^{n} + 1 \) holds.
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Context:
Determine the degree of the image of a curve $C \subset \mathbb{P}^n$ of degree $a$ under the Veronese map $\nu_{n,d}: \mathbb{P}^n \longrightarrow \mathbb{P}^{\binom{d+n}{n}-1}$, where the map is defined by $[...,x_j,...] \longrightarrow [...,M_k,...]$ with $M_k$ being monomials of degree $d$ in $\{...,x_j,...\}$.
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Context:
Calculate the limit $$\lim_{n \to \infty} \frac{f^{n}(0)}{2^{n}n!}$$ given that $\Omega = D(0,2) \setminus \left\{\frac{1}{2}\right\}$, $f$ is a holomorphic function in $\Omega$, and $\frac{1}{2}$ is a simple pole with residue $1$.
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Context:
Find the sum of all prime numbers $p$ such that $\binom{20242024p}{p}\equiv 2024\pmod{p}.$
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Context:
Let $T$ be the sum of all the real coefficients of the expansion of $(1 + ix)^{2018}$. What is $\log_2(T)$?
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Context:
计算下列回路积分．

$\oint _ { l } \frac { d z } { ( z ^ { 2 } + 1 ) ( z - 1 ) ^ { 2 } } ( l$的方程是$x ^ { 2 } + y ^ { 2 } - 2 x - 2 y = 0 )$
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Context:
Given a hyperbola $\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$ where $a>0$ and $b>0$, the right focus is $F(4,0)$. If the line that passes through point F and has an inclination of $60^\circ$ intersects with only one point on the right branch of the hyperbola, determine the range of values for $a$.
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Context:
Mrs. Wilson, the math teacher, has 30 logic puzzles, 18 visual puzzles, and 12 word puzzles that she wants to group into sets for students who finish their tests early. Mrs. Wilson wants each set to be identical, containing the same combination of logic puzzles, visual puzzles, and word puzzles, with no puzzles left over. Additionally, each set must have a minimum of 5 puzzles and the ratio of logic puzzles to visual puzzles should be 2:1. What is the greatest number of sets she can create?
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Context:
For a complex number constant $c$, and real number constants $p$ and $q$, there exist three distinct complex values of $x$ that satisfy the equation:
\[
x^3 + cx + p(1 + qi) = 0
\]
Suppose $c$, $p$, and $q$ were chosen such that all three complex roots $x$ satisfy the inequality:
\[
\frac{5}{6} \leq \frac{\mathrm{Im}(x)}{\mathrm{Re}(x)} \leq \frac{6}{5}
\]
where $\mathrm{Im}(x)$ and $\mathrm{Re}(x)$ are the imaginary and real parts of $x$, respectively. The largest possible value of $|q|$ can be expressed as a common fraction \( \frac{m}{n} \), where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
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Context:
Let \( S \) be the set of all 3-digit numbers with all digits in the set \( \{1,2,3,4,5,6,7\} \) (so in particular, all three digits are nonzero). For how many elements \( \overline{abc} \) of \( S \) is it true that at least one of the (not necessarily distinct) "digit cycles" \( \overline{abc}, \overline{bca}, \overline{cab} \) is divisible by 7? (Here, \( \overline{abc} \) denotes the number whose base 10 digits are \( a, b \), and \( c \) in that order.)
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Context:
已知$f:\mathbf{Z}\rightarrow\mathbf{Z}$为偶函数，且$f(1)=1,f(2017)\neq1$。若对任意的$x、y\in\mathbf{Z}$均满足
$$2f(x+y)-f(x)-f(y)\leqslant|f(x)-f(y)|$$，求$f(2018)$的所有可能值。
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Context:
“斯诺克(Snooker)”是台球比赛的一种，意思是“阻碍、障碍”，所以斯诺克台球有时也被称为障碍台球，是四大“绅士运动”之一，随着生活水平的提高，“斯诺克”也成为人们喜欢的运动之一．$现甲、乙两人进行比赛，比赛采用5局3胜制，各局比赛双方轮流开球(例如：若第一局甲开球，则第二局乙开球，第三局甲开球\ldots \ldots )，没有平局.已知在甲的“开球局”，甲获得该局比赛胜利的概率为 \frac{1}{3} ，在乙的“开球局”，甲获得该局比赛胜利的概率为 \frac{1}{2} ，并且通过“猜硬币”，甲获得了第一局比赛的开球权．$
求甲以3:1赢得比赛的概率；
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Context:
The equation $AB \times CD = EFGH$, where each letter $A$, $B$, $C$, $D$, $E$, $F$, $G$, and $H$ represents a different digit, and the values of $A$, $C$, and $E$ are all nonzero, has many solutions. For example, $46 \times 85 = 3910$. Find the smallest value of the four-digit number $EFGH$ for which there is a solution.
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Context:
Determine the big O notation for the function \(f(n, k)\) defined as follows, given that \(n < k\):

\[f(n, k)=\sum_{i=1}^{n-1}(n-i)(k-i)\log(k-i)\]
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Context:
The figure \( F \) is the intersection of \( n \) circles (\( n \geq 2 \), radii are not necessarily the same). What is the maximum number of curved "sides" that the figure \( F \) can have? (A curved side is a segment of the boundary of \( F \) that belongs to one of the circles and is bounded by intersection points with other circles.)
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Context:
The side surface of a cylinder unfolds into a rectangle with side lengths of $6\pi$ and $4\pi$. The surface area of the cylinder is ______.
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Context:
A five-digit positive integer is created using each of the odd digits $1, 3, 5, 7, 9$ once so that the thousands digit is larger than the hundreds digit, the thousands digit is larger than the ten thousands digit, the tens digit is larger than the hundreds digit, and the tens digit is larger than the units digit. How many such five-digit positive integers are there?
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Context:
Find the limit of the sequence \((b_n)_{n \in \mathbb{N}}\) defined by: \[ b_n = \sum_{k=1}^n\frac{k^3}{n^4+k^2} \] as \( n \to \infty \).
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Context:
Let $\alpha, \beta$ be the roots of $x^2-2$ and $x^2+2$, respectively, over an algebraic closure of $\mathbb{F}_{2023}$. Determine whether $\alpha+\beta$ has degree $2$ or $4$ over $\mathbb{F}_{2023}$.
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Context:
A supermarket has 2355 loaves of bread at the start of the day. By mid-morning, they sell 30% of their loaf stock, and a bakery offers them a 20% discount on a bulk purchase of 750 loaves. The supermarket also purchases 489 loaves with a 15% discount that evening. If the supermarket takes advantage of both discounts, how many loaves of bread are there at the end of the day?
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Context:
From the seven digits $l$, $2$, $3$, $4$, $5$, $6$, $7$, the number of seven-digit positive integers with exactly two even numbers adjacent to each other is $\_\_\_\_\_\_$.
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Context:
Given that $|\overrightarrow{a}|= \sqrt {2}$, $\overrightarrow{b}=(-1,1)$, $\overrightarrow{c}=(2,-2)$, and $\overrightarrow{a}\cdot ( \overrightarrow{b}+ \overrightarrow{c})=1$. Find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.
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Context:
On a square board with a side length of 2017, there is a chess bishop in the lower-left corner. Alexey and Danil take turns making moves with this bishop, with Alexey going first. The players are forbidden from moving the bishop to any cell it has already visited (the bishop moves diagonally for any distance). The player who cannot make a move loses. Which of the boys can guarantee a win regardless of the opponent's actions?
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Context:
Compute the sum of all positive integers $a \leq 26$ for which there exist integers $b$ and $c$ such that $a+23 b+15 c-2$ and $2 a+5 b+14 c-8$ are both multiples of 26.
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Context:
Cube $ABCDEFGH,$ labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\overline{AB}$ and $\overline{CG}$ respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$
[asy]import cse5; unitsize(10mm); pathpen=black; dotfactor=3; pair A = (0,0), B = (3.8,0), C = (5.876,1.564), D = (2.076,1.564), E = (0,3.8), F = (3.8,3.8), G = (5.876,5.364), H = (2.076,5.364), M = (1.9,0), N = (5.876,3.465); pair[] dotted = {A,B,C,D,E,F,G,H,M,N}; D(A--B--C--G--H--E--A); D(E--F--B); D(F--G); pathpen=dashed; D(A--D--H); D(D--C); dot(dotted); label("$A$",A,SW); label("$B$",B,S); label("$C$",C,SE); label("$D$",D,NW); label("$E$",E,W); label("$F$",F,SE); label("$G$",G,NE); label("$H$",H,NW); label("$M$",M,S); label("$N$",N,NE); [/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the condition under which the space \( l^p \) is a subset of the space \( l^q \).
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Context:
Given the basis $B = \{(1, i), (-i, 2)\}$ of $\mathbb{C}^2$ and the linear map $T: \mathbb{C}^2 \to \mathbb{C}^2$ defined by $T(x, y) = (x, 0)$, find the matrix representation of $T$ with respect to basis $B$.
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Context:
In triangle $ABC$, points $D$, $E$, and $F$ are on sides $BC$, $CA$, and $AB$ respectively, such that $BF = BD = CD = CE = 5$ and $AE - AF = 3$. Let $I$ be the incenter of $\triangle ABC$. The circumcircles of $\triangle BFI$ and $\triangle CEI$ intersect at $X \neq I$. Find the length of $DX$.
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Context:
Calculate the flux of the vector field $F(x, y, z) = (x^2, y^2, z^2)$ through the closed surface formed by the paraboloid $z = x^2 + y^2$ and the plane $z = 4$.
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Context:
Jordan noticed that there are 2 cars in his driveway that each have 4 wheels, but one of them has a flat tire, so it's using a spare wheel temporarily. There are 3 bikes; one of them is missing a wheel, and the other two have 2 wheels each. In addition, there is a trash can with 2 wheels, a tricycle, a pair of old roller skates, a wheelchair with 2 additional small front wheels, and a wagon with 4 wheels. Of these items, two of the bikes have a wheel lock installed, and one roller skate is missing a wheel. How many wheels are there?
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Context:
如果一个十位数 $F$ 的各位数字之和为 81 ，则称 $F$ 是一个 "小猿数". 则小猿数的个数为 $\qquad$.
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Context:
A tank can be filled by a tap in 20 minutes and by another tap in 60 minutes. Both the taps are kept open for some minutes and then the first tap is shut off. After this, the tank will be completely filled in 20.000000000000004 minutes. For how long were both taps kept open before the first tap was shut off?
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Context:
Determine the radius of convergence for the power series \(\sum (n^2 \sin(2/n^2))^{n^2+2} x^{n^2}\). Use the formulas \(R=\frac{1}{\limsup_{n\to \infty}\sqrt[n]{|a_n|}}\) and \(R=\lim_{n\to \infty}\left |\frac{a_n}{a_{n+1}} \right |\) to find the radius of convergence.
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Context:
In a football league, whenever a player is transferred from a team  $X$  with  $x$  players to a team  $Y$  with  $y$  players, the federation is paid  $y-x$  billions liras by  $Y$  if  $y \geq x$ , while the federation pays  $x-y$  billions liras to  $X$  if  $x > y$ . A player is allowed to change as many teams as he wishes during a season. Suppose that a season started with  $18$  teams of  $20$  players each. At the end of the season,  $12$  of the teams turn out to have again  $20$  players, while the remaining  $6$  teams end up with  $16,16, 21, 22, 22, 23$  players, respectively. What is the maximal amount the federation may have won during the season?
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Context:
$ S = \{1,2,\dots,n\}$ is divided into two subsets. How the set is divided, if there exist two elements whose sum is a perfect square, then $ n$ is at least ?
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Context:
Consider three different types of computer components with lifetimes following exponential distributions with parameters 1, 2, and 3, respectively. The probability density functions are given by:

