Context:
Consider a new infinite geometric series: $$\frac{7}{4} + \frac{28}{9} + \frac{112}{27} + \dots$$
Determine the common ratio of this series.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A natural number \( n \) has its divisors written in ascending order. After erasing some of the first and last numbers, 151 divisors remain. What is the maximum number of these 151 divisors that could be fifth powers of natural numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A train of length 250 m moving at a speed of 72 kmph takes 54.995600351971845 seconds to cross another train of some length, moving at 36 kmph in the same direction. What is the length of the second train?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function f(x) = ax + sin(x), find the range of the real number a that allows f(x) to have extreme points. Express your answer as an inequality.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A certain unit has set up two buses at a fixed location to pick up employees for work. The departure times and probabilities of the two buses are as follows: The first bus departs at $8:00$, $8:20$, and $8:40$ with probabilities of $\frac{1}{4}$, $\frac{1}{2}$, and $\frac{1}{4}$ respectively. The second bus departs at $9:00$, $9:20$, and $9:40$ with probabilities of $\frac{1}{4}$, $\frac{1}{2}$, and $\frac{1}{4}$ respectively. The departure times of the two buses are independent of each other. If an employee arrives at the boarding point at $8:10$, what is the probability that the employee will wait for more than $50$ minutes? Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The number \( x \) is chosen randomly from the interval \( (0,1] \). Define \( y = \left\lceil \log_{4} x \right\rceil \). Find the sum of the lengths of all subintervals of \( (0,1] \) for which \( y \) is odd. For any real number \( a, \left\lceil a \right\rceil \) is defined as the smallest integer not less than \( a \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the following expression as n approaches infinity:

$$\lim_{n\rightarrow \infty} \sqrt 2 \frac {\Gamma\left(\frac {n+1} 2\right)} {\Gamma\left(\frac {n} 2\right)}-\sqrt n$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many six-letter words formed from the letters of "AMC" do not contain the substring "AMC"? For example, "AMAMMC" has this property, but "AAMCCC" does not.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f$ be a continuous real function such that $f(11) = 10$ and for all $x$, $f(x)f(f(x)) = 1$. What is the value of $f(9)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The remainder of $1991^{2000}$ modulo $10^6$ is \underline{\hspace{2cm}}.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If lines $l_{1}$: $ax+2y+6=0$ and $l_{2}$: $x+(a-1)y+3=0$ are parallel, find the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given 1985 sets, each consisting of 45 elements, and the union of any two sets contains exactly 89 elements. How many elements are in the union of all these 1985 sets?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\diamondsuit$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In preparation for the 2010 Shanghai World Expo, the organizing committee needs to select four volunteers from among Zhang, Liu, Li, Song, and Wang to perform four different tasks: translation, tour guiding, etiquette, and driving. If Zhang and Zhao can only work in the first two roles, while the other three can work in any of the four roles, how many different assignment plans are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Bonnie constructs a rectangular prism frame using 12 pieces of wire, each 8 inches long. Meanwhile, Roark uses 2-inch-long pieces of wire to construct a series of unit rectangular prism frames that are not connected to each other. The total volume of Roark's prisms is equal to the volume of Bonnie's prism. Find the ratio of the total length of Bonnie's wire to Roark's wire. Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the value of the expression:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2)))))))) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is half of the absolute value of the difference of the squares of 21 and 15 added to the absolute value of the difference of their cubes?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the 2020th term of the following sequence:
$$
1, 1, 3, 1, 3, 5, 1, 3, 5, 7, 1, 3, 5, 7, 9, 1, 3, 5, 7, 9, 11, \ldots
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the sum of all positive integers $n$ such that $50 \leq n \leq 100$ and $2 n+3$ does not divide $2^{n!}-1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The area of a cross-section of a regular tetrahedron, which is in the shape of a square, is \(m^2\). Find the surface area of the tetrahedron.
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Context:
Given a pedestrian signal light that alternates between red and green, with the red light lasting for 50 seconds, calculate the probability that a student needs to wait at least 20 seconds for the green light to appear.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Simplify the expression \(\sin 10\alpha \sin 8\alpha + \sin 8\alpha \sin 6\alpha - \sin 4\alpha \sin 2\alpha\).
Please reason step by step, and put your final answer within \boxed{}.

\boxed{2 \cos 2\alpha \sin 6\alpha \sin 10\alpha}


Context:
The value of $\log_{10}{28471}$ is between the consecutive integers $a$ and $b$.  Find $a+b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The time on a cell phone is $3:52$. How many minutes will pass before the phone next shows a time using each of the digits 2, 3, and 5 exactly once?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(A B C D E\) be a square pyramid of height \(\frac{1}{2}\) with a square base \(A B C D\) of side length \(A B = 12\) (so \(E\) is the vertex of the pyramid, and the foot of the altitude from \(E\) to \(A B C D\) is the center of square \(A B C D\)). The faces \(A D E\) and \(C D E\) meet at an acute angle of measure \(\alpha\) (so that \(0^{\circ}<\alpha<90^{\circ}\)). Find \(\tan \alpha\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( A B C D E F \) be a regular hexagon. A frog starts at vertex \( A \). Each time, it can jump to one of the two adjacent vertices. If the frog reaches vertex \( D \) within 5 jumps, it stops jumping; if it cannot reach vertex \( D \) within 5 jumps, it stops after making 5 jumps. How many different sequences of jumps can the frog make from the start until it stops?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $y=a\sin x+b\cos x+c$ has a lowest point $(\frac{11π}{6},1)$ on its graph. If the ordinate of each point on the graph remains unchanged, the abscissa is shortened to $\frac{3}{π}$ of the original, and then shifted to the left by 1 unit length to obtain the graph of $y=f\left(x\right)$, and all roots of $f\left(x\right)=3$ differ by 3 units in increasing order, then $a+b+c=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle intersects an equilateral triangle $\triangle XYZ$ at points $A$, $B$, $C$, $D$, $E$, and $F$ such that the points $X$, $A$, $B$, $Y$, $C$, $D$, $Z$, $E$, and $F$ lie on the equilateral triangle in that order. Given:

- $AC^2 + CE^2 + EA^2 = 1900$
- $BD^2 + DF^2 + FB^2 = 2092$

Compute the positive difference between the areas of triangles $\triangle ACE$ and $\triangle BDF$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Functions $f$ and $g$ are quadratic, $g(x) = - f(100 - x)$, and the graph of $g$ contains the vertex of the graph of $f$. The four $x$-intercepts on the two graphs have $x$-coordinates $x_1$, $x_2$, $x_3$, and $x_4$, in increasing order, and $x_3 - x_2 = 150$. Then $x_4 - x_1 = m + n\sqrt p$, where $m$, $n$, and $p$ are positive integers, and $p$ is not divisible by the square of any prime. Find the value of $m + n + p$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The adult human body has 206 bones. Each foot has 26 bones. Approximately what fraction of the number of bones in the human body is found in one foot?
Please reason step by step, and put your final answer within \boxed{}.


Context:
求最小的正整数 n，使得\sqrt{\df{n-2011}{2012}}-\sqrt{\df{n-2012}{2011}}<\sqrt[3]{\df{n-2013}{2011}}-\sqrt[3]{\df{n-2011}{2013}}.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There is a chain of 5 people - A, B, C, D, and E. A buys a bicycle and includes a 10% tax in the cost price, then, A sells the bicycle to B at a profit of 60%. B sells the bicycle to C at a profit of 25%. C sells the bicycle to D at a loss of 15%. Finally, D sells the bicycle to E at a profit of 30%. If the final price for E is Rs. 500, find the original cost price (excluding tax) of the bicycle for A.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest multiple of $2019$ that is of the form $abcabc\dots abc$, where $a$, $b$, and $c$ are digits.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A wire is cut into two pieces, one of length \(a\) and the other of length \(b\). The piece of length \(a\) is bent to form an equilateral triangle, and the piece of length \(b\) is bent to form a regular hexagon. The triangle and the hexagon have equal area. The original answer is in the form \(\frac{\sqrt{p}}{n}\) where the fraction is in its simplest form. Please give the value of \(p + n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
During a particular month in Orlando, the temperatures recorded were:

Week 1: 55, 62, 58, 65, 54, 60, 56
Week 2: 70, 74, 71, 77, 64, 68, 72
Week 3: 82, 85, 89, 73, 65, 63, 67
Week 4: 75, 72, 60, 57, 50, 55, 58

Additionally, the temperatures were 69, 67, and 70 in the first three days of the following week.

Calculate the average temperature of Orlando for the entire month, including the additional three days.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rectangular piece of paper measures 17 cm by 8 cm. It is folded so that a right angle is formed between the two segments of the original bottom edge. Calculate the area of the new figure formed after folding the paper. Express your answer in square centimeters (cm²) as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
数列 \{a_n\} 为：1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4, \cdots. 即先取 a_1=1，接着复制该项粘贴在后面作为 a_2，并添加后继数 2 作为 a_3，再复制所有项 1, 1, 2 并粘贴在后面作为 a_4, a_5, a_6，并添加后继数 3 作为 a_7, \cdots. 依次继续下去，则 a_{2021}=__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知函数$f(x)=-\sin 2\omega x(\omega >0)$的图象关于点$M(\frac{5\pi }{4},0)$对称，且在区间$[0,\frac{\pi }{2}]$上是单调函数，则$\omega $的值为__________．
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a triangle $ABC$ where the height on side $AB$ is equal to the length of $AB$, find the maximum value of $$\frac{AC}{BC} + \frac{BC}{AC} + \frac{AB^2}{BC \cdot AC}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rectangle with a perimeter of 96 cm is divided by eight vertical cuts and eleven horizontal cuts into 108 identical smaller rectangles. What is the perimeter of each smaller rectangle if the sum of the lengths of all the cuts is 438 cm?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum number of solutions that the equation \(\max \left\{a_{1} x+b_{1}, \ldots, a_{10} x+b_{10}\right\} = 0\) can have, if \(a_{1}, \ldots, a_{10}, b_{1}, \ldots, b_{10}\) are real numbers, with all \(a_{i} \neq 0\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three rods of lengths $1396$, $1439$, and $2018$ millimeters have been hinged from one tip on the ground. What is the smallest value for the radius of the circle passing through the other three tips of the rods in millimeters?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The Fibonacci sequence is defined recursively by $F_{n+2}=F_{n+1}+F_{n}$ for $n \in \mathbb{Z}$ and $F_{1}=F_{2}=1$. Determine the value of:

$$
\left(1-\frac{F_{2}^{2}}{F_{3}^{2}}\right)\left(1-\frac{F_{3}^{2}}{F_{4}^{2}}\right) \cdot \ldots \cdot\left(1-\frac{F_{2019}^{2}}{F_{2020}^{2}}\right)
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sum of the interior angles of a polygon is twice the sum of its exterior angles. Starting from a vertex of this polygon, connect it to the other vertices. How many triangles can this polygon be divided into at most?
Please reason step by step, and put your final answer within \boxed{}.


