Context:
The angle bisector $AL$ and the median $BM$ of triangle $ABC$ intersect at point $X$. The line $CX$ intersects side $AB$ at point $Y$. Find the area of triangle $CYL$, given that $\angle BAC = 60^\circ$ and $AL = x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sets
$$
A=\{1,3,5,7,9\}, B=\{2,4,6,8,10\}
$$

If the set $C=\{x \mid x=a+b, a \in A, b \in B\}$, the number of elements in the set $C$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \( \alpha \in \mathbb{R} \) such that the process \( 3W_t^4 + \alpha t^2 W_t^2 \) is a martingale, where \( W_t \) is a standard Wiener process.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a non-constant function that is harmonic in the punctured unit ball $B(0,1)\setminus\{0\}$ and satisfies $\lim_{x\to 0}u\!\left(x\right)/\lvert x \rvert^{2-N}=0$. Provide the function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A garrison of 2000 men has provisions for 54 days. The initial garrison consumes provisions at a rate of 1x per soldier per day. At the end of 21 days, a reinforcement arrives with a consumption rate of 2x per soldier per day. It is now found that the remaining provisions will last only for 20 days more. What is the size of the reinforcement?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the coordinate plane with the origin $O$, consider points $P(t+2,\ 0),\ Q(0, -2t^2-2t+4)\ (t\geq 0).$ If the $y$-coordinate of $Q$ is nonnegative, then find the area of the region swept out by the line segment $PQ$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
At a market in Egypt, a tourist is negotiating with a seller for a souvenir worth 10,000 Egyptian pounds. The tourist first reduces the price by \( x \) percent \((0 < x < 100)\), then the seller increases the price by \( x \) percent, and so on. The value of \( x \) does not change during the bargaining process, and the seller increases the price at least once. The negotiation continues until one of the participants gets a non-integer value for the price of the souvenir. Find the largest possible number of price changes during such a negotiation (including the last non-integer price change).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a geometric sequence $\{a_n\}$ with all positive terms, if the geometric mean of $a_1$ and $a_{17}$ is 2, find the minimum value of $4a_7+a_{11}$. Express your answer as a single number in the form $\boxed{[answer]}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The semi-latus rectum of a parabola intersects with a hyperbola of the form ${x}^{2}-{y}^{2}={a}^{2}$ at points $A$ and $B$. The distance between these points is $4\sqrt{3}$. Determine the length of the real axis of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 F(x)=\displaystyle\int_0^x\left(\sin\df{1}{t}\right)^4\dd t，则 [100|F'(0)|]=__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\{N(t): t \geq 0\}$ be a Poisson process with rate $1$, and let $T_1 < T_2 < \cdots$ denote the times of the points. Determine the probability density function (pdf) of the random variable $Y = \frac{T_1}{T_3}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest positive integer $\ell$ such that the set $[\ell] = \{1, 2, \dots, \ell\}$ is of type $P(3,1)$. The set $[\ell]$ is of type $P(3,1)$ if any partition of $[\ell]$ into 3 non-empty disjoint subsets yields a good sequence in at least one subset. A sequence $a_1, a_2, \dots, a_n$ is good if for every $3 \leq k \leq n$, there exist indices $1 \leq i \leq j < k$ such that $a_i + a_j = a_k$. Given that $12 < f(3)$ and $13$ is not a valid value for $f(3)$, find the value of $f(3)$. \[ f(m) \leq \left[\sum_{j=0}^m\frac{1}{j!}\right]m! \text{ for all } m \geq 1. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can one arrange the natural numbers from 1 to 9 in a $3 \times 3$ square table so that the sum of the numbers in each row and each column is odd? (Numbers can repeat)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given i.i.d random variables with mean $\mu$ and variance 1, let $\bar{X}_n = \frac{1}{n}(X_1+\cdots+X_n)$. Use the Central Limit Theorem to approximate the probability: $$P(|\bar{X}_n - \mu| \ge \frac{2}{\sqrt {n}}).$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rich emir was admiring a new jewel, a small golden plate in the shape of an equilateral triangle, decorated with diamonds. He noticed that the shadow of the plate forms a right triangle, with the hypotenuse being the true length of each side of the plate.

What is the angle between the plane of the plate and the flat surface of the sand? Calculate the cosine of this angle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of unitary homomorphisms \( \phi : \mathbb{Z}[X]/(X^3+3X+5) \longrightarrow \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the right-angled triangle \(ABC\), the angle at vertex \(B\) is \(30^\circ\). The square is drawn outwardly on the hypotenuse \(BC\) with center \(D\). What is the measure of angle \(ADB\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the right triangle \(ABC\), the leg \(AB = 3\), and the leg \(AC = 6\). The centers of circles with radii 1, 2, and 3 are located at points \(A\), \(B\), and \(C\) respectively. Find the radius of the circle that is externally tangent to each of these three given circles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
记 $M_{100}(\mathbb{R})$ 为所有 100 阶实方阵的集合. 设 $S=\left\{A \in M_{100}(\mathbb{R}) \mid A^{4}+A^{2}=\right.$ $O\}$ ，则 $S$ 中最多可以选出 $\qquad$个互不相似的矩阵. (我们称 $A, B \in M_{100}(\mathbb{R})$ 相似,当且仅当 存在可逆的 $P \in M_{100}(\mathbb{R})$ ，使得 $P^{-1} A P=B$ )
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $ P(x)$  be a nonzero polynomial such that, for all real numbers  $ x$ ,  $ P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x)$ . Determine the maximum possible number of real roots of  $ P(x)$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
An engineer designed a ball so that when it was dropped, it rose with each bounce exactly one-half as high as it had fallen. The engineer dropped the ball from a 16-meter platform and caught it after it had traveled a certain distance. The ball bounced 4 times. How far did the ball travel before it was caught?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How can I express $\frac{1}{\alpha}$ in terms of $\alpha$, where $\alpha = \sqrt{3+\sqrt{20}}$ and its minimal polynomial is $x^4 - 6x^2 - 11$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The construction consists of two weightless rods $AB$ and $CD$, a weightless pulley, and a weight with mass $m=2$ kg that is suspended at point $G$. All strings are weightless and non-stretchable. At point $A$, the rod is connected to a hinge that allows the rod to rotate in the plane of the figure. Moreover, $CG = \frac{1}{4}CD$, $AE = \frac{1}{4}AB$, and point $F$ is the midpoint of rod $AB$.