\[ f_1(x_1) = e^{-x_1}, \quad f_2(x_2) = \frac{1}{2}e^{-\frac{x_2}{2}}, \quad f_3(x_3) = \frac{1}{3}e^{-\frac{x_3}{3}} \]

where the support sets are \(x_i \geq 0\) for \(i = 1, 2, 3\). One component from each type is randomly sampled, and their lifetimes are recorded. Calculate the probability that \(X_1 > 5X_2 > 10X_3\).
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Context:
Given the function f(x) = x^3 + ax^2 - a^2x, it is known to be a decreasing function in the interval (0, 1) and an increasing function in the interval (2, +∞). Determine the range of values for the real number a.
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Context:
Find the number of real roots of the equation $x^3 - 6x^2 + 9x - 4 = 0$. Express your answer as a whole number.
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Context:
In a casino, you and an infinite number of friends each start with $1. The slot machine costs $1 to play and has three possible outcomes: a win, a loss, or a null. The probabilities are: winning (and gaining $2) with $P(Win) = \frac{1}{5}$, losing with $P(Lose) = \frac{1}{5}$, and getting your $1 back with $P(Null) = \frac{3}{5}$. Winners pass their extra dollar to a friend who starts playing, while losers and null outcomes continue playing with the same dollar. Successive plays are independent, and the casino offers an infinite number of machines. Given that the probability of going broke with $2 is the square of the probability of going broke with $1 (i.e., $P(Broke|2\text{ dollars}) = P(Broke|1\text{ dollar})^2$), find the probability that all of you eventually go broke.
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Context:
We color some unit squares in a $ 99\times 99 $ square grid  with one of $ 5 $ given distinct colors, such that each color appears  the same number of times. On each row and on each column there are no differently colored unit squares. Find the maximum possible number of colored unit squares.
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Context:
The intersection of plane $\alpha$ and plane $\beta$ is line $c$. Line $a$ is parallel to plane $\alpha$ and intersects plane $\beta$. What is the positional relationship between line $a$ and line $c$?
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Context:
Determine the total number of ways to achieve four in a row in a 4x4x4 tic-tac-toe game.
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Context:
Calculate the limit: $$\displaystyle \lim_{\alpha \to -1}\left(2\Gamma(-\alpha-1)+\Gamma\left(\frac{\alpha + 1}{2}\right) \right)$$
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Context:
Given proposition P: ∃x₀ ∈ ℝ, x₀² + 2x₀ + 2 ≤ 0, what is the negation of proposition P? Express your answer as a logical statement using proper notation for universal and existential propositions.
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Context:
Given that the Green Park Middle School chess team consists of three boys and four girls, and a girl at each end and the three boys and one girl alternating in the middle, determine the number of possible arrangements.
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Context:
In quadrilateral $EFGH$, $EF = 6$, $FG = 18$, $GH = 6$, and $HE = x$ where $x$ is an integer. Calculate the value of $x$.
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Context:
9.4. Vasya cut out a triangle from cardboard and numbered its vertices with the digits 1, 2, and 3. It turned out that if Vasya's triangle is rotated 15 times clockwise around its vertex numbered 1 by the angle at this vertex, the triangle will return to its original position. If Vasya's triangle is rotated 6 times clockwise around its vertex numbered 2 by the angle at this vertex, the triangle will return to its original position. Vasya claims that if his triangle is rotated $n$ times around the vertex numbered 3 by the angle at this vertex, the triangle will return to its original position. What is the smallest $n$ that Vasya could have named so that his claim is true for at least some cardboard triangle?
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Context:
Tyrone had $97$ marbles and Eric had $11$ marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X_1, \dots, X_n$ be independent and identically distributed (i.i.d.) random samples from the uniform distribution $Unif(\theta, 6\theta)$, where $\theta > 0$. Apply the Rao-Blackwell theorem to find an unbiased estimator of $\theta$, denoted by $\hat{\theta}$, such that its variance is minimized. What is the expression for $\hat{\theta}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the image of the line $y = mx$ under the exponential transformation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the radical ideal \( \sqrt{I} \) where \( I = (x^2y, y^2z) \) in the polynomial ring \( \mathbb{C}[x,y,z] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which values of integers \(r\) and \(k\) does the polynomial \(x^{p^k} - x\) divide \(x^{p^r} - x\) in \(\mathbb{Z}[x]\), where \(p\) is a prime number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $F_{1}$ and $F_{2}$ be the two foci of the hyperbola $C$: $\frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$), and let $P$ be a point on $C$. If $|PF_{1}|+|PF_{2}|=6a$ and the smallest angle of $\triangle PF_{1}F_{2}$ is $30^{\circ}$, then the eccentricity of $C$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ C$ be the circle with radius 1 centered on the origin. Fix the endpoint of the string with length $ 2\pi$ on the point $ A(1, 0)$ and put the other end point $ P$ on the point $ P_0(1,2\pi)$. From this situation, when we twist the string around $ C$ by moving the point $ P$ in anti clockwise with the string streched tightly, find the length of the curve that the point $ P$ draws from sarting point $ P_0$ to reaching point $ A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the Cauchy problem for the heat conduction equation given by:

\[
\begin{gathered}
u_{t}=u_{x x}, \quad x \in(-\infty,+\infty), \quad t \in(0,+\infty) \\
u(x, 0)=e^{-x} \quad x \in(-\infty,+\infty)
\end{gathered}
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the land of Draconia, there are red, green, and blue dragons. Each dragon has three heads, and each head always tells the truth or always lies. Additionally, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said:

- 1st head: "To my left is a green dragon."
- 2nd head: "To my right is a blue dragon."
- 3rd head: "There is no red dragon next to me."

What is the maximum number of red dragons that could have been at the table?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the arithmetic sequence $\{a\_n\}$, $a\_1=2$, $a\_2+a\_5=13$, find $a\_5+a\_6+a\_7=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equation $2mnp=(m+2)(n+2)(p+2)$, where $m$, $n$, and $p$ are natural numbers with $m \leq n \leq p$, determine the maximum possible value of $p$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the computational complexity of calculating the factorial of a number \( m \), given that the complexity of multiplying \( s+1 \) integers is \( O(s^{2}\log^{2}M) \), where \( M \) is the maximum of the integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For each positive integer $n,$ let $\sigma(n)$ denote the sum of the positive integer divisors of $n.$ How many positive integers $n \leq 2021$ satisfy$$\sigma(3n) \geq \sigma(n)+\sigma(2n)?$$
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Context:
In the Cartesian coordinate system $xOy$, a line $l$ passes through the focus $F$ of the parabola $y^{2}=4x$ and intersects the parabola at points $A$ and $B$, where point $A$ is above the $x$-axis. If the angle of inclination of line $l$ is $60^{\circ}$, then $|OA|=$ ______.
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Context:
There are $n \geq 2$ classes organized $m \geq 1$ extracurricular groups for students. Every class has students participating in at least one extracurricular group. Every extracurricular group has exactly $a$ classes that the students  in this group participate in. For any two extracurricular groups, there are no more than $b$ classes with students participating in both groups simultaneously.