Context:
During the Tree Planting Festival, students from a class are planting trees. There are three different types of saplings, labeled as A, B, and C, available in the nursery, with an ample supply of each type. They need to plant 5 saplings in a row in 5 pits, with the condition that the same type of sapling cannot be adjacent, and only type A saplings can be planted in the first and the fifth pits. Calculate the total number of planting methods available.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $S_n$ is the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$, if the ratio of $a_2$ to $a_4$ is $7:6$, then the ratio of $S_7$ to $S_3$ equals \_\_\_\_\_\_\_\_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many convex 33-sided polygons can be formed by selecting their vertices from the vertices of a given 100-sided polygon, such that the two polygons do not share any common sides?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S(n)$ be the sum of the squares of the positive integers less than and coprime to $n$. For example, $S(5) = 1^2 + 2^2 + 3^2 + 4^2$, but $S(4) = 1^2 + 3^2$. Let $p = 2^7 - 1 = 127$ and $q = 2^5 - 1 = 31$ be primes. The quantity $S(pq)$ can be written in the form \( \frac{p^2q^2}{6}\left(a - \frac{b}{c} \right) \) where $a$, $b$, and $c$ are positive integers, with $b$ and $c$ coprime and $b < c$. Find $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the first 100 even numbers divisible by 6 between 100 and 300 inclusive.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of integers $a$ with $1 \le a \le 2012$ for which there exist nonnegative integers $x, y, z$ satisfying the equation:
\[ x^2(x^2 + 2z) - y^2(y^2 + 2z) = a. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that bus types $A$, $B$, and $C$ all depart at 6:00 AM from a certain station, with departure intervals of 10 minutes, 12 minutes, and 15 minutes, respectively, if Xiaoming arrives at the station at some time between 8:00 AM and 12:00 PM to catch one of these buses, what is Xiaoming's average waiting time in minutes?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Cai and Lai are eating cookies. Their cookies are in the shape of $2$ regular hexagons glued to each other, and the cookies have an area of $18$ units. They each make a cut along the $2$ long diagonals of a cookie; this now makes four pieces for them to eat and enjoy. What is the minimum area among the four pieces?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X_i$ be i.i.d. exponential random variables with mean $1$. Compute the variance of the ratio: $$ Y = \dfrac{X_i}{\sum\limits_{j=1,j\neq i}^{n}X_j}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
What integer values can the expression

$$
(\sqrt{5}-\sqrt{2})(\sqrt{a}+\sqrt{b})
$$

take if \(a\) and \(b\) are any non-negative integers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Temerant is a spherical planet with a radius of $1000$ kilometers. The government wants to build twelve towers of the same height on the equator of Temerant, so that every point on the equator can be seen from at least one tower. The minimum possible height of the towers can be written, in kilometers, as $a\sqrt{b} - c\sqrt{d} - e$ for positive integers $a, b, c, d$, and $e$ (with $b$ and $d$ not divisible by the square of any prime). Compute $a + b + c + d + e$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The consignment shop received for sale cameras, clocks, pens, and receivers totaling 240 rubles. The sum of the prices of the receiver and one clock is 4 rubles more than the sum of the prices of the camera and the pen, and the sum of the prices of one clock and the pen is 24 rubles less than the sum of the prices of the camera and the receiver. The price of the pen is an integer not exceeding 6 rubles. The number of accepted cameras is equal to the price of one camera in rubles divided by 10; the number of accepted clocks is equal to the number of receivers, as well as the number of cameras. The number of pens is three times the number of cameras. How many items of the specified types were accepted by the store in total?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Kelvin the Frog is playing the game of Survival. He starts with two fair coins. Every minute, he flips all his coins one by one, and throws a coin away if it shows tails. The game ends when he has no coins left, and Kelvin's score is the *square* of the number of minutes elapsed. What is the expected value of Kelvin's score? For example, if Kelvin flips two tails in the first minute, the game ends and his score is 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three different cars (A, B, and C) have different gasoline consumptions while traveling various distances.

Car A uses 12 gallons of gasoline to travel 180 miles.
Car B uses 15 gallons of gasoline to travel 225 miles.
Car C uses 18 gallons of gasoline to travel 270 miles.

In order for all three cars to travel their respective distances using 10 gallons of gasoline each, by how many miles per gallon must each car's gas mileage be increased on average?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Emma is a straight-A student who loves spending time in the library. She has 5 favorite books that she reads repeatedly. One day, she observes her friend Alex, a social butterfly, confidently talking to a group of 12 students about their favorite books. Inspired by Alex’s confidence, Emma decides to step out of her comfort zone and join the conversation. To prepare, she decides to read 3 new books and share her thoughts on them. Each new book takes her 2 days to read, and she spends 1 day summarizing her thoughts on each book. If Emma starts reading on a Monday, on which day of the week will she be ready to join the conversation with Alex?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A 5x5 square is divided into 25 cells, all initially white. Neighbouring cells share a common edge. On each move, the colors of two neighbouring cells are changed to the opposite color (white cells become black and black ones become white). What is the minimum number of moves required to obtain a chess-like colouring?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( b\left[\frac{1}{1 \times 3}+\frac{1}{3 \times 5}+\cdots+\frac{1}{1999 \times 2001}\right]=2 \times\left[\frac{1^{2}}{1 \times 3}+\frac{2^{2}}{3 \times 5}+\cdots+\frac{1000^{2}}{1999 \times 2001}\right] \), find the value of \( b \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
We say a $2023$-tuple of nonnegative integers $\left(a_{1}, a_{2}, \ldots a_{2023}\right)$ is sweet if the following conditions hold:
- $a_{1}+a_{2}+\ldots+a_{2023}=2023$,
- $\frac{a_{1}}{2^{1}}+\frac{a_{2}}{2^{2}}+\ldots+\frac{a_{2023}}{2^{2023}} \leqslant 1$.

Determine the greatest positive integer $L$ such that
$$
a_{1}+2 a_{2}+\ldots+2023 a_{2023} \geqslant L
$$
holds for every sweet 2023 -tuple $\left(a_{1}, a_{2}, \ldots, a_{2023}\right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of real solutions to the equation \( \frac{x}{100} = \sin x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given integers $m, n \geqslant 2$. Color each cell of an $m$-row by $n$-column grid $S$ with either red or blue such that the following condition is satisfied: For any two cells in the same row, if both are colored red, then one column must have all cells colored red, and the other column must have at least one cell colored blue. Find the number of different coloring methods. Please provide the value when $m = 2$ and $n = 3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral

$$\dfrac{1}{\pi} \int e^{itn}\cos^2\left(\dfrac{1}{2}t \right)dt$$

over the interval $(-\pi, \pi)$ for any positive integer $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the
sequence $1$, $2$, $\dots$ , $n$ satisfying
$$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the definite integral:

$$
\int_{2 \operatorname{arctg} 2}^{2 \operatorname{arctg} 3} \frac{d x}{\cos x(1-\cos x)}
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( A = \{1, 2, 3, \cdots, 4n+2\} \) and \( M = \{2n+1, 4n+3, 6n+5\} \). A subset \( B \) of \( A \) is called an \( M \)-free set if the sum of any two numbers in \( B \) does not belong to \( M \). If \( A = A_1 \cup A_2 \), \( A_1 \cap A_2 = \varnothing \), and both \( A_1 \) and \( A_2 \) are \( M \)-free sets, then the ordered pair \( (A_1, A_2) \) is called an \( M \)-partition of \( A \). Determine the number of all \( M \)-partitions of \( A \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
有长为 $2^{n}(n=0,1, \cdots, 1009)$ 的线段各三条, 则由这 3030 条线段能构成不全等的三角形的个数为 $\qquad$ (用数字作答).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle passes through the vertices \( A \) and \( B \) of triangle \( ABC \) and intersects its sides \( AC \) and \( BC \) at points \( Q \) and \( N \) respectively. Given \( AQ : QC = 5 : 2 \) and \( CN : NB = 5 : 2 \), find \( AB \) if \( QN = 5\sqrt{2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of duplication formulas that exist for the Mordell curve family $Y^2-X^3=c$. Given that Bachet and Réalis have provided two such formulas, how many additional duplication formulas are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On Lineland, there are 2018 bus stations numbered from 1 to 2018 from left to right. A self-driving bus that can carry at most $N$ passengers starts from station 1 and drives all the way to station 2018, making a stop at each bus station. Each passenger that gets on the bus at station $i$ will get off at station $j$ for some $j>i$ (the value of $j$ may vary over different passengers).