The acceleration due to gravity $g=10 \mathrm{~m} / \mathrm{c}^{2}$. Determine the tension force $T_{0}$ in the string attached to rod $AB$ at point $E$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many graphs are there on 10 vertices labeled \(1,2, \ldots, 10\) such that there are exactly 23 edges and no triangles?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The upper base area of a truncated prism is \( S' \). The lower base area is \( S \). A cross-section with area \( Q \), parallel to the bases, divides the height of the prism in the ratio \( n: m \). Find \( \sqrt{Q} = \ \ \ \ \ \ .
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\frac{n \sqrt{S} + m \sqrt{S'}}{n+m}}


Context:
Every cell of a  $3\times 3$  board is coloured either by red or blue. Find the number of all colorings in which there are no  $2\times 2$  squares in which all cells are red.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Of the numbers $\frac{7}{10}, \frac{4}{5}$ and $\frac{3}{4}$, which number is the arithmetic mean of the other two?
Please reason step by step, and put your final answer within \boxed{}.


Context:
9.4. To the number $A$, consisting of eight non-zero digits, a seven-digit number, consisting of identical digits, was added, and the eight-digit number $B$ was obtained. It turned out that the number $B$ can be obtained from the number $A$ by rearranging some of the digits. What digit can the number $A$ start with if the last digit of the number $B$ is 5?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the numerator of \[ \frac{1010\overbrace{11 \ldots 11}^{2011 \text{ ones}}0101}{1100\underbrace{11 \ldots 11}_{2011\text{ ones}}0011} \] when reduced.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy $(ab+1)(bc+1)(ca+1)=84$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a certain middle school, 500 eighth-grade students took the biology and geography exam. There were a total of 180 students who scored between 80 and 100 points. What is the frequency of this score range?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The watchman detained a trespasser and wants to chase him away. However, the trespasser said that he made a bet with friends for 100 coins that the watchman wouldn't chase him away (if he gets chased away, he pays friends 100 coins; otherwise, they pay him). To get rid of the watchman, the trespasser offered to name a price. What is the maximum number of coins the watchman can demand so that the trespasser, guided solely by his own benefit, will definitely pay the watchman?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Olya, after covering one-fifth of the way from home to school, realized that she forgot her notebook. If she does not return for it, she will reach school 6 minutes before the bell rings, but if she returns, she will be 2 minutes late. How much time (in minutes) does the journey to school take?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $\sin \theta$ and $\cos \theta$ are the two roots of the equation $4x^{2}+2mx+m=0$, then the value of $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using toothpicks of equal length, a rectangular grid is constructed. The grid measures 25 toothpicks in height and 15 toothpicks in width. Additionally, there is an internal horizontal partition at every fifth horizontal line starting from the bottom. Calculate the total number of toothpicks used.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given points $P(\cos \alpha, \sin \alpha)$, $Q(\cos \beta, \sin \beta)$, and $R(\cos \alpha, -\sin \alpha)$ in a two-dimensional space, with the cosine distance between $P$ and $Q$ being $\frac{1}{3}$, and the relationship $\tan \alpha \cdot \tan \beta = \frac{1}{7}$, calculate the cosine distance between $Q$ and $R$. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that \[ f(x) + f\left( 1 - \frac{1}{x} \right) = \arctan x \] for all real $x \neq 0$. (As usual, $y = \arctan x$ means $-\pi/2 < y < \pi/2$ and $\tan y = x$.) Find \[ \int_0^1 f(x)\,dx. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
What number should be used to divide 180, and then used as the divisor for 5 * 12, and added to 80 to get the result 81?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a, b, c$ be positive integers such that $\frac{a}{77}+\frac{b}{91}+\frac{c}{143}=1$. What is the smallest possible value of $a+b+c$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
有四所学校的学生参加一项数学竞赛，每所学校派出 3 名选手。组委会要抽选其中若干名选手做一项调研，要求任意两所学校被抽中的选手数之和至少为 1 、至多为 3 ，则不同的抽选方式数为 $\qquad$ （结果用数值表示）。
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the country of knights (who always tell the truth) and liars (who always lie), 10 people are seated at the vertices of a regular decagon around a round table. Among them, there are liars. A traveler can stand somewhere and ask the seated people: "What is the distance from me to the nearest liar among you?" Everyone then responds to the traveler. What is the minimum number of questions that the traveler must ask to definitively identify who the liars are at the table? (There are no bystanders, and the traveler cannot stand on the table. Consider people as points, and assume that everyone, including the traveler, can accurately measure any distance.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
The volume of a right triangular prism is 16. When its surface area is minimized, the length of the base edge $a$ = _______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The school nurse must conduct lice checks, vision tests, and hearing tests at three elementary schools. At School A, She must check 26 Kindergarteners, 19 first graders, 20 second graders, 25 third graders, 30 fourth graders, and 28 fifth graders. At School B, she must check 22 Kindergarteners, 25 first graders, 18 second graders, 24 third graders, 27 fourth graders, and 30 fifth graders. At School C, she must check 24 Kindergarteners, 20 first graders, 22 second graders, 26 third graders, 28 fourth graders, and 26 fifth graders.