Find $m$ when $n = 8, a = 4 , b = 1$.
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Context:
In the plane, there are $2020$ points, some of which are black and the rest are green. For every black point, the following condition holds: there are exactly two green points that are at a distance of $2020$ from that black point. Find the smallest possible number of green dots.
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Context:
Determine the maximal possible dimension of an abelian Lie subalgebra of the Heisenberg Lie algebra \( \mathfrak{h}_n \) of dimension \( 2n+1 \). The Heisenberg Lie algebra \( \mathfrak{h}_n \) has a basis \( x_1, \dots, x_n, y_1, \dots, y_n, c \) with the Lie bracket defined by \([x_i, y_j] = \delta_{ij}c, [x_i, x_j] = [y_i, y_j] = [x_i, c] = [y_j, c] = 0\), where \(1 \le i, j \le n\) and \(\delta_{ij}\) is the Kronecker delta.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an acute triangle $ABC$, the lines tangent to the circumcircle of $ABC$ at $A$ and $B$ intersect at point $D$. Let $E$ and $F$ be points on $CA$ and $CB$ such that $DECF$ forms a parallelogram. Given that $AB = 20$, $CA=25$, and $\tan C = \frac{4\sqrt{21}}{17}$, the value of $EF$ may be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate \((8^{2^{6^{4^{2^{5^{8^9}}}}}}) \mod 10000\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
An ant starts at the origin of a coordinate plane. Each minute, it either walks one unit to the right or one unit up, but it will never move in the same direction more than twice in the row. In how many different ways can it get to the point $(5,5)$ ?
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Context:
There are 10 numbers written on a circle, and their sum equals 100. It is known that the sum of any three consecutive numbers is at least 29.

What is the smallest number \( A \) such that in any such set of numbers, each number does not exceed \( A \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
James injured himself while exercising, pulling his hamstring which caused significant pain for 3 days. Based on his physiotherapist's advice, he realized that this type of injury would actually require at least 5 times as long to fully heal, and he needs to keep in mind that a setback or complication could further delay his recovery. To be cautious, he decided that he would wait another 3 days beyond the initial healing time before even considering light workouts.

When he does get back into it, his physiotherapist wants him to start off with light exercises only for 2 weeks. There's a possibility that complications could arise in this stage, adding an additional week for recovery. If everything goes smoothly, James can then progress to moderate intensity workouts for another week.

Assuming there are no further setbacks, his physiotherapist explained that he would need to wait 3 weeks after transitioning to moderate intensity workouts to start lifting heavy again.

Taking into account the injury, recovery time, potential setbacks, and various workout intensities, how long could it potentially take before James can resume heavy lifting?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the three numbers $x,y=x^x,z=x^{x^x}$ with $.9<x<1.0$. Arranged in order of increasing magnitude, they are:
Please reason step by step, and put your final answer within \boxed{}.


Context:
设数列 a_{n+1}=\left[\df{a_n}{2}\right]+\left[\df{a_n}{3}\right], n=1,2,\cdots,7，这里 [x] 表示不超过 x 的最大整数. 若 a_8=8，则正整数 a_1 有__________种可能的取值情况.
Please reason step by step, and put your final answer within \boxed{}.


Context:
求三条直线 $x+y=60, y=\frac{1}{2} x, y=0$ 所围成三角形上的整点个数;
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two lines $l_{1}$: $(3+m)x+4y=5-3m$ and $l_{2}$: $2x+(5+m)y=8$ are parallel, the value of the real number $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three of the roots of the equation $x^4 -px^3 +qx^2 -rx+s = 0$ are $\tan A, \tan B$, and $\tan C$, where $A, B$, and $C$ are angles of a triangle. Determine the fourth root as a function only of $p, q, r$, and $s.$
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Context:
On the continuation of side \( BC \) of parallelogram \( ABCD \), a point \( F \) is taken beyond point \( C \). Segment \( AF \) intersects diagonal \( BD \) at point \( E \) and side \( CD \) at point \( G \), where \( GF=3 \) and \( AE \) is 1 more than \( EG \). What part of the area of parallelogram \( ABCD \) is the area of triangle \( ADE \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Galois group of the polynomial \(x^5 + x - 1\) over \(\mathbb{Q}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rod has a certain weight. If the weight of a 6 meters long rod is 6.184615384615385 kg, what is the weight of a 13 meters long rod?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $y_n=\sqrt[4]{7+\sqrt[4]{7+\dots+\sqrt[4]{7}}}$ where the expression on the RHS has $n$ roots. Evaluate the limit: $\lim \limits_{n\to \infty}7^n(3-y_n)$.
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Context:
Given functions f(x) and g(x), let a ∈ {x | f(x) = 0} and β ∈ {x | g(x) = 0}. If for all α and β, |α - β| ≤ 1, then f(x) and g(x) are called "adjacent zero-point functions". If f(x) = x - 1 and g(x) = x^2 - ax - a + 3 are "adjacent zero-point functions", find the range of values for the real number a.
Please reason step by step, and put your final answer within \boxed{}.


Context:
集 $A=\{0,1,2, \cdots, 9\}, B_1, B_2, \cdots, B_k$ 为 $A$ 的一族非空子集, 并且 $i \neq j$ 时, $B_i \cap B_j$ 至多有两个元素. 求 $k$ 的最大值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设集合A中元素全为正整数，且对任意x,y∈A(x≠y)，有$\vert x - y \vert \geq \frac { 1 } { 2 5 } x y$，问A中最多有几个数？
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Consider a Brownian motion $X_t$, defined for $t\geqslant 0$. Let $T_a$ denote the stopping time at which $X_t$ reaches level $a$, i.e., $T_a = \inf \{t \mid X_t = a\}$. Compute the probability $\mathbb P\{T_{2}< T_{-1} < T_{3}\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \int_{C}{z\,\mathrm dx+x\,\mathrm dy-y\,\mathrm dz} \) where \( C \) is the path defined by \( \gamma(t)=(t^2, 1-t, t) \) for \( t \in [-1,1] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ ABCD$ be a trapezoid with $ AB$ and $ CD$ parallel, $ \angle D = 2 \angle B, AD = 5,$ and $ CD = 2.$ Then $ AB$ equals
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a random variable $X$ with probability density function $f(X)$, let $Y$ be the sum of $N$ independent draws of $X$. Determine the probability density function of $Y$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x\to0} {\frac{\frac{1}{1+x^3} + \frac{1}{3}\log{\left(1+3x^3\right)}-1}{2\sin{\left(3x^2\right)}-3\arctan{\left(2x^2\right)}}}$$
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Context:
Find the minimum distance of the ternary linear code with the following parity check matrix:

$$H=\begin{pmatrix} 1 & 0 & 2 & 1 & 2 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 2 \\ 2 & 2 & 1 & 1 & 0 \end{pmatrix}.$$
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Context:
Let \(T: \mathbb{R}^4 \to \mathbb{R}^4\) be a linear map with a null space defined by the condition \((x, y, w, z) \in \mathbb{R}^4\) such that \(x + y + w + z = 0\). If the rank of \((T - 4I_4)\) is 3, and the minimal polynomial of \(T\) is \(x(x - 4)^{\alpha}\), find the value of \(\alpha\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
On an infinite chessboard, some bishops and knights are placed such that:
- Each bishop has a knight on the same diagonal.
- Each knight has a bishop exactly \(\sqrt{5}\) units away.
- Removing any piece disrupts at least one of these conditions.

Determine all possible values of \(n\), the total number of pieces on the board.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the residue of \((1-e^{z})^{-n}\) at \(z = 0\) for any positive integer \(n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A positive integer \( N \) is called *apt* if for each integer \( 0 < k < 1009 \), there exists exactly one divisor of \( N \) with a remainder of \( k \) when divided by \( 1009 \). For a prime \( p \), suppose there exists an *apt* positive integer \( N \) such that \( \frac{N}{p} \) is an integer but \( \frac{N}{p^2} \) is not. Determine the number of possible remainders when \( p \) is divided by \( 1009 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the set of values for \( \alpha \) such that the integral \( \int_1^{+\infty} \frac{x^{\alpha} \cdot \sin x}{1+x^3} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the greatest common divisor (GCD) of the complex numbers $6-17i$ and $18+i$ in the ring of Gaussian integers $\Bbb Z[i]$. Provide your answer in the form of a complex number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a positive integer $n$, denote by $\tau(n)$ the number of positive integer divisors of $n$, and denote by $\phi(n)$ the number of positive integers that are less than or equal to $n$ and relatively prime to $n$. Call a positive integer $n$ good if $\varphi(n)+4 \tau(n)=n$. For example, the number 44 is good because $\varphi(44)+4 \tau(44)=44$. Find the sum of all good positive integers $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression \( s_1 t_k \delta_{ii} \delta_{k1} \delta_{nn} \) where all indices run from 1 to 3. What is the result?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the probability that a group generated by two random invertible $2 \times 2$ matrices $A$ and $B$ over the integers is isomorphic to the free group $F_2$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A point is randomly selected on the segment [12, 17]. Let $k$ be the selected value. Determine the probability that the roots of the quadratic equation $(k^2 + k - 90)x^2 + (3k - 8)x + 2 = 0$ satisfy the condition $x_1 \leq 2x_2$. Provide your answer as a probability.
Please reason step by step, and put your final answer within \boxed{}.