Define any group of four distinct stations $i_1, i_2, j_1, j_2$ with $i_u < j_v$ for all $u,v \in \{1,2\}$ as a good group. Suppose that in any good group $i_1, i_2, j_1, j_2$, there is a passenger who boards at station $i_1$ and de-boards at station $j_1$, or there is a passenger who boards at station $i_2$ and de-boards at station $j_2$, or both scenarios occur. Compute the minimum possible value of $N$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Construct a curve of constant width \( h \), composed of five, seven, or any odd number of arcs of circles with radius \( h \). What is the length of each of these curves?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Five numbers, $a_1$, $a_2$, $a_3$, $a_4$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 50\}$. Four other numbers, $b_1$, $b_2$, $b_3$, $b_4$, are then drawn randomly and without replacement from the remaining set of 46 numbers. Let $p$ be the probability that, after a suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3 \times a_4$ can be enclosed in a box of dimensions $b_1 \times b_2 \times b_3 \times b_4$, with the sides of the brick parallel to the sides of the box. Compute $p$ in lowest terms and determine the sum of the numerator and denominator.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A deck consists of 25 cards: 10 black cards, and 5 sets of 3 identical cards each in green, red, yellow, blue, and pink. How many different hands of 10 cards can be drawn from this deck?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the vectors $\overrightarrow{a} \cdot (\overrightarrow{a}+2\overrightarrow{b})=0$ and the magnitudes $|\overrightarrow{a}|=|\overrightarrow{b}|=2$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There is a store that sells each of the 26 letters. All letters of the same type cost the same amount (i.e. any 'a' costs the same as any other 'a'), but different letters may or may not cost different amounts. For example, the cost of spelling "trade" is the same as the cost of spelling "tread," even though the cost of using a 't' may be different from the cost of an 'r.' If the letters to spell out 1 cost \$ $1001$, the letters to spell out 2 cost \$ $1010$, and the letters to spell out 11 cost \$ $2015$, how much do the letters to spell out 12 cost?
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Context:
如果 $\cos ^{5} \theta-\sin ^{5} \theta<7\left(\sin ^{3} \theta-\cos ^{3} \theta\right), \theta \in[0,2 \pi)$, 那么 $\theta$ 的取值范围是 \qquad. 请给出 m + n + k + j 的和，其中原始答案格式为$\left(\frac{m\pi}{n}, \frac{k \pi}{j}\right)$。
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Context:
Given two random variables $X$ and $Y$ with joint distribution function $f_{X,Y}(x,y) = \frac{8xy}{3}$ for $0 < x < 1$ and $x < y < 2x$, determine the covariance of $X$ and $Y$. Consider how the inequality constraint $x < y < 2x$ should be applied when integrating over $Y$.
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Context:
Let $S$ be the set of integer triplets $(a, b, c)$ with $1 \le a \le b \le c$ that satisfy $a + b + c = 77$ and:
\[\frac{1}{a} +\frac{1}{b}+\frac{1}{c}= \frac{1}{5}.\]
What is the value of the sum $\sum_{a,b,c \in S} a\cdot b \cdot c$?
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Context:
Let $f$ be continuous for all $x$. Find the value of the double integral $$\int_0^1\int_x^{1-x}f(t)dt\ dx$$
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Context:
Let $N = 12!$ and denote by $X$ the set of positive divisors of $N$ other than $1$. A [i]pseudo-ultrafilter[/i] $U$ is a nonempty subset of $X$ such that for any $a,b \in X$:
\begin{itemize}
\item If $a$ divides $b$ and $a \in U$ then $b \in U$.
\item If $a,b \in U$ then $\gcd(a,b) \in U$.
\item If $a,b \notin U$ then $\operatorname{lcm} (a,b) \notin U$.
\end{itemize}
How many such pseudo-ultrafilters are there?

[i]Proposed by Evan Chen[/i]
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Context:
Let  $S=\{1,2,\ldots ,98\}$ . Find the least natural number  $n$  such that we can pick out  $10$  numbers in any  $n$ -element subset of  $S$  satisfying the following condition: no matter how we equally divide the  $10$  numbers into two groups, there exists a number in one group such that it is coprime to the other numbers in that group, meanwhile there also exists a number in the other group such that it is not coprime to any of the other numbers in the same group.
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Context:
A wire of length $4$ meters is bent into a square. A bug starts crawling from one corner of the square to the adjacent corner at time $t = 0$, maintaining a constant speed. It continues along the remaining sides until it returns to its starting point at $t = 1$ second. Let $A$ be the starting point of the bug, $O$ the center of the square, and $B$ its position at time $t$. Calculate the rate at which the angle $\angle AOB$ is changing at $t = 0.3$ seconds, with angle measures in radians.
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Context:
Given that \(f(x) = x^3 + x + 3\) is irreducible over \(\mathbb{Q}[x]\) and has exactly one real root \(\theta\), determine the degree of the extension \(\mathbb{Q}(\theta, \phi')\) over \(\mathbb{Q}(\phi)\), where \(\phi\) and \(\phi'\) are the other two roots of \(f(x)\).
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Context:
At the parade, the drummers stand in an even square formation of 50 rows by 50 drummers. The drummers are dressed in either blue or red costumes. What is the largest number of drummers that can be dressed in blue costumes such that each drummer dressed in blue can only see drummers in red costumes? Consider the drummers to be looking in all directions (360 degrees) and as point-like.
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Context:
Determine the subspace of polynomials of degree at most 2, denoted as \( P^2 \), that is orthogonal to the polynomial \( y(x) = x \) with respect to the inner product \( \langle f, g \rangle = \int_{-1}^{1} f(x)g(x) \, dx \).
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\boxed{\{ a x^2 + c \mid a, c \in \mathbb{R} \}}


Context:
For how many ordered triples $(x, y, z)$ of nonnegative integers less than $20$ are there exactly two distinct elements in the set $\{i^x, (1 + i)^y, z\}$, where $i = \sqrt{-1}$?
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Context:
The first two terms of a sequence are $a_1 = 1$ and $a_2 = \frac {1}{\sqrt3}$. For $n\ge1$, \[a_{n + 2} = \frac {a_n + a_{n + 1}}{1 - a_na_{n + 1}}.\] What is $|a_{2009}|$?
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Context:
Let \( n > 1 \) be an odd positive integer and \( A = {\left( {a}_{ij}\right) }_{i,j = 1..n} \) be the \( n \times  n \)
 matrix with
 \( {a}_{ij} = \left\{  \begin{array}{ll} 2 & \text{ if }i = j \\  1 & \text{ if }i - j \equiv   \pm  2\;\left( {\;\operatorname{mod}\;n}\right) . \\  0 & \text{ otherwise } \end{array}\right. \)
 Find det \( A \) .
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Context:
Determine the maximum size of a set \( S \) of irrational real numbers such that among any subset of five numbers in \( S \), there exist two numbers whose sum is irrational.
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Context:
A person's age at the time of their death was one 31st of their birth year. How old was this person in 1930?
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Context:
As shown in the figure, in the triangular prism \(P-ABC\), \(PA \perp \) plane \(ABC\), and \(\angle ABC = 120^\circ\), with \(PA = 4\). If the radius of the circumscribed sphere of the triangular prism \(P-ABC\) is \(2\sqrt{2}\), then the tangent of the angle between line \(PC\) and plane \(ABC\) is \(\quad\).
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Context:
If Yen has a 5 × 7 index card and reduces the length of the shorter side by 1 inch, the area becomes 24 square inches. Determine the area of the card if instead she reduces the length of the longer side by 2 inches.
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Context:
Avery puts 25 shirts in the donation box. He adds eight times as many pants as shirts in the box. He throws in one-third as many shorts as pants in the box. In total, how many pieces of clothes is Avery donating?
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Context:
I randomly pick an integer $p$ between $1$ and $20$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 6p - 3q = 3$? Express your answer as a common fraction.
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Context:
If a real number $a$ satisfies $a^3 < a < a^2$, then the solution set of the inequality $x + a > 1 - ax$ is.
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Context:
A waiter at 'The Gourmet Kitchen' restaurant had 70 customers to wait on during breakfast. Among these, 20 customers came in early at 8 am while the remaining arrived later. Then during the lunch rush he served another 100 customers, 60 of whom were priority customers. In the evening, he served only 45 customers, half of whom were regulars.

If 30% of the early morning customers, 60% of the priority customers, 50% of the regular customers and 25% of all remaining customers did not leave him a tip, how many customers in total left him a tip?
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Context:
If the angles $a, b, c, d,$ and $e$ are known, what is the expression for the angle $u$ in terms of $a$, $b$, $c$, $d$, and $e$? The original answer is in the form of m*a + n*c + p*d. Please provide the value of m + n + p.
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Context:
Find the integers \( n \) such that 5 divides \( 3n - 2 \) and 7 divides \( 2n + 1 \).
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Context:
If the vector $\lambda \overrightarrow{e_{1}}- \overrightarrow{e_{2}}$ is collinear with $\overrightarrow{e_{1}}-\lambda \overrightarrow{e_{2}}$, where $\overrightarrow{e_{1}}, \overrightarrow{e_{2}}$ are non-collinear unit vectors, then the value of the real number $\lambda$ is ______.
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Context:
Farmer James invents a new currency, such that for every positive integer $n \leq 6$, there exists an $n$-coin worth $n$ ! cents. Furthermore, he has exactly $n$ copies of each $n$-coin. An integer $k$ is said to be nice if Farmer James can make $k$ cents using at least one copy of each type of coin. How many positive integers less than 2018 are nice?
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Context:
Given the positive real numbers \(x\) and \(y\) such that

\[ x^5 + 5x^3y + 5x^2y^2 + 5xy^3 + y^5 = 1, \]

what positive values can \(x + y\) take?
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Context:
Given the set $P=\{x|x^2-8x-20\leqslant 0\}$ and non-empty set $S=\{x|1-m\leqslant x\leqslant 1+m\}$, if $x\notin P$ is a necessary but not sufficient condition for $x\notin S$, find the range of values for the real number $m$.
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Context:
Given an arithmetic-geometric sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, if $a_1 = 1$, and $S_6 = 4S_3$, find the value of $a_4$.
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Context:
What is the smallest positive integer that ends in 3 and is divisible by 11?
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Context:
Buses travel along a country road, at equal intervals in both directions and at equal speeds. A cyclist, traveling at $16 \, \mathrm{km/h}$, begins counting buses from the moment when two buses meet beside him. He counts the 31st bus approaching from the front and the 15th bus from behind, both meeting the cyclist again.