Each lice check takes two minutes, each vision test takes two minutes, and each hearing test takes three minutes. Each school has a mandatory one-hour lunch break for the nurse and ten-minute breaks between grades. The nurse spends ten minutes traveling between School A and School B and fifteen minutes traveling between School B and School C. How many hours will it take the nurse to complete all the checks and tests for all students in the three schools?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alex has a $20 \times 16$ grid of lightbulbs, initially all off. He has 36 switches, one for each row and column. Flipping the switch for the $i$th row will toggle the state of each lightbulb in the $i$th row (so that if it were on before, it would be off, and vice versa). Similarly, the switch for the $j$th column will toggle the state of each bulb in the $j$th column. Alex makes some (possibly empty) sequence of switch flips, resulting in some configuration of the lightbulbs and their states. How many distinct possible configurations of lightbulbs can Alex achieve with such a sequence? Two configurations are distinct if there exists a lightbulb that is on in one configuration and off in another.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There exist positive integers $N$ and $M$ such that the remainders of $N$ when divided by the four integers $6$, $36$, $216$, and $M$ form an increasing nonzero geometric sequence in that order. Find the smallest possible value of $M$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
When flipping a coin, if we denote heads as $Z$ and tails as $F$, then a sequence of coin flips can be represented as a sequence composed of $Z$ and $F$. We can count the occurrences of heads followed by tails $(F Z)$, heads followed by heads $(Z Z)$, etc. For example, the sequence
\[ ZZFFZZZZFZZFFFF \]
is the result of flipping a coin 15 times. In this sequence, there are 5 occurrences of $ZZ$, 3 occurrences of $ZF$, 2 occurrences of $FZ$, and 4 occurrences of $FF$. How many sequences of 15 coin flips have exactly 2 occurrences of $ZZ$, 3 occurrences of $ZF$, 4 occurrences of $FZ$, and 5 occurrences of $FF$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the value of

\(\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1?\)

Provide your answer in the form of \(\frac{k}{m}\), where the fraction is in the simplest form. Calculate the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
On weekdays, the Absent-Minded Scientist travels to work on the ring line of the Moscow Metro from "Taganskaya" station to "Kievskaya" station in the morning and back in the evening (see the diagram).

Upon entering a station, the Scientist boards the first arriving train. It is known that trains travel in both directions at approximately equal intervals. The travel time from "Kievskaya" to "Taganskaya" or back is 17 minutes via the northern route (through "Belorusskaya"), and 11 minutes via the southern route (through "Paveletskaya"). As per his long-standing habit, the Scientist always makes calculations. One day, he calculated that based on many years of observations:

- The train going counterclockwise arrives at "Kievskaya" on average 1 minute and 15 seconds after the train going clockwise arrives at "Kievskaya". The same is true for "Taganskaya".
- The Scientist takes on average 1 minute less time traveling from home to work than from work to home.

Find the mathematical expectation of the interval between trains traveling in one direction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f : \mathbb{N} \to \mathbb{N}$ be a strictly increasing function such that $f(f(n))= 3n$, for all $n \in \mathbb{N}$. Find $f(2010)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $15^{-1} \equiv 31 \pmod{53}$, find $38^{-1} \pmod{53}$, as a residue modulo 53.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $bcosB$ is the arithmetic mean of $acosC$ and $ccosA$. Then, angle $B$ equals \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
1. Indicate the largest possible number, in decimal notation, in which all digits are different, and the sum of its digits is 37.

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


Context:
A rectangular prism has dimensions 4 x 4 x 3, and the vertices A, B, C, and D are adjacent. Calculate the shortest distance from vertex D to the plane ABC. Express your answer as a decimal value rounded to one decimal place.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Raymonde's financial institution publishes a list of service charges as follows: Autodebit costs $0.60 per transaction, cheque costs $0.50 per transaction, and cash withdrawal costs $0.45 per transaction. For her first twenty-five transactions, she uses Autodebit three times as often as she writes cheques, and she writes as many cheques as she makes cash withdrawals. After her twenty-fifth transaction, she begins to make single transactions. Calculate the smallest number of transactions she needs to make so that her monthly service charges will exceed the $15.95 "all-in-one" fee.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sum \[\sum_{i=0}^{1000} \frac{\binom{1000}{i}}{\binom{2019}{i}}\] can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Compute $p+q$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A man buys a house for $10,000 and rents it. He puts $12\frac{1}{2}\%$ of each month's rent aside for repairs and upkeep; pays  $325 a year taxes and realizes $5\frac{1}{2}\%$ on his investment. The monthly rent (in dollars) is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \( x_{i}=\frac{i}{101} \), find the value of \( S=\sum_{i=1}^{101} \frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
设等边三角形 $A B C$ 的内切圆半径为 2, 圆心为 $I$. 若点 $P$ 满足 $P I=1$, 则 $\triangle A P B$ 与 $\triangle A P C$ 的面积之比的最大值为 $\qquad$. 题目原始答案为\frac{m+\sqrt{n}}{k}的格式，请给出m + n + k的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mark had a box of chocolates. He consumed $\frac{1}{4}$ of them and then gave $\frac{1}{3}$ of what remained to his friend Lucy. Mark and his father then each ate 20 chocolates from what Mark had left. Finally, Mark's sister took between five and ten chocolates, leaving Mark with four chocolates. How many chocolates did Mark start with?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A company is lining a 900-foot path with a fence. Fence poles are placed at a certain distance apart. The entire path is lined with a fence except for a 42-foot bridge. There are 286 fence poles placed along the whole path on both sides. What is the distance between each fence pole?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Santa Claus has 36 identical gifts distributed among 8 bags. Each bag contains at least 1 gift, and the number of gifts in each of the 8 bags is unique. From these bags, select some bags such that the total number of gifts in the selected bags can be evenly divided among 8 children, with each child receiving at least one gift. Determine how many different ways the selection can be made.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of a square is 24, and the perimeter of a small rectangle is 16. What is the perimeter of the large rectangle?