Context:
6. In a rectangle $A B C D$ (including the boundary) with an area of 1, there are 5 points, among which no three points are collinear. Find the minimum number of triangles, with these 5 points as vertices, whose area is not greater than $\frac{1}{4}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the upper bound time complexity (Big-O notation) for the recurrence relation: \[ T(n) = T\left(\frac{n}{3}\right) + T\left(\frac{2n}{3} + 4\right) + a \times n \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jonathan wants to buy a dictionary that costs $11, a dinosaur book that costs $19, a children's cookbook that costs $7, an atlas that costs $15, and a set of picture books costing $32. He finds a coupon that offers a 5% discount on his total purchase before taxes. The tax rate is 8%. He has already saved $8 from his allowance. How much more money does Jonathan need to buy all five books, considering the discount and tax?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A line passing through the focus of the parabola $y^2 = ax$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. If $x_1 + x_2 = 8$ and the length of segment $AB$ is $12$, then find the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知不等式 |ax^2+bx+a|\leq x 对 x\in[1,2] 恒成立，则 3a+b 的最大值为__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
3. 3.1. Petya marked 5 points on a face of a cube, turned it and marked 6 points on an adjacent face, then turned it again and marked 7 points, and so on. He marked points on each face this way. What is the maximum number of points that can be on two opposite faces?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of integers $2 \le n \le 2016$ such that $n^n - 1$ is divisible by $2$, $3$, $5$, and $7$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many group homomorphisms are there from $\Bbb Z/2\Bbb Z \times \Bbb Z/2\Bbb Z$ to $S_3$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
底面半径为 1 cm 的圆柱形容器里放有四个半径为 $\frac{1}{2} \mathrm{~cm}$ 的实心铁球, 四个球两两相切, 其中底层两球与容器底面相切. 现往容器里注水, 使水面恰好浸没所有铁球, 则需要注水 $\qquad$ $\mathrm{cm}^{3}$. 原始答案为$\left(\frac{m}{n}+\frac{\sqrt{o}}{p}\right) \pi$的格式，请给出m+n+o+p的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the following infinite series:

$$\sum_{n=1}^\infty \frac{1\cdot3 \cdot ... \cdot (2n-1)}{4^n 2^n n!}$$
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Context:
Given a mechanical device with a lifetime distribution characterized by the hazard rate function \( \lambda(t) = 3.9 \cdot t^2 \) for \( t > 0 \), calculate the probability that the device's lifetime is between 1 and 2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the volume of the figure cut off from a cylinder by a plane that passes through the diameter of its base. Given the radius of the cylinder $R$ and the height of the resulting figure $h$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If a person walks at 35 km / hr instead of 15 km / hr and covers 60% of the distance uphill with a 10% decrease in speed, while the rest of the distance is flat, he would have walked 45 km more. What is the actual distance traveled by him?
Please reason step by step, and put your final answer within \boxed{}.


Context:
设S={1,2,⋯,7}，则以S的两个不相交的子集为元素的不同2元集的个数为
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of palindromes that are less than 1000.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Six congruent copies of the parabola $y = x^2$ are arranged in the plane so that each vertex is tangent to a circle, and each parabola is tangent to its two neighbors.  Find the radius of the circle.

[asy]
unitsize(1 cm);

real func (real x) {
  return (x^2 + 3/4);
}

path parab = graph(func,-1.5,1.5);

draw(parab);
draw(rotate(60)*(parab));
draw(rotate(120)*(parab));
draw(rotate(180)*(parab));
draw(rotate(240)*(parab));
draw(rotate(300)*(parab));
draw(Circle((0,0),3/4));
[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jade had $38 and her sister Julia had half as much money she had. On New Year's eve, their aunt gave each of them an equal amount of money. How much money did they have in total after their aunt gave them money?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the  $4\times4$  array of  $16$  dots, shown below.

[asy]
size(2cm);
dot((0,0));
dot((1,0));
dot((2,0));
dot((3,0));
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((0,3));
dot((1,3));
dot((2,3));
dot((3,3));
[/asy]

Counting the number of squares whose vertices are among the  $16$  dots and whose sides are parallel to the sides of the grid, we find that there are nine  $1\times1$  squares, four  $2\times2$  squares, and one  $3\times3$  square, for a total of  $14$  squares. We delete a number of these dots. What is the minimum number of dots that must be deleted so that each of the  $14$  squares is missing at least one vertex?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be the set $A = \{ 1,2, \ldots, n\}$. Determine the maximum number of elements of a subset $B\subset A$ such that for all elements $x,y$ from $B$, $x+y$ cannot be divisible by $x-y$.

[i]Mircea Lascu, Dorel Mihet[/i]
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Context:
A pile of candies consists of three types of flavors: chocolate, fruit, and milk. If you take away 5 pieces of chocolate candy, the remaining candies can be divided into 2 identical portions; if you take away 2 pieces of fruit candy, the remaining candies can be divided into 3 identical portions; if you take away 3 pieces of milk candy, the remaining candies can be divided into 5 identical portions. What is the minimum number of candies in this pile?
Please reason step by step, and put your final answer within \boxed{}.


Context:
John decides to get a new apartment with a roommate. His former rent was $2 per square foot for a 750 square foot apartment. In the first six months of the year, the rent for the new apartment is $2,800 per month and increases by 5% for the last six months. The apartment's utilities cost is $200 per month during the winter months (December, January, and February) and $150 per month during the other months. If John and his roommate split the rent and utilities cost evenly each month, how much money does John save in a year compared to living in his former apartment?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \(ABC\), angle \(C\) equals \(30^\circ\), and angle \(A\) is acute. A line perpendicular to side \(BC\) is drawn, cutting off triangle \(CNM\) from triangle \(ABC\) (point \(N\) lies between vertices \(B\) and \(C\)). The areas of triangles \(CNM\) and \(ABC\) are in the ratio \(3:16\). Segment \(MN\) is half the height \(BH\) of triangle \(ABC\). Find the ratio \(AH:HC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a particular country, the state of Sunland issues license plates with a format of one letter, followed by three digits, and then two letters (e.g., A123BC). Another state, Moonland, issues license plates where the format consists of two digits, followed by two letters, and then two more digits (e.g., 12AB34). Assuming all 10 digits and all 26 letters are equally likely to appear in their respective positions, calculate how many more license plates Sunland can issue compared to Moonland, given that Sunland always uses the letter 'S' as the starting letter in their license plates.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A magician has one hundred cards numbered from 1 to 100. They place each card into one of three boxes - a red box, a white box, and a blue box - in such a way that each box contains at least one card.

An audience member selects two of the three boxes and takes one card from each. They then announce the sum of the numbers on the two cards. Knowing this sum, the magician can determine from which box no card was taken.

How many ways are there to distribute the cards into the boxes so that this trick always works? (Two distributions are considered different if at least one card is placed into a different box.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the additive group of ordered pairs of integers $(m,n) [$with addition defined componentwise: $\left.(m,n)+\left(m^{\prime},n^{\prime}\right)=\left(m+m^{\prime}, n+n^{\prime}\right)\right]$ consider the subgroup $H$ generated by the three elements$$(3,8),\quad(4,-1),\quad(5,4).$$Then $H$ has another set of generators of the form$$(1,b),\quad(0,a)$$for some integers $a,b$ with $a>0$. Find $a$.
[Elements $g_{i},\ldots,g_{k}$ are said to generate a subgroup $\boldsymbol H$ if $\mathrm{(i)}$ each $g_{i}\in\boldsymbol H$, and $\mathrm{(ii)}$ every $h \in\boldsymbol H$ can be written as a sum $h=n_{1}g_{1}+\cdots+n_{k}g_{k}$ where the $n_{i}$ are integers (and where, for example, $3g_{1}-2g_{2}$ means $\left.\left.g_{1}+g_{1}+g_{1}-g_{2}-g_{2}\right).\right]$
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Context:
In the village, there are 100 houses. What is the maximum number of closed, non-intersecting fences that can be built so that each fence encloses at least one house and no two fences enclose the same set of houses?

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


Context:
A parliament consists of $n$ senators. These senators are organized into 10 parties and 10 committees. Each senator belongs to exactly one party and one committee.