How fast were the buses traveling?
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Context:
Given a rhombus with diagonals of length $12$ and $30$, find the radius of the circle inscribed in this rhombus.
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Context:
Find the third derivative of the inverse function $g(x)$ of $f(x)=x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\frac{x^5}{5}$ at $x=0$.
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Context:
A group of fifty people went to a theater performance consisting of twenty-five adults, fifteen seniors, and ten children. Adult tickets cost $15 each, senior tickets have a 25% discount off the adult price, and the total bill for the entire group was $600. If x dollars is the price of one child's ticket, how much does one child's ticket cost?
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Context:
The graph of the functions $y=-3x$ and $y=kx+1$ intersect at point $A(m,4)$. If $m=4$, then the solution set for the inequality $kx+b \lt -3x$ in terms of $x$ is ______.
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Context:
Compute the number of positive integers that divide at least two of the integers in the set $\{1^{1}, 2^{2}, 3^{3}, 4^{4}, 5^{5}, 6^{6}, 7^{7}, 8^{8}, 9^{9}, 10^{10}\}$.
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Context:
For the final act, the circus brought out dancing animals wearing crowns. There are two groups of animals performing. The first group of animals is wearing crowns made with 7 different colored feathers, while the second group of animals is wearing crowns with 5 additional feathers, which are gold. If there are 934 animals in the first group and 425 animals in the second group, how many feathers are needed in total to create crowns for all the animals?
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Context:
Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$s that may occur among the $100$ numbers.
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Context:
Using the digits 1, 2, 3, 4, and 5 without repetition, find the number of unique five-digit "wave numbers" that can be formed, where the tens and thousands digits are greater than the digits immediately adjacent to them. Express your answer as a single integer.
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Context:
Kaleen filled a bucket with 4 quarts of water. A few minutes later, she realized only 0.33 gallons of water remained. _____ water had leaked out of the bucket.
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Context:
Call a positive integer \( N \geq 2 \) "special" if for every \( k \) such that \( 2 \leq k \leq N \), \( N \) can be expressed as a sum of \( k \) positive integers that are relatively prime to \( N \) (although not necessarily relatively prime to each other). How many special integers are there less than 100?
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Context:
If there exist a real number $a$ and a positive integer $n$, such that $f\left(x\right)=\cos 2x-a\sin x$ has exactly $2022$ zeros in the interval $\left(0,n\pi \right)$, then the total number of positive integers $n$ that satisfy this condition is ______.
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Context:
Given 1987 sets, each containing exactly 45 elements, and the union of any two sets contains exactly 89 elements, calculate the number of elements in the union of these 1987 sets. Express your answer as a single integer.
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Context:
Given that \( I \) is the incenter of \( \triangle ABC \), and the following equation holds:
\[
3 \overrightarrow{IA} + 4 \overrightarrow{IB} + 5 \overrightarrow{IC} = \mathbf{0}
\]

Determine the measure of \( \angle C \).
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Context:
(10 points) A swimming pool sells student swim cards for 240 yuan each. The usage rules are as follows: no name registration, each card allows only one person per use, and each card can be used only once a day. A class consists of 48 students, and the teacher plans to organize a swimming trip. Besides purchasing a number of swim cards, they need to rent a bus for each trip. The cost of renting a bus is 40 yuan each time, regardless of the number of passengers. If each student is to swim 8 times, what is the minimum amount of money each student should contribute?
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Context:
Define an odd function $f(x)$ on $\mathbb{R}$. When $x \in (-\infty, 0)$, $f(x) + xf'(x) < 0$ always holds. Let $a = 3f(3)$, $b = (\log_{\pi}e)f(\log_{\pi}e)$, and $c = -2f(-2)$. Determine the relationship among $a$, $b$, and $c$.
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Context:
Given the parabola $C$: ${y}^{2}=2px (p > 0)$ with focus $F$, latus rectum $l$: $x=-\frac{5}{4}$, point $M$ on parabola $C$, point $A$ on latus rectum $l$. If $MA \perp l$, and the angle of inclination of line $AF$ is $\frac{\pi}{3}$, then $|MF|=$ ________.
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Context:
Let $f'(x)$ be the derivative of the function $f(x)$ on $\mathbb{R}$, and for $\forall x \in \mathbb{R}$, $f(-x) + f(x) = x^2$, and for $\forall x \in [0, +\infty)$, $f'(x) > x$. If $f(2-a) - f(a) \geq 2 - 2a$, then the range of the real number $a$ is \_\_\_\_\_\_.
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Context:
Determine the number of idempotents in the ring $\mathbb{Z}[\sqrt{-11}] / (1+\sqrt{-11})$.
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Context:
How many distinct binary codes with parameters $(n, 2, n)$ exist for $n \geq 2$?
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Context:
A painting $18$" X $24$" is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, the ratio of the smaller to the larger dimension of the framed painting is:
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Context:
Given vectors $\overrightarrow{a}=(2,1)$ and $\overrightarrow{b}=(\sin α-\cos α,\sin α+\cos α)$, and $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, calculate the value of $\cos 2α+\sin 2α$. Provide your answer as a single number.
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Context:
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\frac{{\sin B \sin C}}{{3 \sin A}} = \frac{{\cos A}}{a} + \frac{{\cos C}}{c}$, and the area of $\triangle ABC$ is ${S_{\triangle ABC}} = \frac{{\sqrt{3}}}{4}({a^2} + {b^2} - {c^2})$, then the range of $\frac{c}{{a+b}}$ is ______.
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Context:
If \( x \in \left[-\frac{5 \pi}{12}, -\frac{\pi}{3}\right] \), what is the maximum value of \( y = \tan \left( x + \frac{2 \pi}{3} \right) - \tan \left( x + \frac{\pi}{6} \right) + \cos \left( x + \frac{\pi}{6} \right) \)?
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Context:
Four vehicles were traveling on the highway at constant speeds: a car, a motorcycle, a scooter, and a bicycle. The car passed the scooter at 12:00, encountered the bicyclist at 14:00, and met the motorcyclist at 16:00. The motorcyclist met the scooter at 17:00 and caught up with the bicyclist at 18:00.

At what time did the bicyclist meet the scooter?
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Context:
Suppose that $X_1 , X_2 , X_3$ are independent $U (0, 1)$-distributed random variables and let $(X_{(1)} , X_{(2)} , X _{(3)} )$ be the corresponding order statistic. Compute $\mathbb{P}\{X_{(1)}+X_{(2)} > X_{(3)}\}$.
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Context:
Danny collects bottle caps and wrappers. He found 46 wrappers and 50 bottle caps at the park. Now he has some bottle caps and 52 wrappers in his collection. He found 4 more bottle caps than wrappers at the park. How many bottle caps does Danny have in his collection now?
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Context:
Half of the population of the island of Nevezhenia consists of hares, and all the rest are rabbits. If a resident of the island asserts something, he always sincerely believes what he is saying. At the same time, hares are mistaken in an average of every fourth case, whereas rabbits are mistaken in an average of every third case.

One day, an animal emerged in the center of the island and shouted: "I am not a hare!" Then thought for a moment and said sadly: "I am not a rabbit."

What is the probability that it actually is a hare?
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Context:
What is the cardinality of the quotient ring $\frac{\mathbb{Z}_5[X,Y]}{(X^2+4Y^4)}$?
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Context:
A store offers an initial discount of 25% on all its merchandise. Subsequently, an additional discount of 10% is applied to the already reduced prices. The store advertises that the final price of the goods is 35% less than the original price. What is the actual percentage difference between the store's claimed discount and the true discount from the original price?
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Context:
Define the *bigness*of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer  $N$  for which there exists a rectangular prism with integer side lengths and *bigness* $N$  and another one with integer side lengths and *bigness* $N + 1$ .
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Context:
Let $X\sim N(0,\sigma^2)$ and $C>0$. For each $r>0$ define the interval centered at $C$ with radius $r$; that is $I_r:=[C-r,C+r]$. Let $E[X|I_r]$ be the conditional expectation of $X$ on $I_r$. What is the radius $r$ which maximizes $E[X|I_r]$?
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Context:
Find the domain of the function $y=\tan ( \frac {π}{2}x+ \frac {π}{3})$.
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\boxed{\{x|x\neq 2k+ \frac {1}{3},k\in\mathbb{Z}\}}


Context:
A sequence of natural numbers $\left\{x_{n}\right\}$ is constructed according to the following rules:
$$
x_{1}=a, x_{2}=b, x_{n+2}=x_{n}+x_{n+1}, \text{ for } n \geq 1.
$$

It is known that some term in the sequence is 1000. What is the smallest possible value of $a+b$?
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Context:
Bob Barker decided to raise the intellectual level of The Price is Right by having contestants guess how many objects exist of a certain type, without going over. The number of points you will get is the percentage of the correct answer, divided by 10, with no points for going over (i.e., a maximum of 10 points).