The perimeter of a shape is the sum of its side lengths.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If x is greater than 3 and less than 6, and y is greater than 6 and less than a certain number, then the greatest possible positive integer difference of x and y is 4. What is the upper bound for y?
Please reason step by step, and put your final answer within \boxed{}.


Context:
John chops down 30 trees. He can make 20 planks from the first 10 trees, 25 planks from the next 10 trees, and 30 planks from the last 10 trees. The labor cost for the first 10 trees is $120 per tree, $80 per tree for the next 10 trees, and $60 per tree for the last 10 trees. A table takes 15 planks to make, and the selling price of the tables varies; the first 10 tables sell for $350 each, the next 20 tables for $325 each, and every table after that for $300 each. How much profit did John make?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x) = x^2 + 4mx + n$ is decreasing on the interval $[2, 6]$, find the range of real numbers $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the parabola $C: y^2 = 2px \ (0 < p < 4)$ with focus $F$, and a moving point $P$ on $C$. Let $A(4, 0)$ and $B(p, \sqrt{2}p)$ be such that the minimum value of $|PA|$ is $\sqrt{15}$. Find the value of $|BF|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the value of \[(\log 5)^{3}+(\log 20)^{3}+(\log 8)(\log 0.25)\] where $\log$ denotes the base-ten logarithm?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ephram is growing $3$ different variants of radishes in a row of $13$ radishes total, but he forgot where he planted each radish variant and he can't tell what variant a radish is before he picks it. Ephram knows that he planted at least one of each radish variant, and all radishes of one variant will form a consecutive string, with all such possibilities having an equal chance of occurring. He wants to pick three radishes to bring to the farmers market, and wants them to all be of different variants. Given that he uses optimal strategy, the probability that he achieves this can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the complex plane, let $z_1, z_2, z_3$ be the roots of the polynomial $p(x) = x^3- ax^2 + bx - ab$. Find the number of integers $n$ between $1$ and $500$ inclusive that are expressible as $z^4_1 +z^4_2 +z^4_3$ for some choice of positive integers $a, b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a 5x7 grid with 1 gray square, how many rectangles with an area of 15 can be drawn such that they do not contain or intersect the gray square? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Carol spends ten hours filling up her pool. During the first hour, the pool fills at a rate of 8 gallons of water per hour. For the next two hours, the pool fills at 10 gallons of water per hour. From the fourth hour to the fifth hour, the pool fills at a rate of 14 gallons of water per hour. During the sixth hour, the pool starts filling at a rate of 12 gallons of water per hour.

At the start of the seventh hour, the pool develops a leak, losing 8 gallons of water per hour. From the seventh hour to the ninth hour, two more leaks occur. The first leak starts during the eighth hour, and the pool loses 10 gallons of water per hour. The second leak starts during the ninth hour, and the pool loses 6 gallons of water per hour.

In the tenth hour, all leaks have been fixed, but the filling rate decreases to 6 gallons of water per hour. At the end of the ten hours, how many gallons of water are still left in the pool?
Please reason step by step, and put your final answer within \boxed{}.


Context:
James and his younger sibling, Sarah, both have growth spurts. Initially, James was 2/3s as tall as his 72-inch tall uncle, and Sarah was only 3/4 as tall as James. During the growth spurt, James gains 10 inches, while Sarah gains 12 inches. What is the combined height difference between the uncle and both James and Sarah now?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Scott, Mary, Ken, and Amy committed to raising $15,000 for their local children's cancer hospital. Currently, Mary's collection is five times what Ken has and three times Scott’s funds, while Amy's collection is two-thirds of the total raised by Ken and Scott together. They also agreed that for every $100 collected, they will each contribute another 10% of their funds as a bonus. If Ken has raised $900, by how much have the four of them exceeded or fallen short of their goal, including the bonus contribution?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Lil writes one of the letters \( \text{P}, \text{Q}, \text{R}, \text{S} \) in each cell of a \( 2 \times 4 \) table. She does this in such a way that, in each row and in each \( 2 \times 2 \) square, all four letters appear. In how many ways can she do this?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find (in terms of \( n \geq 1 \)) the number of terms with odd coefficients after expanding the product:
\[ \prod_{1 \leq i<j \leq n}(x_{i} + x_{j}) \]