Determine the smallest possible value of $n$ such that it is possible to label the parties and the committees with numbers from 1 to 10, ensuring that there are at least 11 senators for which the party and committee numbers match.
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Context:
On Ming's way to the swimming pool, there are 200 trees. On his round trip, Ming marked some trees with red ribbons. On his way to the swimming pool, he marked the 1st tree, the 6th tree, the 11th tree, and so on, marking every 4th tree. On his way back, he marked the 1st tree he encountered, the 9th tree, the 17th tree, and so on, marking every 7th tree. How many trees are unmarked when he returns home?
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Context:
How long will it take a train traveling at 68 kmph to pass an electric pole if the train is 170 m long, given a 5% incline on the track and a headwind of 20 kmph affecting the train's speed?
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Context:
Given that the volume of pool \(M\) is 1.5 times that of pool \(N\), there are three water pipes \(A\), \(B\), and \(C\). Pipe \(A\) alone can fill pool \(M\) in 5 hours; pipe \(B\) alone can fill pool \(N\) in 5 hours; pipe \(C\) alone can fill pool \(N\) in 6 hours. If pipes \(A\), \(B\), and \(C\) are opened simultaneously, with \(A\) always filling pool \(M\), \(C\) always filling pool \(N\), and \(B\) starting by filling pool \(M\) and then switching to fill pool \(N\), the two pools are both filled exactly at the same time. How many hours does pipe \(B\) fill pool \(M\)?
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Context:
A parallelepiped $A B C D A_1 B_1 C_1 D_1$ is given. Points $M, L$, and $K$ are taken on the edges $A D, A_1 D_1$, and $B_1 C_1$ respectively, such that $B_1 K = \frac{1}{3} A_1 L$ and $A M = \frac{1}{2} A_1 L$. It is known that $K L = 2$. Find the length of the segment by which the plane $K L M$ intersects the parallelogram $A B C D$.
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Context:
Let $G$ be the Galois group of a field with nine elements over its subfield with three elements. Determine the number of orbits for the action of $G$ on the fields with nine elements.
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Context:
Write the numbers \(1234567891011 \cdots \cdots 19941995\) on the blackboard to form the integer \(N_{1}\). Erase the digits at the even positions of \(N_{1}\). The remaining digits form the integer \(N_{2}\). Erase the digits at the odd positions of \(N_{2}\). The remaining digits form the integer \(N_{3}\). Erase the digits at the even positions of \(N_{3}\). The remaining digits form the integer \(N_{4}\). This process continues until only one digit remains on the blackboard. Determine this digit (Note: Counting positions from left to right, for example, in 12345, 1 is in the 1st position, 2 is in the 2nd position, and so on).
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Context:
A 是 10 阶实矩阵, 且 $A^{5}=O$, 则 $\operatorname{rank}(A)$ 的所有可能值之和为 $\qquad$.
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Context:
A train of a certain length moving at a speed of 72 kmph takes 105.99152067834574 seconds to cross another train of length 640 m, moving at 36 kmph in the same direction. What is the length of the first train?
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Context:
Find the largest prime number less than $2008$ that is a divisor of some integer in the infinite sequence \[ \left\lfloor\frac{2008^1}{1}\right\rfloor, \left\lfloor\frac{2008^2}{2}\right\rfloor, \left\lfloor\frac{2008^3}{3}\right\rfloor, \left\lfloor\frac{2008^4}{4}\right\rfloor, \ldots. \]
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Context:
In a 7x7 grid, each unit square can be part of a Swiss cross shape consisting of 5 unit squares. Determine the minimum number of squares that must be marked on the grid so that every possible placement of a Swiss cross covers at least one marked square.
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Context:
John watches a TV show and they announce they are going to do 1 more season. Each season has a certain number of episodes except for the last season which is 4 episodes longer. There were 9 seasons before the announcement. If each episode is .5 hours, it will take 112 hours to watch them all after the last season finishes. How many episodes are there in each season?
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Context:
In convex quadrilateral \(EFGH\), \(\angle E = \angle G\), \(EF = GH = 150\), and \(EH \neq FG\). The perimeter of \(EFGH\) is 580. Find \(\cos E\).
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Context:
Calculate the work done by the force field $F(x, y) = (-y^2, x^2)$ along the boundary of the rectangle defined by the lines $x - y = 0$, $x - y = 1$, $x + y = 1$, and $x + y = 2$, where the path is taken in the positive direction (counter-clockwise) using Green's theorem.
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Context:
甲、乙两人进行乒乓球比赛，约定每局胜者得 1 分，负者得 0 分，比赛进行到有一人比对方多 2 分或打满 6 局时停止。设甲在每局中获胜的概率为 \(\frac{2}{3}\)，乙在每局中获胜的概率为 \(\frac{1}{3}\)，且各局胜负相互独立。求比赛停止时已打局数 \(\xi\) 的期望 \(E \xi\)。答案形式为\(\frac{k}{m}\)，请给出k+m的值。
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Context:
A sequence of positive integers with $n$ terms satisfies $\sum_{i=1}^{n} a_i = 2007$. Find the least positive integer $n$ such that there exist some consecutive terms in the sequence with their sum equal to $30$.
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Context:
Let $G=S_5$. Consider the set $L$ of subgroups of $G$ that are isomorphic to the non-cyclic group of order 4. Let us call two subgroups $H$ and $K$ belonging to $L$ as equivalent if they are conjugate (that is $gHg^{-1}=K$ for some $g\in G$). How many equivalence classes are there in $L$?
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Context:
a side of beef lost 35 percent of its weight in processing . if the side of beef weighed 560 pounds after processing , how many pounds did it weigh before processing ?
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Context:
Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first meet at the point A again, calculate the number of times they meet, excluding the start and finish. Express your answer as a whole number.
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Context:
Given the function f(x) = 7 + a^(x-1), where a > 0 and a!= 1, find the coordinates of the fixed point P that the graph of this function passes through for any valid value of a. Express your answer as an ordered pair (x, y).
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Context:
19. Fill in the following squares with $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ respectively, so that the sum of the two five-digit numbers is 99999. How many different addition equations are there? (Consider $a+b$ and $b+a$ as the same equation)
$\square \square \square \square \square+$ $\square$ $=99999$
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Context:
Determine the maximum power of 2 that completely divides \(3^{1024} - 1\).
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Context:
Given two circles ${C}_{1}:(x-a)^{2}+{y}^{2}=36$ and ${C}_{2}:{x}^{2}+(y-2)^{2}=4$ are tangent internally, then $a=\_\_\_\_\_\_.$
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Context:
Let \( f(x) = x^3 - x^2 - 3x - 1 \) and \( h(x) = \frac{f(x)}{g(x)} \) be a rational function such that:

(a) \( h(x) \) is continuous everywhere except at \( x = -1 \).
(b) \( \lim_{x \to \infty} h(x) = \infty \).
(c) \( \lim_{x \to -1} h(x) = \frac{1}{2} \).

Find \( \lim_{x \to 0} (3h(x) + f(x) - 2g(x)) \).
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Context:
When triangle $EFG$ is rotated by an angle $\arccos(_{1/3})$ around point $O$, which lies on side $EG$, vertex $F$ moves to vertex $E$, and vertex $G$ moves to point $H$, which lies on side $FG$. Find the ratio in which point $O$ divides side $EG$.
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Context:
Evaluate the limit: $$\lim_{x\rightarrow\infty} \frac{1}{x}\left(\frac{\Gamma(2x-1)}{\Gamma(x-1)}\right)^{1/x}$$
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Context:
Evaluate the limit: \[ \lim_{k \rightarrow \infty} \int_{-\infty}^\infty \frac{1}{1+x^2+\cdots+x^{2k}} \, \text{dx} = \lim_{k \rightarrow \infty} \int_{-\infty}^\infty \frac{1}{\sum_{n=0}^k x^{2n}} \, \text{dx}. \]
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Context:
Point $P(2,16)$ and $Q(\log_{2}3,t)$ both lie on the graph of the same exponential function. Find $t$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an election, there are three candidates - A, B, and C. Candidate A gets 50% of the votes, candidate B gets 35% of the votes, and candidate C gets 15% of the votes. Candidate A is elected by a majority of 1400 votes over candidate B. What is the total number of votes polled?
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Context:
Bobby needs to buy a new pair of fancy shoes. He decides to go to a cobbler and get them handmade. The cobbler charges $250 to make the mold. He then charges $75 an hour for 8 hours to make the shoes. The cobbler agrees to charge a certain percentage of the cost for work to make the shoes, since it is his first pair of shoes. Bobby paid $730. What percentage of the cost for work did the cobbler charge?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A point is randomly selected on a plane, where its Cartesian coordinates are integers with absolute values less than or equal to 4, and all such points are equally likely to be chosen. Calculate the probability that the selected point is at most 2 units away from the origin. Express your answer as a fraction in simplest form.
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Context:
Given the hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, $F_{1}$, $F_{2}$ are its left and right foci, point $P$ is a point on the right branch of the hyperbola, point $M$ is the center of the circle, circle $M$ is the inscribed circle of triangle $PF_{1}F_{2}$, $PM$ line intersects the $x$-axis at the point $(1, 0)$, and is parallel to one of the asymptotes of the hyperbola and is at a distance of $\frac{\sqrt{2}}{2}$. Calculate the eccentricity of the hyperbola $C$. Express your answer as a single value.
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Context:
Find the closure of the set of irrational numbers in the cocountable topology on the real numbers.
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Context:
How many nine-digit numbers composed of eight 3's and one 0 satisfy the condition that they leave a remainder of 1 when divided by 4? Express your answer as a single integer.
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Context:
An geometric sequence $\{a_n\}$ has 20 terms, where the product of the first four terms is $\frac{1}{128}$, and the product of the last four terms is 512. The product of all terms in this geometric sequence is \_\_\_\_\_\_.
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Context:
Given the set A = {1, 2, 3, 4, 5, 6}, count the number of bijections f: A -> A that satisfy the condition f(f(f(x))) = x for all x in A. Express your answer as a single integer.
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Context:
Samson writes down the number 123456789 on a piece of paper. He can insert multiplication signs between any two adjacent digits, any number of times at different places, or none at all. By reading the digits between the multiplication signs as individual numbers, he creates an expression made up of the products of these numbers. For example, 1234$\cdot$56$\cdot$789. What is the maximum possible value of the resulting number?
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Context:
Given the equation \(1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{23} = \frac{a}{23!}\), find the value of \(a \mod 13\).
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Context:
Calculate the probability \( P(p=0.5|X=7) \) given that \( X \sim \text{Bin}(8, p) \) and \( p \) is either 0.5 or uniformly distributed over [0.5, 1]. Use Bayes' Rule to find the solution.
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Context:
Maria buys computer disks at a price of 5 for $6 and sells them at a price of 4 for $7. Find how many computer disks Maria must sell to make a profit of $120.
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Context:
Suppose $X, Y$ are independent and identically distributed random variables with $X,Y\sim N(0,1)$. What is $P(X > 0 \mid X + Y > 0)?$
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Context:
There are 10 consecutive natural numbers written on the board. What is the maximum number of them that can have a digit sum equal to a perfect square?
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Context:
We are given a coin of diameter $\frac{1}{2}$ and a checkerboard of $1 \times 1$ squares with an area of $2010 \times 2010$. We toss the coin such that it lands completely on the checkerboard. If the probability that the coin doesn't touch any of the lattice lines is $\frac{a^2}{b^2}$, where $\frac{a}{b}$ is a reduced fraction, find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the genus of the curve given by the intersection \( \{X^4+Y^4+Z^4=0\} \cap \{X^3+Y^3+(Z-tW)^3=0\} \subset \mathbb{P}^3 \). Use the adjunction formula and the fact that \( K_{\mathbb{P}^3}=O(-4) \).
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Context:
A merchant purchases a gadget for $30$ less $15\%$. He aims to sell the gadget at a gain of $25\%$ on his cost after allowing a $10\%$ discount on his marked price. At what price, in dollars, should the gadget be marked?
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Context:
Compute the intersection number of the pullbacks of the line bundles \(\iota^*\mathcal{O}(2,3)\) and \(\iota^*\mathcal{O}(2,3)\) on the smooth subvariety \(X = V_+(f) \subset \mathbb{P}^2 \times \mathbb{P}^1\), where \(f \in H^0(\mathcal{O}(1,2))\).
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Context:
A sequence \( i_{1}, i_{2}, \ldots, i_{n} \) is formed from the first \( n > 3 \) natural numbers, arranged in random order. What is the probability that for all \( k = 1, 2, \ldots, n \), the inequality \( i_{k} \geq k - 3 \) holds true?
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Context:
Evaluate the limit \( \lim_{\epsilon\to 0^+} \frac{3}{4\pi\epsilon^3}\oint_{S_\epsilon} G \cdot\hat{N}dS \), where \( S_\epsilon \) is the sphere centered at \((a,b,c)\) with radius \( \epsilon \), and the vector field is given by \( G(x,y,z) = (x^2y, z^2x, y^2z) \). Here, \( \hat{N} \) is the outward unit normal vector to the sphere.
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Context:
If the circle $(x+2)^2 + (y-a)^2 = 1$ intersects with the circle $(x-a)^2 + (y-5)^2 = 16$, the range of the real number $a$ is _________.
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Context:
Calculate the probability that a random variable \( \xi \) from a Beta distribution with parameters \((1,2)\) is less than a random variable \( \eta \) from an Exponential distribution with parameter \(1\), given that \( \xi \) and \( \eta \) are independent.
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Context:
A drink vendor has some liters of Maaza, 144 liters of Pepsi and 368 liters of Sprite. He wants to pack them in cans, so that each can contains the same number of liters of a drink, and doesn't want to mix any two drinks in a can. The least number of cans required is 143. How many liters of Maaza does the vendor have?
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Context:
Find the dimension of the quotient ring \( \mathbb{K}[x,y,z]/(xy,xz,yz) \) where \( \mathbb{K} \) is a field.
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Context:
正整数 $9+95+995+\cdots+\underbrace{99 \cdots 9}_{2016} 5$ 的十进制表示中数值 1 的个数为多少？
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Context:
The decreasing sequence $a, b, c$ is a geometric progression, and the sequence $19a, \frac{124b}{13}, \frac{c}{13}$ is an arithmetic progression. Find the common ratio of the geometric progression.
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Context:
Given the sequence $\left\{a_{n}\right\}$ satisfying:
$$
a_{0}=\sqrt{6}, \quad a_{n+1}=\left[a_{n}\right]+\frac{1}{\left\{a_{n}\right\}} \text {, }
$$