The first object for our contestants is a table of shape \((5,4,3,2,1)\). This table is an arrangement of the integers 1 through 15 with five numbers in the top row, four in the next, three in the next, two in the next, and one in the last, such that each row and each column is increasing (from left to right, and top to bottom, respectively). For instance:
```
6
13 14
15
```
is one such table. How many tables are there?
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Context:
Solve the indefinite integral: $$\int \frac{2}{(x^2+2)\sqrt{x^2+4}} dx$$
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Context:
A function f(x) defined on R satisfies f(0)=0, f(x)+f(1-x)=1, f(x/5)=1/2f(x), and f(x1) ≤ f(x2) when 0 ≤ x1 < x2 ≤ 1. Find the value of f(1/2007). Express your answer as a fraction in simplest form.
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Context:
In the sequence $\left\{a_{n}\right\}$, $a_{1}=2 \sqrt{3}$ and $a_{n-1}=\frac{8 a_{n}}{4-a_{n}^{2}}$ for $n \geq 2$, find $a_{n}$.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{2 \tan \frac{\pi}{3 \cdot 2^{n-1}}}


Context:
A positive integer with \( n+3 \) digits, \( 144\cdots430 \) (with \( n \) number of 4s), is a multiple of 2015. What is the smallest value of \( n \)?
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Context:
Consider triangle \(ABC\) where \(BC = 7\), \(CA = 8\), and \(AB = 9\). \(D\) and \(E\) are the midpoints of \(BC\) and \(CA\), respectively, and \(AD\) and \(BE\) meet at \(G\). The reflection of \(G\) across \(D\) is \(G'\), and \(G'E\) meets \(CG\) at \(P\). Find the length \(PG\).
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Context:
Given that $\alpha, \beta \in \left(0, \frac{\pi}{2}\right)$, and $\sin^2 \alpha = \cos(\alpha - \beta)$, determine the relationship between $\alpha$ and $\beta$. Express your answer as a direct comparison between $\alpha$ and $\beta$.
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Context:
An ant starts out at $(0,0)$. Each second, if it is currently at the square $(x, y)$, it can move to $(x-1, y-1),(x-1, y+1),(x+1, y-1)$, or $(x+1, y+1)$. In how many ways can it end up at $(2010,2010)$ after 4020 seconds?
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Context:
What is the largest integer that must divide the product of any 5 consecutive integers?
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Context:
Given the sequence defined by $x_{n+1} = \frac{x_n^2 + x_n + 1}{x_n + 1}$ with $x_1 = \frac{1}{2}$, find the general expression for the sum $\sum_{n=1}^{p}\frac{1}{1+x_n}$.
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Context:
Find a positive integer that is divisible by 18 and has a square root between 22 and 22.5.
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Context:
The pentagon \(PQRST\) is divided into four triangles with equal perimeters. The triangle \(PQR\) is equilateral. \(PTU\), \(SUT\), and \(RSU\) are congruent isosceles triangles. What is the ratio of the perimeter of the pentagon \(PQRST\) to the perimeter of the triangle \(PQR\)?
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Context:
对任意实数 $a_1, a_2, \cdots, a_5$, 不等式

$$
\min _{1 \leqslant i<j \leqslant 5}\left\{\left(a_i-a_j\right)^2\right\} \leqslant k\left(a_1^2+a_2^2+\cdots+a_5^2\right)
$$

恒成立, 求 $k$ 的最小值.
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Context:
Find the absolute value of the difference of single-digit integers $A$ and $B$ such that  $$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c} & & & B& B & A_6\\ & & & \mathbf{4} & \mathbf{1} & B_6\\& & + & A & \mathbf{1} & \mathbf{5_6}\\ \cline{2-6} & & A & \mathbf{1} & \mathbf{5} & \mathbf{2_6} \\ \end{array} $$Express your answer in base $6$.
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Context:
We have a rectangular piece of paper. We want to produce 20-sided polygons using the following procedure. In each step, we select a piece of paper (initially, only the starting piece of paper is available), and cut it into two along a straight line. Continuing this process, what is the minimum number of cuts needed to obtain at least one hundred 20-sided polygons?
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Context:
The sum of the largest number and the smallest number of a triple of positive integers $(x,y,z)$ is called the power of the triple. Find the sum of powers of all triples $(x,y,z)$ where $x,y,z \leq 9$.
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Context:
Fix a sequence  $ a_1,a_2,a_3,... $  of integers satisfying the following condition:for all prime numbers  $ p $  and all positive integers  $ k $ , we have  $ a_{pk+1}=pa_k-3a_p+13 $ .Determine all possible values of  $ a_{2013} $ .
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Context:
If the function $$f(x)=(2m+3)x^{m^2-3}$$ is a power function, determine the value of $m$.
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Context:
Given three different numbers are randomly selected from the set $\{-2, -1, 0, 0, 3, 4, 5\}$ and multiplied together, calculate the probability that the product is $0$.
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Context:
Given $a>0$, let proposition $p$: the function $y=a^x$ is monotonically increasing on $\mathbb{R}$; proposition $q$: the inequality $ax^2-ax+1>0$ holds for all $x \in \mathbb{R}$. If both $p$ and $q$ are false, and either $p$ or $q$ is true, find the range of values for $a$.
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Context:
Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$. Circle $\Omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$. If $QR=3\sqrt3$ and $\angle QPR=60^\circ$, then the area of $\triangle PQR$ equals $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. What is $a+b+c$?
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Context:
A driver goes on a trip of 100 kilometers divided into three segments. The first segment is 40 kilometers at a speed of 50 kilometers per hour, the second segment is 35 kilometers at a speed of 30 kilometers per hour, and the third segment is 25 kilometers at a speed of 60 kilometers per hour. What is the average speed of the entire trip in kilometers per hour?
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Context:
In the right triangle \( \triangle ABC \) with right angle at \( C \), the altitude \( CH \) and the median \( CM \) trisect the right angle. If the area of \( \triangle CHM \) is \( k \), then what is the area of \( \triangle ABC \)?
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Context:
What digit should be placed instead of the question mark in the number 888...88?99...999 (with the eights and nines written 50 times each) so that it is divisible by 7?
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Context:
When the Black King is asleep, he judges everything incorrectly. In other words, everything the Black King considers true in his sleep is actually false, and vice versa. On the other hand, when the Black King is awake, he judges everything correctly, considering what is true as true, and what is false as false. Yesterday evening at exactly ten o'clock, the Black King believed that he and the Black Queen were already sleeping at that time.

Was the Black Queen asleep or awake at ten o'clock yesterday evening?
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Context:
Given the sequence \(\left\{a_{n}\right\}\) defined by \(a_{n}=(\sqrt{2}+1)^{n}-(\sqrt{2}-1)^{n}(n \in \mathbb{N})\), using \([x]\) to denote the greatest integer less than or equal to the real number \(x\), what is the units digit of \(\left[a_{2017}\right]\)?
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Context:
Given that the domain of the function $f(x)$ is $R$, and $f(x) > f'(x) + 1$, $f(0) = 3$, find the solution set of the inequality $f(x) > 2e^{x} + 1$. Express your answer as an interval or a description of the solution set.
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Context:
Express the area of the quadrilateral determined by the midpoints of the segments \( AB \), \( BE \), \( FC \), and \( DG \), in terms of \( a \) and \( b \), given that \( A B C D \) is a square with side length \( a \), and \( B E F G \) is a square with side length \( b \).
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Context:
Given that ${{a_n}}$ is an arithmetic sequence with a non-zero common difference, and ${{a_9}}$, ${{a_1}}$, ${{a_5}}$ form a geometric sequence, and ${{a_1} + 3{{a_5}} + {{a_9}} = 20}$, determine the value of ${{a_{13}}}$.
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Context:
In the diagram, square \(PQRS\) has side length 40. Points \(J, K, L,\) and \(M\) are on the sides of \(PQRS\), so that \(JQ = KR = LS = MP = 10\). Line segments \(JZ, KW, LX,\) and \(MY\) are drawn parallel to the diagonals of the square so that \(W\) is on \(JZ\), \(X\) is on \(KW\), \(Y\) is on \(LX\), and \(Z\) is on \(MY\). What is the area of quadrilateral \(WXYZ\)?
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Context:
Given that the asymptotic line of the hyperbola $\frac{x^2}{a}+y^2=1$ has a slope of $\frac{5π}{6}$, determine the value of $a$.
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Context:
Evaluate the double integral $$\iint_R ye^{xy}~\mathrm{d}A$$ over the rectangular region defined by $0\le x\le 2$ and $0 \le y \le 1$.
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Context:
给定正整数k≤m≤n，求

$\sum _ { i = 0 } ^ { n } ( - 1 ) ^ { i } \frac { 1 } { n + k + i } \cdot \frac { ( m + n + i ) ! } { i ! ( n - i ) ! ( m + i ) ! }$
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Context:
How many Young diagrams exist with at most $p$ rows and $q$ columns, without any restrictions on the weight of the diagrams?
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Context:
In a right-angled triangle $ABC\left(\angle C=90^{\circ}\right)$ with side $BC=a$, points $M$ and $N$ are the midpoints of sides $AB$ and $BC$, respectively. The angle bisector of $\angle A$ intersects the line $MN$ at point $L$. Find the radius of the circumcircle of triangle $ACL$ if the angle $\angle CBL=\alpha$.
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Context:
There are 5 tables in a classroom. Each table has 4 chairs with a child sitting on it. All the children get up and randomly sit in a seat. Two people that sat at the same table before are not allowed to sit at the same table again. Assuming tables and chairs are distinguishable, if the number of different classroom arrangements can be written as $2^a3^b5^c$, what is $a+b+c$?
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Context:
Given the ellipse \( C: \frac{x^{2}}{4}+y^{2}=1 \) with the upper vertex \( M \) and the lower vertex \( N \), \( T(t, 2) (t \neq 0) \) is a point on the line \( y=2 \). The lines \( TM \) and \( TN \), passing through point \( T \), intersect the ellipse \( C \) at points \( E \) and \( F \), respectively. If the area of triangle \( TMN \) is \( k \) times the area of triangle \( TEF \), find the value of \( t \) that maximizes \( k \).
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Context:
Person 1 and Person 2 enter a queue independently. Person 1's waiting time is exponentially distributed with rate $\theta$, while Person 2's waiting time is exponentially distributed with rate $\mu$. If Person 1 enters the queue $s$ minutes before Person 2, what is the probability that Person 2 leaves the queue before Person 1?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{1-\frac{\theta  e^{-\mu  s}}{\theta +\mu }}


Context:
In triangle $XYZ$, angle $XZY$ is $45$ degrees, and angle $ZYX$ is $80$ degrees. Let $P$ be the foot of the perpendicular from $X$ to $YZ$, $Q$ the center of the circle circumscribed around triangle $XYZ$, and $R$ the other end of the diameter which goes through $X$. Determine the measure of angle $XRP$ in degrees.
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Context:
The numbers 407 and 370 equal the sum of the cubes of their digits. For example, \( 4^3 = 64 \), \( 0^3 = 0 \), and \( 7^3 = 343 \). Adding 64, 0, and 343 gives you 407. Similarly, the cube of 3 (27), added to the cube of 7 (343), gives 370.