For example, for \( n = 3 \), the expanded product is given by \( x_{1}^{2} x_{2} + x_{1}^{2} x_{3} + x_{2}^{2} x_{3} + x_{2}^{2} x_{1} + x_{3}^{2} x_{1} + x_{3}^{2} x_{2} + 2 x_{1} x_{2} x_{3} \), resulting in 6 terms with odd coefficients.
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Context:
Two trains, 200 m and 160 m long, run at the speeds of 68 kmph and 40 kmph, in opposite directions on parallel tracks. However, the tracks are not on the same level: one track is at ground level and the other is elevated. The angle of elevation between the tracks is 15 degrees. What is the time they take to cross each other, taking into account the difference in heights between the tracks?
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Context:
Calculate the L.C.M of 4/9, 5/7, 7/15, 9/13, 13/21, 16/35, and 19/45.
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Context:
In an opaque bag, there are a total of 50 glass balls in red, black, and white colors. Except for the color, everything else is the same. After several trials of drawing balls, Xiaogang found that the probability of drawing a red or black ball is stable at 15% and 45%, respectively. What could be the possible number of white balls in the bag?
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Context:
Find the number of ordered triples $(x,y,z)$ of real numbers such that
\[x^4 + y^4 + z^4 - 4xyz = -1.\]
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Context:
A cistern is two-thirds full of water. There are three pipes: Pipe A can fill the remaining part in 12 minutes, Pipe B in 8 minutes, and Pipe C can empty the full cistern in 24 minutes. If Pipes A and B fill while Pipe C is open, how much time will they take to fill the cistern completely?
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Context:
Let $f(x)$ be a real function with smallest period $5$, and $g(x)$ be defined as $f(2^x)$. Given that $g(x)$ has $25$ zeros in the range $[0,5)$, find the number of zeros that $g(x)$ has in the range $[1,4)$.
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Context:
Find the number of ordered pairs of positive integers $(x, y)$ that satisfy the equation
$$
x \sqrt{y} + y \sqrt{x} + \sqrt{2009 x y} - \sqrt{2009 x} - \sqrt{2009 y} - 2009 = 0.
$$
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Context:
Gold is 19 times as heavy as water and copper is 9 times as heavy as water. These should be mixed in a ratio of 9 to get an alloy that is a certain times as heavy as water. How heavy is the alloy compared to water?
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Context:
Given the sequence $\left\{a_{n}\right\}$ defined by
$$
a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right] \quad (n \in \mathbf{Z}_{+}),
$$
where $[x]$ denotes the greatest integer not exceeding the real number $x$, determine the minimum value of the constant $C$ such that for any positive integer $n$, the following inequality holds:
$$
\sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C.
$$
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Context:
Let $ABC$ be a scalene triangle whose side lengths are positive integers. It is called [i]stable[/i] if its three side lengths are multiples of 5, 80, and 112, respectively. What is the smallest possible side length that can appear in any stable triangle?

[i]Proposed by Evan Chen[/i]
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Context:
How many injective functions are there from an uncountable set to the set of bounded functions from the natural numbers to the natural numbers?
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Context:
The squares of a $3\times3$ grid are filled with positive integers such that $1$ is the label of the upper- leftmost square, $2009$ is the label of the lower-rightmost square, and the label of each square divides the  ne directly to the right of it and the one directly below it. How many such labelings are possible?
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Context:
29. Find the largest natural number in which each non-edge digit is less than the arithmetic mean of the adjacent digits.
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Context:
A frog is at the center of a circular shaped island with radius $r$. The frog jumps $\frac{1}{2}$ meters at first. After the first jump, it turns right or left at exactly $90^\circ$, and it always jumps one half of its previous jump. After a finite number of jumps, what is the least $r$ that ensures the frog can never fall into the water? The original answer is in \frac{\sqrt{m}}{n} format, please give the value of m + n.
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Context:
Given that Jones is waiting for one of two telephone booths occupied by A and B, where the duration of their calls follows an exponential distribution with λ = 1/8, what is the probability that Jones is not the last to finish their call?
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Context:
Compare the following expressions: $-|-3|\_\_\_\_\_\_-\pi$.
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Context:
55. Three faces of an $8 \times 8 \times 8$ cube are painted blue, and the other three faces are painted red, such that no vertex has three faces of the same color. How many small cubes from this large cube have both a blue and a red face?
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Context:
Given a real number $a$, the floor of $a$, written $\lfloor a \rfloor$, is the greatest integer less than or equal to $a$. For how many real numbers $x$ such that $1 \le x \le 20$ is \[ x^2 + \lfloor 2x \rfloor = \lfloor x^2 \rfloor + 2x \? \]
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Context:
Given a right triangular pyramid with pairwise perpendicular edges, its three side faces have areas of 4, 6, and 12 square units. Calculate the surface area of the circumscribing sphere. Express your answer in terms of pi (π) and include the unit'square units'.
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Context:
Determine the probability that both banknotes are counterfeit given that one of the two randomly selected 100 yuan banknotes from a stack of 20 (which contains 5 counterfeit notes) is verified to be counterfeit upon examination with a money detector.
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Context:
Let $a, b, c, x, y,$ and $z$ be complex numbers such that:
\[ a = \frac{b+c}{x-2}, \quad b = \frac{c+a}{y-2}, \quad c = \frac{a+b}{z-2}. \]
Given that:
\[ xy + yz + zx = 1000 \quad \text{and} \quad x + y + z = 2016, \]
find the value of $xyz$.
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Context:
Let \( F_{1}, F_{2}, F_{3}, \ldots \) be a sequence of convex quadrilaterals where \( F_{k+1} \) (for \( k=1, 2, 3, \ldots \)) is obtained as follows: \( F_{k} \) is cut along a diagonal, one of the parts is flipped over, and then glued along the cut line with the other part. What is the maximum number of distinct quadrilaterals that can be contained in this sequence? (Polygons are considered distinct if they cannot be made to coincide by motion.)
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Context:
设$Z = \{ 1 , 2 , \cdots , n \}$，k为正整数，$\frac { n } { 2 } \leq k \leq n$，求满足下列条件的映射$f : Z \rightarrow Z$的个数：$( i ) f ^ { 2 } = f ;( i i ) f ( x )$的值域$f ( Z )$、中恰有k个不同元素；（iii）对每个$y \inf ( Z ) .$，使$f ( x ) = y$的$x \in Z$至多两个.
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Context:
How many positive integers divide $5n^{11}-2n^5-3n$ for all positive integers $n$.
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Context:
Calculate: $\int_{-1}^{1}\left(2 \sqrt{1-{x}^{2}-\sin x}\right)dx = $______．
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Context:
Let the function $f(x) = x^3 - 3ax + b$ $(a \neq 0)$. If the curve $y = f(x)$ is tangent to the line $y = 8$ at the point $(2, f(2))$, find the value of the product $ab$.
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Context:
Let $ABCD$ be a convex quadrilateral with $AC=20$, $BC=12$ and $BD=17$. If $\angle{CAB}=80^{\circ}$ and $\angle{DBA}=70^{\circ}$, then find the area of $ABCD$.