where $[a]$ denotes the greatest integer less than or equal to the real number $a$, and $\{a\}=a-[a]$. Determine the value of $a_{2020}$.
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Context:
If \(\sin \left(c^{2}-3 c+17\right)^{\circ}=\frac{4}{b-2}\), where \(0 < c^{2}-3 c +17 < 90\) and \(c > 0\), find the value of \(c\).
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Context:
In this version of SHORT BINGO, a $5\times5$ card is again filled by marking the middle square as WILD and placing 24 other numbers in the remaining 24 squares. Now, the card is made by placing 5 distinct numbers from the set $1-15$ in the first column, 5 distinct numbers from $11-25$ in the second column, 4 distinct numbers from $21-35$ in the third column (skipping the WILD square in the middle), 5 distinct numbers from $31-45$ in the fourth column, and 5 distinct numbers from $41-55$ in the last column. How many distinct possibilities are there for the values in the first column of this SHORT BINGO card?
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Context:
A square flag features a symmetric red cross with uniform width and two identical blue squares in the center on a white background. The entire cross, including the red arms and the two blue centers, occupies 40% of the flag's area. Determine what percent of the flag's area is occupied by the two blue squares.
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Context:
In the year 1990, there were 5000 men, 3000 women, and 2000 boys. In 1994, the number of men increased by a certain percentage and the number of women increased by the ratio of boys to women. The total population in 1994 was 13000. What was the percentage increase in the number of men?
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Context:
If you multiply two integers together and then add 4, the result is 40. What sum of the two numbers could not be possible?
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Context:
A sequence $a_0,a_1,a_2,\cdots,a_n,\cdots$ satisfies that $a_0=3$, and $(3-a_{n-1})(6+a_n)=18$, then the value of $\sum_{i=0}^{n}\frac{1}{a_i}$ is________.
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Context:
The mathematician John is having trouble remembering his girlfriend Alicia's 7-digit phone number. He remembers that the first four digits consist of one 1, one 2, and two 3s. He also remembers that the fifth digit is either a 4 or 5. While he has no memory of the sixth digit, he remembers that the seventh digit is 9 minus the sixth digit. If this is all the information he has, how many phone numbers does he have to try if he is to make sure he dials the correct number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum \( \sum_{n=0}^\infty \frac{1}{n!(n^4+n^2+1)}. \)
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Context:
Determine the number of possible bases for a field extension $M/K$ where $|M|=p^3$ and $K=\mathbb{F}_{p}$.
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Context:
Calculate the least number of intervals \(n\) required to approximate the definite integral of \(5e^{x^2}\) from 0 to 1 using the Error Bound formula with an error less than \(10^{-9}\). The given error bound formula is:
\[Error(S_n) \le \frac{k(b-a)^5}{180N^4}\]
where \(f^{(4)}(x) = 20e^{x^2}(4x^4 + 12x^2 + 13)\), and the maximum of \(f^{(4)}(x)\) occurs at \(x = 1\).
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Context:
A coffee express machine has the following probabilities for each request: it brews coffee with probability \( p_c = 0.9 \), pours hot water with probability \( p_w = 0.09 \), and gets out of order with probability \( p_f = 0.01 \). Let \( X \) denote the number of coffees the machine prepares before it gets out of order. Compute \( \operatorname{E}[X] \).
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Context:
Determine if there exists a distinct non-zero integer $k$ such that the equation $x^4 + y^4 = k$ has more than one solution in terms of distinct non-zero integer pairs $(x, y)$. If such a $k$ exists, provide an example.
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Context:
How many unique $3$-letter sequences with no spaces can be made using the letters in "AUGUSTIN LOUIS CAUCHY", which contains $19$ letters? For example, "GAA" is one acceptable sequence, but "GGA" is not an acceptable sequence because there is only one G available. The original ordering of the letters does not have to be preserved.
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Context:
In a 3-dimensional space, there are 2016 spheres, none of which coincide. Some of the spheres are red, and the others are green. Each contact point between a red sphere and a green sphere is painted blue. Find the maximum possible number of blue points.
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Context:
Two vertical towers, \( AB \) and \( CD \), are located \( 16 \mathrm{~m} \) apart on flat ground. Tower \( AB \) is \( 18 \mathrm{~m} \) tall and tower \( CD \) is \( 30 \mathrm{~m} \) tall. Ropes are tied from \( A \) to \( C \) and from \( B \) to \( C \). Assuming the ropes are taut, calculate the total length of rope in meters.
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Context:
Calculate the residue of the function \( f(z)=\frac{1}{1+9z^2} \) at the pole \( z=\frac{i}{3} \).
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Context:
How many positive integers $N$ satisfy all of the following three conditions?