Could you find a number, not containing zero and having the same property? Of course, we exclude the trivial case of the number 1.
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Context:
A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. The original answer is in the form \frac{a\sqrt{b}- c}{d}, please provide the value of a + b + c + d.
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Context:
16. $[7]$ Let $\mathbb{R}$ be the set of real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that for all real numbers $x$ and $y$, we have
$$
f\left(x^{2}\right)+f\left(y^{2}\right)=f(x+y)^{2}-2 x y \text {. }
$$

Let $S=\sum_{n=-2019}^{2019} f(n)$. Determine the number of possible values of $S$.
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Context:
From the set \( M = \{1, 2, \cdots, 2008\} \) of the first 2008 positive integers, a \( k \)-element subset \( A \) is chosen such that the sum of any two numbers in \( A \) cannot be divisible by the difference of those two numbers. What is the maximum value of \( k \)?
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Context:
You have a circular necklace with  $10$  beads on it, all of which are initially unpainted. You randomly select  $5$  of these beads. For each selected bead, you paint that selected bead and the two beads immediately next to it (this means we may paint a bead multiple times). Once you have finished painting, what is the probability that every bead is painted?
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Context:
In how many ways can Alice, Bob, Charlie, David, and Eve split $18$ marbles among themselves so that no two of them have the same number of marbles?
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Context:
Uncle Chernomor assigns 9 or 10 of his thirty-three bogatyrs to duty every evening. What is the minimum number of days required so that all the bogatyrs have been on duty the same number of times?
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Context:
In triangle $PQR$, $\angle Q$ is a right angle, and the altitude from $Q$ meets $\overline{PR}$ at $S$. The lengths of the sides of triangle $PQR$ are integers, and $PR=3^{5}, PS=3^3$. If $\tan P = m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
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Context:
设 S=\{1,2,\cdots,98\}. 求最小正整数 n，使得 S 的任一 n 元子集中都可以选出 10 个数，无论怎样将这 10 个数均分成两组，总有一组中存在一个数与另外 4 个数都互质，而另一组中总有一个数与另外 4 个数都不互质.
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Context:
5. Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, a sphere is constructed with vertex $A$ as the center and $\frac{2 \sqrt{3}}{3}$ as the radius. The length of the curve formed by the intersection of the sphere's surface with the cube's surface is $\qquad$
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Context:
Compute the integral $$I = \int_1^\infty \log^2 \left(1-\frac{1}{x}\right) \, dx$$ using a suitable substitution and appropriate techniques.
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Context:
Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how many ways can this be done?
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Context:
A certain commodity has a cost price of 200 yuan and a marked price of 400 yuan. What is the maximum discount that can be offered to ensure that the profit margin is not less than 40%?
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Context:
Given the equations:
\[
\log_{2}\left[\log_{\frac{1}{2}}\left(\log_{2} x\right)\right]=\log_{3}\left[\log_{\frac{1}{3}}\left(\log_{3} y\right)\right]=\log_{5}\left[\log_{\frac{1}{5}}\left(\log_{5} z\right)\right]=0
\]
Determine the relationship among x, y, and z. Express your answer in the format x? y? z, where? represents <, >, or =.
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Context:
Evaluate the expression $-20 + 15 \times (4^{\div -1} \times 2)$.
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Context:
Given the function $f\left(x\right)=\cos 2x\cdot \ln x$, find the derivative of $f\left(x\right)$. Express your answer as a mathematical expression.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{-2\sin 2x\cdot \ln x + \frac{\cos 2x}{x}}


Context:
Given the function $f(x) = (x+1)^3 + 1$, a positive geometric sequence $\{a_{n}\}$ satisfies $a_{1012} = \frac{1}{10}$. Find $\sum_{k=1}^{2023} f(\log{a}_{k})$.
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Context:
Two small rulers form a set square, where the small ruler can slide along the large ruler. Each unit on the large ruler is marked with natural numbers. The first small ruler divides 11 units on the large ruler into 10, and the second small ruler divides 9 units on the large ruler into 10. The starting points of both small rulers are 0, marked from 1 to 10 respectively. Now measure the distance between points \(A\) and \(B\), with point \(A\) at 0 units on the large ruler, and point \(B\) between 18 and 19 units on the large ruler. When the 0 unit of the first small ruler is placed at point \(B\), its 3rd unit coincides with a certain unit on the large ruler. If the 0 unit of the second small ruler is placed at point \(B\), then which unit of the second small ruler coincides with a certain unit on the large ruler?
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Context:
Use the Newton-Raphson method to find the value of $t$ near $0.04$, correct to four decimal places, for which the amplitude $y$ is $0.880$ in the equation:
\[ y = \left(t + \frac{\pi}{4}\right) + \sin(t) + \frac{1}{8}\sin(3t) \]
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Context:
A quadrilateral is inscribed in a circle with a radius of 13. The diagonals of the quadrilateral are perpendicular to each other. One of the diagonals is 18, and the distance from the center of the circle to the point where the diagonals intersect is \( 4 \sqrt{6} \).

Find the area of the quadrilateral.
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Context:
Determine the rank of the linear function $f: V \to W$, where $V$ is the vector space of all square matrices with trace 0, and $W$ is the set of real numbers.
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Context:
Rectangle $W X Y Z$ has $W X=4, W Z=3$, and $Z V=3$. The rectangle is curled without overlapping into a cylinder so that sides $W Z$ and $X Y$ touch each other. In other words, $W$ touches $X$ and $Z$ touches $Y$. The shortest distance from $W$ to $V$ through the inside of the cylinder can be written in the form $\sqrt{\frac{a+b \pi^{2}}{c \pi^{2}}}$ where $a, b$ and $c$ are positive integers. What is the smallest possible value of $a+b+c$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the domain of the function $f(x) = \ln(x - 1)$. Express your answer as a set of all x satisfying a certain condition.
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Context:
The owner of an apartment rents it out for five years and receives 240,000 rubles from tenants at the end of each year. He does not spend his rental income, but keeps it in a term deposit at 10% annual interest (capitalized once at the end of each year). What amount will the owner have in the account after five years?
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Context:
What is the smallest constant $\alpha$ such that, given 19 points on a solid, regular hexagon with side 1, there will always be 2 points with distance at most $\alpha$?
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Context:
设点 $A$ 的坐标为 $(0,3)$, 点 $B, C$ 为圆 $O: x^{2}+y^{2}=25$ 上的两动点, 满足 $\angle B A C=90^{\circ}$.  $\triangle A B C$ 面积的最大值为$\frac{a+b \sqrt{c}}{d}$，求$a+b+c+d$的值。
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Context:
Pizzas are sized by diameter. What percent increase in area results if Lorrie’s pizza increases from a 16-inch pizza to an 18-inch pizza?
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Context:
The numbers \(a, b, c\) lie in the interval \((0 ; \pi / 2)\) and satisfy the equations: \(\cos a = a\), \(\sin (\cos b) = b\), \(\cos (\sin c) = c\). Arrange these numbers in ascending order.
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Context:
In triangle \(ABC\), the angle bisector \(AP\) is drawn. It is known that \(BP = 16\), \(PC = 20\), and the circumcenter of triangle \(ABP\) lies on segment \(AC\). Find side \(AB\).
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Context:
在正六棱锥中，侧面与底面的夹角均为 \df{\pi}{3}，则其外接球与内切球的半径的比值为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
18. Let $A B C$ be an equilateral triangle. Let $\overrightarrow{A B}$ be extended to a point $D$ such that $B$ is the midpoint of $\overline{A D}$. A variable point $E$ is taken on the same plane such that $D E=A B$. If the distance between $C$ and $E$ is as large as possible, what is $\angle B E D$ ?
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Context:
Evaluate the integral: \[ \int_0^{2\pi} - \frac{\cos t \; ( 2 (\sin t)^2 + (\cos t)^2)}{(\cos t)^4 + (\sin t)^2} \, dt \]
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Context:
Let \( \omega \) be a circle, and let \( ABCD \) be a quadrilateral inscribed in \( \omega \). Suppose that \( BD \) and \( AC \) intersect at a point \( E \). The tangent to \( \omega \) at \( B \) meets line \( AC \) at a point \( F \), so that \( C \) lies between \( E \) and \( F \). Given that \( AE = 6 \), \( EC = 4 \), \( BE = 2 \), and \( BF = 12 \), find \( DA \).
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Context:
Let (1+√2)^2012 = a + b√2, where a and b are integers. Find the greatest common divisor of b and 81. Express your answer as a single integer.
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Context:
Consider the set $\mathcal F$ consisting of all holomorphic functions $f$ in the unit disc $\Bbb D$ such that $f(\Bbb D)\subset\Bbb D$, and satisfying $f\left(\frac12\right)=f'\left(\frac12\right)=0$. Determine the supremum of $|f(0)|$ over all functions $f$ in $\mathcal F$.
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Context:
David has a collection of 40 rocks, 30 stones, 20 minerals and 10 gemstones. An operation consists of removing three objects, no two of the same type. What is the maximum number of operations he can possibly perform?
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Context:
Evaluate the integral \( \int \frac{5z^2 - 3z + 2}{(z-1)^3} \, dz \) using the Cauchy Integral Formula.
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Context:
Determine the maximal interval of definition for solutions of the autonomous differential equation $\dot{x}=f(x)$, where $f:\mathbb{R}\rightarrow \mathbb{R}$ is locally Lipschitz continuous, $f(0)=0=f(1)$, and at time zero, $\dot{x}$ lies in the interval $[0,1]$. What is this interval?
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Context:
Let $x$ and $y$ be positive real numbers and $\theta$ an angle such that $\theta \neq \frac{\pi}{2}n$ for any integer $n$.  Suppose
\[\frac{\sin\theta}{x}=\frac{\cos\theta}{y}\]
and
\[
\frac{\cos^4 \theta}{x^4}+\frac{\sin^4\theta}{y^4}=\frac{97\sin2\theta}{x^3y+y^3x}.
\]
Compute $\frac xy+\frac yx.$
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Context:
7. There are 5 power poles, 2 of which will leak electricity. There are 5 birds each randomly choosing one of the poles to rest (if a bird touches a leaking pole, it will be stunned by the electric shock and fall to the ground). The probability that only 2 of the poles have birds on them is $\qquad$
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Context:
Find the principal on a certain sum of money at 5% per annum for 2 2/5 years if the amount being Rs. 2120. What is the principal amount in rupees?
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Context:
A fat coin is one which, when tossed, has a \(\frac{2}{5}\) probability of being heads, \(\frac{2}{5}\) of being tails, and \(\frac{1}{5}\) of landing on its edge. Mr. Fat starts at 0 on the real line. Every minute, he tosses a fat coin. If it's heads, he moves left, decreasing his coordinate by 1; if it's tails, he moves right, increasing his coordinate by 1. If the coin lands on its edge, he moves back to 0. If Mr. Fat does this indefinitely, what fraction of his time will he spend at 0?
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Context:
Determine the number of relatively prime dates in the month with the second fewest relatively prime dates.
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Context:
There are 30 people sitting around a round table - knights and liars (knights always tell the truth, and liars always lie). It is known that each of them has exactly one friend at the table, and if one is a knight, their friend is a liar, and vice versa (friendship is always mutual). In response to the question "Is your friend sitting next to you?" the individuals who are sitting in every other seat answered "Yes." How many of the remaining individuals could also answer "Yes"?
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Context:
Evaluate the integral \( \int_{0}^{\infty} \frac{\log(x) }{\sqrt{x} (x+1)^{2}}\,dx \).
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Context:
When Betty makes cheesecake, she sweetens it with a ratio of one part sugar to four parts cream cheese. She flavors it with a certain ratio of vanilla to cream cheese. For every one teaspoon of vanilla, she uses two eggs. She used two cups of sugar and 8 eggs in her latest cheesecake. What is the ratio of vanilla to cream cheese?
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Context:
Determine the smallest finite field in which the polynomial \((x^2+x+1)(x^5+x^4+1)\) is reducible.
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Context:
In a configuration of 16 students, each positioned at the vertices of concentric regular pentagons, a total of 3360 coins are distributed such that students equidistant from the center receive the same number of coins. After a redistribution process where each student gives all their coins equally to their neighbors, each student ends up with the same number of coins they started with. How many coins did the student in the central circle originally have?
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Context:
Let $F$ be a field of order $32$. Determine the number of non-zero solutions $(a, b) \in F \times F$ to the equation $x^2 + xy + y^2 = 0$.
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Context:
The diagonals of convex quadrilateral $BSCT$ meet at the midpoint $M$ of $\overline{ST}$.  Lines $BT$ and $SC$ meet at $A$, and $AB = 91$, $BC = 98$, $CA = 105$.  Given that $\overline{AM} \perp \overline{BC}$, find the positive difference between the areas of $\triangle SMC$ and $\triangle BMT$.