[i]2017 CCA Math Bonanza Team Round #7[/i]
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Context:
Given a positive integer \( k \), a sequence \( \{a_n\} \) satisfies the following conditions:
\[ a_1 = 3, \quad a_{n+1} = \left(3^{\frac{2}{2x-1}} - 1\right) S_n + 3 \quad (n \in \mathbb{Z}_+) \]
where \( S_n \) is the sum of the first \( n \) terms of the sequence \( \{a_n\} \).

Let
\[ b_n = \frac{1}{n} \log_3 (a_1 a_2 \ldots a_n) \quad (n \in \mathbb{Z}_+) \]
and define
\[ T_k = \sum_{i=1}^{2k} \left| b_i - \frac{3}{2} \right|. \]

If \( T_k \in \mathbb{Z}_+ \), find all possible values of \( k \).
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Context:
The cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ shown in Figure 2 is cut into two parts by a plane passing through point $A$, the trisection point $H$ on the edge $C C^{\prime}$ closer to $C$, and the midpoint $F$ on the edge $D D^{\prime}$. Determine the volume ratio of the two parts.
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Context:
Given the number of elements of a list ($n$) and the quadratic mean ($QM$) of that list, can we find a lower bound for the arithmetic mean ($AM$) of that list? If so, what is the greatest lower bound for the $AM$ that we can find?
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Context:
Evaluate the expression: $\left(2\left(3\left(2\left(3\left(2\left(3 \times (2+1) \times 2\right)+2\right)\times 2\right)+2\right)\times 2\right)+2\right)$.
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Context:
A train running at a speed of 36 km/h passes an electric pole in 14 seconds. How long is the platform if it takes the train 48.997200223982084 seconds to pass it?
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Context:
A couple agreed to split their expenses equally, including the salary of their house help. While doing her job, the house help had an accident; thus, she underwent three medical procedures which cost her $128, $256, and $64 respectively. The couple decided to pay half of the total medical expenses and the other half will be deducted from her $160 salary. Additionally, they also deducted a 5% tax from her salary. If the husband paid for the entire half of the hospital bill, how much does he still need to pay so that they split the expenses for the house help equally?
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Context:
Let $M$ be the set of all polynomials $P(x)$ with pairwise distinct integer roots, integer coefficients, and all absolute values of the coefficients less than $2007$. What is the highest degree among all the polynomials in the set $M$?
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Context:
A solid in the shape of a right circular cone is $4$ inches tall and its base has a $3$-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum -shaped solid $F,$ in such a way that the ratio between the areas of the painted surfaces of $C$ and $F$ and the ratio between the volumes of $C$ and $F$ are both equal to $k$ . Given that $k=\frac m n,$ where $m$ and $n$ are relatively prime positive integers , find $m+n.$
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Context:
According to the relevant tax laws of the country, the author's remuneration should be subject to personal income tax. The specific regulations are as follows: For each individual remuneration income, the balance after a fixed amount or fixed rate minus the specified expenses is the taxable income. For each income not exceeding 4,000 yuan, first deduct an expense of 800 yuan per remuneration income; for each income exceeding 4,000 yuan, first deduct 20% of the expenses, and in both cases, a tax rate of 20% is applied, and according to the regulations, a 30% tax is levied on the taxable amount. It is known that a person has published a manuscript and paid a tax of 280 yuan, the pre-tax remuneration this person should receive is \_\_\_\_\_\_.
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Context:
Given the first term, the last term, and the number of terms in a geometric sequence, determine the product of the terms.
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Context:
Given that all faces of the tetrahedron P-ABC are right triangles, and the longest edge PC equals $2\sqrt{3}$, the surface area of the circumscribed sphere of this tetrahedron is \_\_\_\_\_\_.
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Context:
A technician makes a round-trip to and from a certain service center by the same route that consists of three stops. If the technician completes the drive to the first stop and then completes 75% of the drive from the first stop to the second stop, 50% from the second stop to the third stop, and 25% from the third stop to the service center, what percent of the complete round-trip journey has the technician completed?
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Context:
Compute the expected value $E[{X \choose 3}]$ for a geometrically distributed random variable $X$ with parameter $p$.
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Context:
如果三角形 $\triangle A B C$ 的三个内角 $A, B, C$ 的余切 $\cot A, \cot B, \cot C$ 依次成等差数列, 则角 $B$ 的最大值是以 $\frac{m\pi}{n}$ 表示，求 $m + n$ 的值。
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Context:
Troy's and Emily's homes are 75 meters and 98 meters away from school, respectively. However, there is a park and a river along their routes, which force them to take different detours each day.

On Monday, Troy takes a detour adding 15 meters to his route, while Emily takes a detour adding 10 meters. On Tuesday, Troy takes another detor adding 20 meters, while Emily takes one adding 25 meters. On Wednesday, both take a detour adding 10 meters each. On Thursday, Troy has no detours while Emily takes one adding 15 meters. On Friday, Troy takes a detour adding 5 meters, and Emily takes one adding 10 meters.