1. $N$ is divisible by $2020$.
2. $N$ has at most $2020$ decimal digits.
3. The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros.
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Context:
Charles bought 20 papers to draw, consisting of 10 regular papers and 10 watercolor papers. Today, Charles drew 6 pictures, 4 on regular papers, and 2 on watercolor papers. Yesterday, he drew 6 pictures before going to work, all of which were on regular papers. He also committed to using only watercolor papers for all the drawings when he came back from work yesterday. If he has 2 regular papers left, how many watercolor pictures did Charles draw after coming back from work yesterday, given that at least one of the remaining drawings must be on a watercolor paper?
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Context:
You can lower the price by 20% if you buy more than fifteen units of iPhone cases. If you pay $500 to buy 18 units, what is the original price?
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Context:
Suppose \( f(x+1) = 2f(x) \) for all \( x \in \mathbb{R} \), and \( f(x) = x(x-1) \) for \( 0 < x \leq 1 \). Find the value of \( m \) such that \( f(x) \geq -\frac{8}{9} \) for all \( x \leq m \).
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Context:
There is an unlimited number of chips in six colors. What is the minimum number of chips that need to be arranged in a row so that for any two different colors in the row, there are two adjacent chips of those colors?
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Context:
A safe has three locks, each with 8 possible settings (1 through 8). The safe opens if any two of the three locks are set correctly. What is the minimum number of combinations needed to guarantee opening the safe?
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Context:
已知函数$f(x)=\sin (\omega x+\frac{\pi }{6})+\frac{1}{2}(\omega >0)$，点$P$，$Q$，$R$是直线$y=m(m>0)$与函数$f(x)$的图象自左至右的某三个相邻交点，且$2|PQ|=|QR|=\frac{3\pi }{2}$，则$\omega +m=$____．
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Context:
Given acute angles $α$ and $β$ that satisfy: $\cos α= \frac {1}{3}$ and $\cos (α+β)=- \frac {1}{3}$, find $\cos (α-β)$.
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Context:
Let $A B C$ be an acute triangle with $A$-excircle $\Gamma$. Let the line through $A$ perpendicular to $B C$ intersect $B C$ at $D$ and intersect $\Gamma$ at $E$ and $F$. Suppose that $A D=D E=E F$. If the maximum value of $\sin B$ can be expressed as $\frac{\sqrt{a}+\sqrt{b}}{c}$ for positive integers $a, b$, and $c$, compute the minimum possible value of $a+b+c$.
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Context:
Suppose  $A=\{1,2,\dots,2002\}$  and  $M=\{1001,2003,3005\}$ .  $B$  is an non-empty subset of  $A$ .  $B$  is called a  $M$ -free set if the sum of any two numbers in  $B$  does not belong to  $M$ . If  $A=A_1\cup A_2$ ,  $A_1\cap A_2=\emptyset$  and  $A_1,A_2$  are  $M$ -free sets, we call the ordered pair  $(A_1,A_2)$  a  $M$ -partition of  $A$ . Find the number of  $M$ -partitions of  $A$ .
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Context:
Find the number of pairs of integers $x, y$ with different parities such that $\frac{1}{x}+\frac{1}{y} = \frac{1}{2520}$.
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Context:
Determine the cardinality of the set of all functions that can be constructed as iterations of pointwise convergence starting from continuous functions.
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Context:
Determine the conditions under which the linear operator $T$ on $\mathbb{R}^4$, represented by the matrix \( Q = \begin{bmatrix} 0 & 0 & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & b & 0 & 0\\0 & 0 & c & 0 \end{bmatrix} \) relative to the standard basis, is diagonalizable.
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Context:
Let \( x \) be a real number. How many of the first 1000 positive integers can be expressed in the form \( [2x] + [4x] + [6x] + [8x] \)?
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Context:
Find the least natural number $n$, such that the following inequality holds:
\[
\sqrt{\frac{n-2011}{2012}} - \sqrt{\frac{n-2012}{2011}} < \sqrt[3]{\frac{n-2013}{2011}} - \sqrt[3]{\frac{n-2011}{2013}}
\]
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Context:
Determine the cardinality of the set $M$, where $M$ consists of all subsets of the natural numbers $\mathbb{N}$ such that both the subset and its complement are infinite.
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Context:
设$S = \{ 1 , 2 , 3 , 4 \} , n$项数列$a _ { 1 } , a _ { 2 } , \cdots , a _ { n }$有以下性质：对于$S$的任何一个非空子集$B$，在该数列中有相邻的$|B|$项恰好组成集合$B$.求$n$的最小值.
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Context:
Consider a sequence of independent and identically distributed random variables drawn from either a uniform or normal distribution. Calculate the probability that any point in the sequence, excluding the first and last points, is a local extremum (either a local maximum or minimum).
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Context:
Consider a sequence  $\{a_n\}$  of integers, satisfying  $a_1=1, a_2=2$  and  $a_{n+1}$  is the largest prime divisor of  $a_1+a_2+\ldots+a_n$ . Find  $a_{100}$ .
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Context:
He then went to see the oranges being harvested. He found out that they harvest 82 sacks of ripe oranges and some sacks of unripe oranges per day. If they have 2050 sacks of ripe oranges after 25 days of harvest, how many sacks of unripe oranges do they harvest per day?
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Context:
fantasticbobob is proctoring a room for the SiSiEyMB with $841$ seats arranged in $29$ rows and $29$ columns. The contestants sit down, take part 1 of the contest, go outside for a break, and come back to take part 2 of the contest. fantasticbobob sits among the contestants during part 1, also goes outside during the break, but when he returns, he finds that his seat has been taken. Furthermore, each of the $840$ contestants now sit in a chair horizontally or vertically adjacent to their original chair. How many seats could fantasticbobob have started in?
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Context:
In a certain school cafeteria, there are only four main dishes available for breakfast: steamed buns, baozi (stuffed buns), noodles, and fried rice with eggs. There are 5 students going to have breakfast, each choosing only one of these dishes, and each type of dish is chosen by at least one student. It is known that there are not enough baozi for more than one person to eat, and Student A, due to stomach issues, will not choose fried rice with eggs. The number of different meal choice combinations for these 5 students is ______. (Answer with a number)
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Context:
Given non-negative real numbers \( a, b, c, x, y, z \) that satisfy \( a+b+c = x+y+z = 1 \), find the maximum value of \( \left(a-x^{2}\right)\left(b-y^{2}\right)\left(c-z^{2}\right) \).
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Context:
Evaluate the integral:

$$\int_0^\infty \frac{\tan^{-1}\left(\frac{x^2-x^{1/2}}{1+x^{5/2}}\right)}{1+x^2}dx$$
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Context:
Determine the global maximum value of the function \( f(x) = \frac{J_2(x)}{x^2} \), where \( J_2(x) \) is a Bessel function of the first kind of order 2.
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Context:
Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$. Pick points $E, F, G, H$ such that the angles $\angle AEB, \angle BFC, \angle CGD, \angle AHD$ are all right. What is the largest possible area of quadrilateral $EFGH$?
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Context:
Given three 3D vectors $W$, $T_1$, and $T_2$ that lie on the same plane, and the equation $W = T_1 \cos^2(\theta) + T_2 \sin^2(\theta)$, find the angle between $W$ and $T_1$. Express your answer in terms of $\theta$.
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Context:
Find a continuous and absolutely integrable function \( f(x) \) that satisfies the integral equation \( \int_{-\infty}^{\infty} e^{-|t-x|}f(x) \, dx = e^{-|t|} + |t|e^{-|t|} \) using the Convolution Theorem.
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Context:
What is the smallest number which when diminished by 24, is divisible by 5, 10, 15 and a certain number and the result is 84?
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Context:
A snowball with a temperature of $0^{\circ} \mathrm{C}$ is launched at a speed $v$ towards a wall. Upon impact, $k=0.02\%$ of the entire snowball melts. Determine what percentage of the snowball will melt if it is launched towards the wall at a speed of $\frac{v}{2}$? The specific heat of fusion of snow is $\lambda = 330$ kJ/kg. Assume that all the energy released upon impact is used for melting.
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Context:
The sum of several (not necessarily different) real numbers from $[0,1]$ doesn't exceed $S$.
Find the maximum value of $S$ such that it is always possible to partition these numbers into two groups with sums $A\le 8$ and $B\le 4$.

[i](I. Gorodnin)[/i]
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Context:
If the graph of the function $f(x) = \sin \omega x$ is shifted to the right by $\frac{\pi}{6}$ units to obtain the graph of $f(x) = \sin(\omega x - \frac{4}{3}\pi)$, then the minimum value of $|\omega|$ is \_\_\_\_\_\_.
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Context:
Find the value of $\lfloor 1 \rfloor + \lfloor 1.7 \rfloor + \lfloor 2.4 \rfloor + \lfloor 3.1 \rfloor + \cdots + \lfloor 99 \rfloor$.
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Context:
A dealer purchases 15 articles of type A for Rs. 25 each and 20 articles of type B for Rs. 35 each. He sells 12 articles of type A for Rs. 33 each and 18 articles of type B for Rs. 45 each. Find the overall profit or loss percentage he made from these transactions?
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Context:
Given points D and E on sides AB and AC of ∠CAB respectively, the angle bisectors of ∠BDE and ∠CED intersect at F. If ∠BAC = 38°, find the measure of ∠DFE. Express your answer in degrees as a whole number.
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Context:
A certain middle school assigns numbers to each student, where the last digit indicates the gender of the student: 1 for male and 2 for female. If 028432 represents "a female student who is number 43 in class 8 and enrolled in the year 2002," then the number for a male student who is number 23 in class 6 and enrolled in the year 2008 is.
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Context:
Determine the values of $m$ for which the integral \( \int_0^{\pi/2} \frac{1-\cos(x)}{x^m} \, dx \) converges.
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Context:
If \(\cos^5 \theta - \sin^5 \theta < 7 (\sin^3 \theta - \cos^3 \theta)\), \(\theta \in [0, 2\pi)\), then what is the range of values for \(\theta\)?
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Context:
Evaluate the surface integral $\iint_S f(x,y,z)dS$ where $f(x,y,z) = x^2+y^2 + z^2 $ and $S$ is the part of the cone $z = \sqrt{x^2+y^2} $ above the circle $x^2+y^2 \leq 2x$.
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Context:
Consider a finite group $S$ with identity element $1$. Let $G$ and $H$ be subgroups of $S$ with orders $m$ and $n$, respectively. If the least common multiple of $m$ and $n$ is $k$, determine the maximum possible order of $G \cap H$. \( \text{Express your answer as a number.} \)
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Context:
What is the range of values for \( m \) such that the equation \( m\left(x^{2}+y^{2}+2 y+1\right) = (x - 2y + 3)^{2} \) represents an ellipse?
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Context:
A train is 800 meters long and is running at a speed of 115 km/hour. In what time will it pass a bridge of 375 meters length?
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Context:
Four princesses each guessed a two-digit number, and Ivan guessed a four-digit number. After they wrote their numbers in a row in some order, they got the sequence 132040530321. Find Ivan's number.
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Context:
In a setup with 5 blue nails in one column and 3 red nails in another column, you can attach a string between any red nail and any blue nail. Determine the number of ways to attach strings such that every nail has at least one string attached to it.
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Context:
Find the highest lower bound for the probability of getting exactly one head when tossing $d$ pairwise independent coins, each with a probability of $1/d$ of landing heads.
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Context:
Calculate the center of gravity for a solid cone with a circular base of radius 1 and height 3, where the top of the cone is located at the origin and the base is in the xy-plane.
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Context:
Calculate the expected number of coin tosses needed to obtain the sequence H-T-H-T-T-H with a fair coin, where each toss is independent.
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Context:
Find the polar coordinates (ρ, θ) of the center of the circle given by the polar equation ρ = -2cosθ. Express your answer in the format (ρ, θ), where θ is in radians.
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Context:
Determine the maximum value of \( \sum_{i=1}^{k} a_{i} \) for integers \( a_{1}, a_{2}, \ldots, a_{k} \) greater than 1, such that \( (a_{1}!) (a_{2}!) \cdots (a_{k}!) \mid 2017! \).
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Context:
In the "Three Kingdoms Kill" game, there is 1 'Monarch', 3 'Loyalists', 5 'Rebels', and 1 'Traitor', with a total of 10 people playing the game. The Monarch and the Loyalists are on the same team, the Rebels are on another team, and the Traitor is on his own team. If two brothers participate in the same round of the "Three Kingdoms Kill" game, what is the probability that the two brothers are on the same team? (Express the result as a simplified fraction)
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Context:
Compute the integral \(\displaystyle \int_{\gamma} \frac{z-1}{z(z-i)(z-3i)}dz\) where \(\gamma\) is the curve defined by \(|z-i| = \frac{1}{2}\). Decompose the integrand into partial fractions to aid in the computation.
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Context:
A dart is thrown at a square dartboard of side length 2 so that it hits completely randomly. What is the probability that it hits closer to the center than any corner, but within a distance of 1 of a corner?
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Context:
Consider numbers of the form  $1a1$ , where  $a$  is a digit. How many pairs of such numbers are there such that their sum is also a palindrome?