[i]Proposed by Evan Chen[/i]
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Context:
Random variable $Z$ has a mean of $15$ and a standard deviation of $2$. What is the minimum percentage of $Z$ values that lie between $8$ and $17$?
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Context:
Let $G=(L\cup R,E)$ be a bipartite graph with $n$ nodes on each side. Determine the maximum value of $n$ such that if any two vertices in $L$ have at least two neighbors in $R$, then $G$ has a perfect matching.
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Context:
Strain A had 9000 cells and decreased at a rate of 3000 cells per hour. Strain B started with 4000 cells and decreased at a rate of 2000 cells per hour. When will the strains have the same number of cells? Give the ordered pair please!
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Context:
Consider an ellipse given by \(\frac{x^2}{4} + y^2 = 1\). A point \(P\) on the ellipse has an eccentric angle of \(45^\circ\). Through this point, variable pairs of chords are drawn such that they are perpendicular to each other and meet the ellipse at points \(A\) and \(B\). If the line joining \(A\) and \(B\) always passes through a fixed point \(Q(a, b)\), find the value of \(a^2 + b^2\).
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Context:
Pikachu, Charmander, and Vulpix are three of the four equally-skilled players in a Pokemon bracket tournament. Because they are equally skilled, whenever any two of the players battle, they are equally likely to win. In the bracket tournament, the four players are randomly paired into two rounds, each round consisting of two players. The winners of the first two rounds then play each other in the final round. The winner of the final match ranks first; the loser of the final round ranks second; and the two losers of the previous rounds jointly rank third. What is the probability that Charmander plays Vulpix in a round, but ranks lower than Pikachu? Express your answer in the form \(\frac{k}{m}\) where the fraction is in simplest form, and provide the value of \(k + m\).
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Context:
The algebraic expression $4x^{2}+3mx+9$ is a perfect square trinomial. Then, $m=$ \_\_\_\_\_\_ .
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Context:
Calculate the expected number of turns required to jump from the ground floor (floor 0) to the 5th floor in a building with 5 floors. The probability of jumping from the ground to floor 1 is 1. For each floor i (where i ranges from 1 to 4), the probability of jumping to the next floor (i+1) is \( \frac{1}{3^i} \), the probability of staying on the same floor is \( \frac{3^i-1}{2 \times 3^i} \), and the probability of jumping back to the previous floor (i-1) is \( \frac{3^i-1}{2 \times 3^i} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A quadrilateral has side lengths of 70, 90, 130, and 110, in that order. It is both inscribed in a circle and has an incircle. The points where the incircle touches the side of length 130 divide it into segments of lengths \( x \) and \( y \). Determine the value of \( |x - y| \).
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Context:
In a circuit with 2000 components, every pair of components is initially connected by a wire. Vasya and Petya take turns cutting wires, with Vasya starting. Vasya cuts one wire per turn, while Petya cuts two or three wires per turn. The player who cuts the last wire connected to any component loses. Determine which player has a winning strategy.
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Context:
To make the inequality $0\leqslant x^{2}+ax+6\leqslant 4$ have exactly one solution with respect to $x$, then $a=\_\_\_\_\_\_$.
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Context:
Find the value of \(x\) that satisfies the equation: \[ 1 + x + \frac{1\cdot3}{2!}x^2 + \frac{1\cdot3\cdot5}{3!}x^3 + \ldots = \sqrt{2} \]
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Context:
Yan is at a point between his house and a park. He has two options to reach the park: He can either walk directly to the park or he can walk back to his house and use his scooter to reach the park. He scoots 10 times faster than he walks, and both routes take the same amount of time. Determine the ratio of Yan's distance from his house to his distance to the park.
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Context:
Consider a Wiener-Hopf integral equation of the first kind with unknown function $\phi(x)$ for $x\geq 0$: $$f(x)=\int_0^\infty \phi(y)\min\{x,y\}\,\mathrm{d}y.$$ If $f(x)=\sin(x)$, find the solution for $\phi(x)$.
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Context:
Let $a_d$ be the number of non-negative integer solutions $(a, b)$ to $a + b = d$ where $a \equiv b \pmod{n}$ for a fixed $n \in \mathbb{Z}^+$. Consider the generating function $M(t) = a_0 + a_1t + a_2t^2 + \ldots$ Consider
$$P(n) = \lim_{t\to 1} \left( nM(t) - \frac{1}{(1 - t)^2} \right).$$
Then $P(n)$, $n \in \mathbb{Z}^+$ is a polynomial in $n$, so we can extend its domain to include all real numbers while having it remain a polynomial. Find $P(0)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An even perfect square in the decimal system is of the form: $\overline{a b 1 a b}$. What is this perfect square?
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Context:
Let $S$ be the set of monic polynomials in $x$ of degree 6, all of whose roots are members of the set $\{-1, 0, 1\}$. Let $P$ be the sum of the polynomials in $S$. What is the coefficient of $x^4$ in $P(x)$?
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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 $x(x - 4)^{\alpha}$, what is the value of $\alpha$?
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Context:
Given a cyclic quadrilateral \(ABCD\). The rays \(AB\) and \(DC\) intersect at point \(K\). It turns out that points \(B\), \(D\), and the midpoints of segments \(AC\) and \(KC\) lie on the same circle. What values can the angle \(ADC\) take?
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Context:
设函数$f\left( x \right)$在定义域(0，+∞)上是单调函数，$\forall x\in \left( 0,+\infty  \right),f\left[ f\left( x \right)-{{e}^{x}}+x \right]=e$，若不等式$f\left( x \right)+{f}'\left( x \right)\ge ax$对$x\in \left( 0,+\infty  \right)$恒成立，则实数$\textit{a}$的取值范围是______．
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Context:
Given the rational numbers $a$, $b$, $c$, $d$ satisfy $2006 = 9a = 15b = 32c = 68d$, arrange the numbers $a$, $b$, $c$, $d$ in descending order.
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Context:
Car A and Car B start simultaneously from locations A and B, respectively, and they travel between these two locations at constant speeds. After the first time they meet, Car A takes 4 hours to reach B, while Car B takes 1 hour to reach A. How many hours have both cars traveled by the 15th time they meet (excluding meetings at locations A and B)?
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Context:
The workday at the enterprise was 8 hours. During the first six hours, labor productivity met the planned level, but then it fell by 25%. The director, in agreement with the workforce, extended the shift by an hour. It turned out that for the added hour, the first six hours were again at the planned productivity level, but afterward, productivity dropped by 30%. By what percent did the overall productivity for the shift increase as a result of extending the workday?
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Context:
In a movie theater, five friends took seats from 1 to 5 (the leftmost seat is number 1). During the movie, Anya went to get popcorn. When she returned, she found that Varya had moved one seat to the right, Galya had moved two seats to the left, and Diana and Ella had swapped seats, leaving the end seat for Anya. In which seat was Anya sitting before she got up?
Please reason step by step, and put your final answer within \boxed{}.