Given the varying detours and distances added on each day, find out how much farther Emily walks to school and back in five days.
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Context:
Given that the domain of the function f(x) is \u211d, and the following conditions hold:
- When x < 0, f(x) = x^3 - 1
- When -1 \u2264 x \u2264 1, f(-x) = -f(x)
- When x > 1/2, f(x + 1/2) = f(x - 1/2)
Find the value of f(6). Express your answer as a single integer.
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Context:
In an isosceles triangle \(ABC\) with base \(AC\), the side \(AB\) is equal to 2, segment \(AD\) is the angle bisector. A tangent \(DH\) to the circumcircle of triangle \(ADB\) passes through point \(D\) and point \(H\) lies on side \(AC\). Find the area of triangle \(ABC\) if \(CD = \sqrt{2} \cdot CH\).
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Context:
For any real number $x$, let $\lfloor x \rfloor$ denote the integer part of $x$; $\{ x \}$ be the fractional part of $x$ ($\{x\} = x - \lfloor x \rfloor$). Let $A$ denote the set of all real numbers $x$ satisfying:
\[\{x\} = \frac{x + \lfloor x \rfloor + \lfloor x + \frac{1}{2} \rfloor}{20}\]
If $S$ is the sum of all numbers in $A$, find $\lfloor S \rfloor$.
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Context:
If the function $y=f(x)$ ($x \in \mathbb{R}$) satisfies $f(x+2)=f(x)$, and when $x \in [-1,1)$, $f(x)=|x|$, then the number of intersection points between the graph of the function $y=f(x)$ and the graph of the function $y=\log_{4}|x|$ is.
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Context:
Given that $a, b \in \mathbb{R}$, and $f(x) = e^x - ax + b$, if $f(x) \geq 1$ always holds, then the range of $\frac{b - a}{a}$ is $\_\_\_\_\_\_$.
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Context:
A student drew a $50 \times 50$ empty table and wrote a number above each column and to the left of each row. It turned out that all 100 numbers written are different, with 50 of them being rational and 50 being irrational. Then, in each cell of the table, he wrote the product of the numbers written next to its row and column (like in a multiplication table). What is the maximum number of products in this table that could be rational numbers?
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Context:
Find the number of permutations \((b_1, b_2, b_3, b_4, b_5, b_6)\) of \((1,2,3,4,5,6)\) such that
\[
\frac{b_1 + 6}{2} \cdot \frac{b_2 + 5}{2} \cdot \frac{b_3 + 4}{2} \cdot \frac{b_4 + 3}{2} \cdot \frac{b_5 + 2}{2} \cdot \frac{b_6 + 1}{2} > 6!.
\]
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Context:
A subset $X$ of $\{1, 2, 3, \ldots, 10000\}$ has the following property: If $a, b$ are distinct elements of $X$, then $ab \not\in X$. What is the maximal number of elements in $X$?
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Context:
Find the amount on Rs. 5000 in 2 years, with the rate of interest being 10% for the first year and some percentage for the second year, if the final amount is Rs. 6160. What was the rate of interest for the second year?
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Context:
Given that points $P$ and $Q$ are located on the circle $({x}^{2}+{{\left( y-6 \right)}^{2}}=2)$ and the ellipse $(\dfrac{{{x}^{2}}}{10}+{{y}^{2}}=1)$ respectively, find the maximum distance between $P$ and $Q$.
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Context:
在平面直角坐标系 xOy 中，点集 K=\{(x,y)\ |\ x,y=-1,0,1\}. 在 K 中随机取出三个点，则这三个点两两之间的距离均不超过 2 的概率为________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Bogosiov I.I.

On the reverse sides of 2005 cards, different numbers are written (one on each). In one question, it is allowed to point to any three cards and find out the set of numbers written on them. What is the minimum number of questions needed to find out what numbers are written on each card?

#
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Context:
The graphs of the equations
$y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k,$
 are drawn in the coordinate plane for $k=-10,-9,-8,\ldots,9,10.\,$ These 63 lines cut part of the plane into equilateral triangles of side length $\tfrac{2}{\sqrt{3}}.\,$ How many such triangles are formed?
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Context:
There are five chain links, each with 3 rings. What is the minimum number of rings that need to be unlinked and relinked to connect these links into one chain?