*Note: A palindrome is a number which reads the same from left to right and from right to left. Examples:  $353$ ,  $91719$ .*
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Context:
Determine the cardinality of the set of all convergent sequences in \(\mathbb{Q}\) that approach 0.
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Context:
Determine the relationship among the three numbers $2011^{0.99}$, $0.99^{2011}$, and $\log_{0.99}2011$. Express your answer as an ordering of the three functions in the format $a < b < c$.
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Context:
In triangle $ABC$, points $D$ and $E$ lie on the interior of segments $AB$ and $AC$, respectively, such that $AD = 1$, $DB = 2$, $BC = 4$, $CE = 2$, and $EA = 3$. Let $DE$ intersect $BC$ at $F$. Determine the length of $CF$.
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Context:
In a game played by two players, A and B, they take turns filling six boxes with digits 1-9, forming a six-digit number. If this six-digit number is divisible by an integer N, B wins; otherwise, A wins. Find the value of N less than 15 that guarantees a winning strategy for B. Express your answer as a single integer.
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Context:
Determine the asymptotic behavior, in terms of the length L, for the number of topologically different knots that can be formed using a perfectly flexible, non-self-intersecting rope of length L and radius 1, with the ends glued together after knotting.
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Context:
In a $9 \times 9$ grid, 29 squares are colored black. If $m$ represents the number of rows that contain at least 5 black squares, and $n$ represents the number of columns that contain at least 5 black squares, determine the maximum value of $m + n$.
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Context:
In the polynomial quotient ring \(B = \mathbb{Z}_2[x]/(x^3+x^2+x+1)\), find a polynomial \([g(x)]\) such that \([x^2+1] \cdot [g(x)] = [0]\).
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Context:
27.7. (New York, 76). The sequence ( $i_{1} ; i_{2} ; \ldots ; i_{n}$ ) is formed from the first $n>3$ natural numbers, arranged in a random order. What is the probability that for all $k=1,2, \ldots, n$ the inequality $i_{k} \geqslant k-3$ holds?
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Context:
Evaluate the limit: $$\lim_{x\to0}\frac{\int_{0}^{x^2} t^{t-1} dt}{x^4}$$
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Context:
Given the polynomial $f(x) = x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64$, calculate the value of $v_4$ when $x = 2$ using Horner's method.
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Context:
Determine the values of $a \in \mathbb{R}$ for which the function $f(x) = x^{100} - ax^{97} + 5$ has multiple local minimum values.
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Context:
On a two-dimensional coordinate plane, the line a = x^2 - x^3 touches the x-axis in 2 places. What is the first term of the equation representing the line?
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Context:
How many permutations \( \pi \) of \( \{ 1,2,\ldots ,7\} \) are there such that \( \pi \left( k\right)  \leq  {2k} \) for \( k = 1,\ldots ,7 \) ? A permutation \( \pi \) of a set \( S \) is a
 function from \( S \) to itself such that if \( a \neq  b \) ,then \( \pi \left( a\right)  \neq  \pi \left( b\right) \) . For example, \( \pi \left( x\right)  = x \) and \( \pi \left( x\right)  = 8 - x \) are permutations
 of \( \{ 1,2,\ldots ,7\} \) but \( \pi \left( x\right)  = 1 \) is not.
 2019 CCA Math Bonanza Individual Round #7
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Context:
Given $\sin \alpha + \cos \alpha = - \frac{\sqrt{5}}{2}$, and $\frac{5\pi}{4} < \alpha < \frac{3\pi}{2}$, calculate the value of $\cos \alpha - \sin \alpha$.
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Context:
Sharik and Matroskin ski on a circular track, half of which is an uphill slope and the other half is a downhill slope. Their speeds are identical on the uphill slope and are four times less than their speeds on the downhill slope. The minimum distance Sharik falls behind Matroskin is 4 km, and the maximum distance is 13 km. Find the length of the track.
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Context:
Given a triangular prism \( S-ABC \) with a base that is an equilateral triangle. The projection of point \( A \) on the side \( S-BC \), denoted as \( H \), is the orthocenter of \( \triangle SBC \). The dihedral angle between planes \( H-AB \) and \( BC \) is \( 30^\circ \). Additionally, \( SA = 2\sqrt{3} \). What is the volume of the triangular prism \( S-ABC \)?
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Context:
Suppose you and I each have a box of 600 marbles. In my box, 4 of the marbles are black, while 3 of your marbles are black. We each draw 300 marbles with replacement from our own boxes. Using Poisson approximation, calculate the probability that you and I draw the same number of black marbles.
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Context:
3. The mathematical committee consists of $2 n$ members, $n \geqslant 3$. It is known that each member of the committee is in a dispute with exactly one other member of the committee (this relationship is symmetric). In how many ways can the committee be divided into three subcommittees: one for creating problems, one for evaluating problems, and one for organizing the competition, such that each subcommittee has at least two members and no two members of the committee who are in a dispute are in the same subcommittee?
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Context:
Let \( f(x) = \frac{1}{(1+x)(1+x^2)(1+x^4)} \). Find the coefficient of the term \( x^{50} \) in \( (f(x))^3 \).
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Context:
For any positive integer  $x$ , let  $f(x)=x^x$ . Suppose that  $n$  is a positive integer such that there exists a positive integer  $m$  with  $m  \neq 1$  such that  $f(f(f(m)))=m^{m^{n+2020}}$ . Compute the smallest possible value of  $n$ .

*Proposed by Luke Robitaille*
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Context:
A driver departed from point A to point B, which are 100 km apart. When the navigator indicated that 30 minutes of travel remained, the driver reduced speed by 10 km/h for the first time. When the navigator indicated that 20 minutes of travel remained, the driver reduced speed by another 10 km/h. (The navigator determines the remaining time and distance based on the current speed.) Determine the initial speed of the car, given that the car traveled 5 minutes longer at the reduced speed than it would have at the twice-reduced speed.
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Context:
Let $E$ be a point inside the rhombus $ABCD$ such that $|AE| = |EB|$, $m(\widehat{EAB}) = 12^\circ$, and $m(\widehat{DAE}) = 72^\circ$. What is the value of $m(\widehat{CDE})$ in degrees?
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Context:
Let $p=2^{16}+1$ be a prime. A sequence of $2^{16}$ positive integers $\{a_n\}$ is monotonically bounded if $1 \leq a_i \leq i$ for all $1 \leq i \leq 2^{16}$. We say that a term $a_k$ in the sequence with $2 \leq k \leq 2^{16}-1$ is a mountain if $a_k$ is greater than both $a_{k-1}$ and $a_{k+1}$. Evan writes out all possible monotonically bounded sequences. Let $N$ be the total number of mountain terms over all such sequences he writes. Find the remainder when $N$ is divided by $p$.
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Context:
An object travels 90 feet in 3 seconds. What is the object's approximate speed in miles per hour if 1 mile is equal to some number of feet?
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Context:
Evaluate the surface integral \( \iint_S \nabla \times \vec{v} \cdot \vec{N} \, dS \) where \( \vec{v} = (xz, -y, x^2y) \) and \( S \) consists of the five faces of the cube \([0,2] \times [0,2] \times [0,2]\) that are not on the xy-plane. The vector \( \vec{N} \) points outward from the surface. Use Stokes' theorem to find the value of the integral.
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