Context:
1. Milan moved 10 steps forward, then returned 2 steps back, then 10 forward and 1 back, then again 10 forward and 2 back, etc. How many steps does Milan need to take to be 1000 steps away from the starting point?
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Context:
Evaluate the limit: \[ \lim_{n \to \infty} \sum_{k=1}^{\infty}\frac{1}{k^{2}\sqrt[k]{n}}\sin^{2}\left(\frac{n \pi}{k}\right) \]
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Context:
Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20\%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5\%$ of the difference between $A$ and the cost of his movie ticket. To the nearest whole percent, what fraction of $A$ did Roger pay for his movie ticket and soda?

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


Context:
Calculate the limit using Taylor expansions or the Generalized Binomial Theorem:

$$\lim_{x \rightarrow 0} \frac{(1+3x)^{\frac{1}{2x}} - (1+2x)^{\frac{3}{x}}}{x}$$
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Context:
Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20\%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5\%$ of the difference between $A$ and the cost of his movie ticket. To the nearest whole percent, what fraction of $A$ did Roger pay for his movie ticket and soda?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Starting with a $5 \times 5$ grid, choose a $4 \times 4$ square in it. Then, choose a $3 \times 3$ square in the $4 \times 4$ square, and a $2 \times 2$ square in the $3 \times 3$ square, and a $1 \times 1$ square in the $2 \times 2$ square. Assuming all squares chosen are made of unit squares inside the grid, in how many ways can the squares be chosen so that the final $1 \times 1$ square is the center of the original $5 \times 5$ grid?
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Context:
At the beginning of each hour from 1 o'clock AM to 12 NOON and from 1 o'clock PM to 12 MIDNIGHT a coo-coo clock's coo-coo bird coo-coos the number of times equal to the number of the hour. In addition, the coo-coo clock's coo-coo bird coo-coos a single time at :30 minutes past each hour. How many times does the coo-coo bird coo-coo from 12:42 PM on Monday until 3:42 AM on Wednesday?
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Context:
Calculate the current price of a 2-year derivative contract with a maturity payoff of £90 × I(S2 > £90), where the current share price is £100, the share price can increase or decrease by 20% annually, and the continuously compounded constant annual risk-free interest rate is such that e^r = 1.1.
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Context:
How many partitions are there for a set with an uncountable number of elements?
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Context:
Given that $A=\{x\in \mathbb{N} | -2 \lt x \lt 2\}$ and $B=\{-1,0,1,2\}$, find the intersection $A\cap B$. Express your answer in standard set notation.
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Context:
Given $x < y < 0$, let $M = |x|$, $N = |y|$, and $P = \frac{|x + y|}{2}$. Determine the relationship among $M$, $N$, and $P$.
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Context:
Identify a group that is isomorphic to the additive group $\mathbb{R} / \mathbb{Z}$.
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Context:
Calculate the intersection number of the curves given by \((x^2+y^2)^3-4x^2y^2=0\) and \(x=0\) at the point \((0,0)\). The intersection number is defined as \(\dim_k(O_p/(F,G))\), where \(O_p\) is the local ring at the point \(P\).
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Context:
Given an alphabet with \( n > 1 \) letters, a word is defined as any finite sequence of letters where no two neighboring letters are the same. A word is considered good if it is impossible to delete all letters except for four such that the remaining sequence is of the form \( a a b b \), where \( a \) and \( b \) are different letters. Determine the maximum possible number of letters in a good word.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Petya and Vasya came up with ten quadratic trinomials. Then Vasya sequentially named consecutive natural numbers (starting from a certain number), and Petya substituted each named number into one of the trinomials of his choice and wrote the resulting values on the board from left to right. It turned out that the numbers written on the board formed an arithmetic progression (in that exact order). What is the maximum number of numbers that Vasya could name?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the integral

$$ \mathcal{I}[y] = \int_{-\infty}^{\infty} \left( \frac{1}{2} y'^2 + 1 - \cos y \right) \, dx $$

subject to the boundary conditions $y(-\infty) = 0$ and $y(\infty) = 2\pi$.
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Context:
Two math students play a game with $k$ sticks. Alternating turns, each player chooses a number from the set \{1, 3, 4\} and removes exactly that number of sticks from the pile. If the pile has fewer sticks than the chosen number, the player must choose a number that can be removed. The winner is the player who takes the last stick. For $1 \leq k \leq 100$, determine the number of cases in which the first player can guarantee a win.
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Context:
Suppose $x, y, z$, and $w$ are positive reals such that:
\[ x^2 + y^2 - \frac{xy}{2} = w^2 + z^2 + \frac{wz}{2} = 36 \]
\[ xz + yw = 30. \]
Find the largest possible value of $(xy + wz)^2$.
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Context:
Let $r$ be a real number such that $\sqrt[3]{r} - \frac{1}{\sqrt[3]{r}} = 2$. Find $r^3 - \frac{1}{r^3}$.
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Context:
Use the Lagrange Multipliers to find the maximum value of the function $f(x,y,z)= x+2y+3z$ on the curve of intersection of the plane $x-y+z=1$ and the cylinder $x^2+y^2=1$.
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Context:
How many boolean functions $F(w, x, y, z)$ satisfy the property $F(\bar{w}, x, \bar{y}, z) = F(w, \bar{x}, y, \bar{z}) = F(w, x, \bar{y}, \bar{z})$ for all values of the Boolean variables $w, x, y,$ and $z$?
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Context:
Calculate the value of $8\cos ^{2}25^{\circ}-\tan 40^{\circ}-4$.
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Context:
Find the minimal polynomial of $\zeta = e^{2\pi i / 7}$ over the field $\mathbb{Q}(\cos(2\pi/7))$.
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Context:
Three boys played a "Word" game in which they each wrote down ten words. For each word a boy wrote, he scored three points if neither of the other boys had the same word; he scored one point if only one of the other boys had the same word. No points were awarded for words which all three boys had. When they added up their scores, they found that they each had different scores. Sam had the smallest score (19 points), and James scored the most. How many points did James score?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A group of 40 boys and 28 girls stand hand in hand in a circle facing inwards. Exactly 18 of the boys give their right hand to a girl. How many boys give their left hand to a girl? Express your answer as a whole number.
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Context:
In △ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. If a=1 and the height on side BC is equal to tanA, then the range of the perimeter of △ABC is _______.
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Context:
Let \( p(x) \) be a polynomial of degree \( 3n \) such that \( p(0)=p(3)=\cdots=p(3n)=2 \), \( p(1)=p(4)=\cdots=p(3n-2)=1 \), and \( p(2)=p(5)=\cdots=p(3n-1)=0 \). Given that \( p(3n+1)=730 \), find \( n \).
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Context:
What is the  $111^{\text{st}}$  smallest positive integer which does not have  $3$  and  $4$  in its base- $5$  representation?
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Context:
The diagram shows two equilateral triangles with side length $4$ mounted on two adjacent sides of a square, also with side length $4$. The distance between the two vertices marked $A$ and $B$ can be written as $\sqrt{m}+\sqrt{n}$ for two positive integers $m$ and $n$. Find $m + n$.
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Context:
A group of hikers is planning a trip that will take them up a mountain using one route and back down using another route. The upward route consists of various terrains: the first half is a well-maintained trail with tree cover, while the second half is a rocky, open terrain that exposes them to changing weather conditions. The downward route, on the other hand, is a more consistent trail through the forest, which shields them from the elements, allowing them to travel 1.5 times their rate on the way up.

On the first day, the hikers move at a rate of 4 miles per day on the well-maintained trail, but their speed drops to 2 miles per day when they reach the rocky terrain on the second day due to the rough ground and unpredictable weather. The total time it takes for them to climb the mountain is two days.

Considering the varying terrain and weather conditions, how many miles long is the route down the mountain, given that the hikers will take the same amount of time to complete the descent?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A group of adventurers is displaying their loot. It is known that exactly 13 adventurers have rubies; exactly 9 have emeralds; exactly 15 have sapphires; exactly 6 have diamonds. Additionally, the following conditions are known:

- If an adventurer has sapphires, then they have either emeralds or diamonds (but not both simultaneously);
- If an adventurer has emeralds, then they have either rubies or sapphires (but not both simultaneously).

What is the minimum number of adventurers that can be in such a group?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each month a retailer sells some identical items. On each item, he makes a profit of $30 that constitutes 16% of the item's price to the retailer. If the retailer contemplates giving a 5% discount on the items he sells, he will have to sell at least 156.86274509803923 items each month to justify the policy of the discount. How many items does the retailer sell each month without the discount?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many real solutions are there to the equation $|||| x|-2|-2|-2|=|||| x|-3|-3|-3|$?
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Context:
Find the area of the region \(D\) bounded by the curves

\[ x^{2} + y^{2} = 12, \quad x \sqrt{6} = y^{2} \quad (x \geq 0) \]
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Context:
Joe will randomly select two letters from the word CAMP, four letters from the word HERBS, and three letters from the word GLOW.  What is the probability that he will have all of the letters from the word PROBLEM?  Express your answer as a common fraction.
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Context:
Given a rectangular prism \(ABCD - A'B'C'D'\), how many pairs of skew lines are there among the following twelve lines: \(AB', BA', CD', DC', AD', DA', BC', CB', AC, BD, A'C', B'D'\)? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.