#
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Context:
Find the product of uncommon real roots of the two polynomials $x^4 + 2x^3 - 8x^2 - 6x + 15$ and $x^3 + 4x^2 - x - 10$.
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Context:
Five persons wearing badges with numbers  $1, 2, 3, 4, 5$  are seated on  $5$  chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different)
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Context:
As shown in the figure, points \( M \) and \( N \) lie on sides \( AB \) and \( AC \) of \(\triangle ABC\) respectively, and \(\overrightarrow{AM} = x \overrightarrow{AB}, \overrightarrow{AN} = y \overrightarrow{AC}\), \( D \) is the midpoint of line segment \( BC \), and \( G \) is the intersection point of line segments \( MN \) and \( AD \). If \(\overrightarrow{AG}=\frac{3}{5} \overrightarrow{AD}\), find the minimum value of \(\frac{1}{x^{2}}+\frac{4}{y^{2}}\).
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Context:
Given that \( a \) is a positive real number and \( b \) is an integer between \( 2 \) and \( 500 \), inclusive, find the number of ordered pairs \( (a,b) \) that satisfy the equation \( (\log_b a)^{1001}=\log_b(a^{1001}) \).
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Context:
The number $2027$ is prime. Let $T = \sum \limits_{k=0}^{72} \binom{2024}{k}$. What is the remainder when $T$ is divided by $2027$?
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Context:
For $-1<r<1$, let $T(r)$ denote the sum of the geometric series \[20 + 10r + 10r^2 + 10r^3 + \cdots.\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b)=5040$. Find $T(b)+T(-b)$.
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Context:
The function \( y = \tan(2015x) - \tan(2016x) + \tan(2017x) \) has how many zeros in the interval \([0, \pi]\)?
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Context:
We say that a subset of $\mathbb{R}^{n}$ is $k$-almost contained by a hyperplane if there are less than $k$ points in that set which do not belong to the hyperplane. We call a finite set of points $k$-generic if there is no hyperplane that $k$-almost contains the set. For each pair of positive integers $k$ and $n$, find the minimal number $d(k, n)$ such that every finite $k$-generic set in $\mathbb{R}^{n}$ contains a $k$-generic subset with at most $d(k, n)$ elements. Calculate the value of $d(k, n)$ when $k = 2$ and $n = 2$, and provide the result of k + n + d(k, n).
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Context:
If (a certain number ^ 4 * 3.456789) ^ 10 is written as a single term, there would be 20 digits to the right of the decimal place. What is the certain number?
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Context:
In the addition problem, each digit has been replaced by a letter.  If different letters represent different digits then what is the value of $C$?
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Context:
Sabrina went to the library and found a historical series novel called The Rangers Apprentice. There are 14 books in the series, and each book has 200 pages. She read four books in a month and a certain fraction of the number of books remaining in the second month. Sabrina has to read a total of 1000 pages to finish the whole series. What fraction of the remaining books did she read in the second month?
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Context:
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines?
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Context:
Given a sequence $\{a_{n}\}$ that satisfies $a_{1}+2a_{2}+4a_{3}+\ldots +2^{n-1}a_{n}=\frac{n}{2}$, rearrange the sequence $\{a_{n}\}$ into a new sequence as follows: $a_{1}$, $a_{2}$, $a_{2}$, $a_{2}$, $a_{3}$, $a_{3}$, $a_{3}$, $a_{3}$, $a_{3}$, $\ldots$, $\underbrace{{a_n},{a_n},…,{a_n}}_{({2n-1}) terms}$, $\ldots$. Find the sum of the first 70 terms of the new sequence.
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Context:
Calculate the variance of the difference between the maximum and minimum of n independent and identically distributed uniform random variables on the interval (0, 1).
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Context:
一次会议共有 24 名代表参加，每两名代表之间要么握手一次，要么不握手. 会议结束后发现，总共出现了 216 次握手，且对任意相互握过手的两名代表 P,Q，在剩下的 22 名代表中，恰与 P,Q 之一握过手的代表不超过 10 名. 一个"朋友圈"指的是会议中三名两两之间握过手的代表所构成的集合. 求这 24 名代表中朋友圈个数的最小值.
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Context:
The Mictlán is an  $n\times n$  board and each border of each  $1\times 1$  cell is painted either purple or orange. Initially, a catrina lies inside a  $1\times 1$  cell and may move in four directions (up, down, left, right) into another cell with the condition that she may move from one cell to another only if the border that joins them is painted orange. We know that no matter which cell the catrina chooses as a starting point, she may reach all other cells through a sequence of valid movements and she may not abandon the Mictlán (she may not leave the board). What is the maximum number of borders that could have been colored purple?
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Context:
In a convex pentagon \(ABCDE\), the diagonals \(BE\) and \(CE\) are the angle bisectors at vertices \(B\) and \(C\) respectively, \(\angle A = 35^{\circ}\), \(\angle D = 145^{\circ}\), and the area of triangle \(BCE\) is 11. Find the area of the pentagon \(ABCDE\).
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Context:
Consider a random permutation of the set  $\{1, 2, . . . , 2015\}$ . In other words, for each  $1 \le i \le 2015$ ,  $i$  is sent to the element  $a_i$  where  $a_i \in \{1, 2, . . . , 2015\}$  and if  $i \neq j$ , then  $a_i \neq a_j$ . What is the expected number of ordered pairs  $(a_i, a_j )$  with  $i - j > 155$  and  $a_i - a_j > 266$ ?
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Context:
14 bags of candy have an average weight rounded to the nearest tenth of a gram equal to 90.2 grams. Given that the weight of each bag of candy is an integer, find the total weight of these 14 bags of candy.
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Context:
In a survey on snoring and heart disease, a total of 2000 people were surveyed, and the calculated $\chi^2 = 20.87$. Given the critical values for the chi-squared statistic: 3.841 and 6.635, with corresponding confidence levels of 95% and 99%, determine the confidence level that snoring and heart disease are related based on the calculated chi-squared value. Express your answer as a percentage (%).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x, y, z, t$ be real numbers satisfying $x y+y z+z t+t x=1$. Find the minimum of the expression

$$
5 x^2+4 y^2+5 z^2+t^2 .
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A can give B a start of 50 meters or some seconds in a kilometer race. A takes 190 seconds to complete the race. How many seconds of a head start does A give B?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the set \( A \) formed by exponential functions, there are 10 odd functions, 8 increasing functions defined on \((-\infty, \infty)\), and 12 functions whose graphs pass through the origin. Determine the minimum number of elements in set \( A \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
An ice skating rink charges 5 dollars for admission and 2.50 dollars to rent skates. Jill can purchase a new pair of skates for 65 dollars. However, she also needs to consider the sales tax that applies to the purchase of new skates, which is 9%. If she purchases skates, she can only expect them to remain in good condition for a period of two years. Taking into account the cost and life expectancy of new skates as well as the rental fees, how many times would she need to go to the rink to justify buying the skates rather than renting a pair over a period of two years?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A smaller rectangle is made from three squares, each of area 25 cm², and is placed inside a larger rectangle. Two of the vertices of the smaller rectangle lie on the mid-points of the shorter sides of the larger rectangle, and the other two vertices of the smaller rectangle lie on the other two sides of the larger rectangle. Find the area of the larger rectangle. Express your answer in square centimeters (cm²) as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An industrial vehicle requires maintenance checks every 25 days. Three different protective additives, A, B, and C, can be applied to the vehicle, each increasing the time between maintenance checks as follows:

- Additive A increases the time by 10%.
- Additive B increases the time by 15%.
- Additive C increases the time by 5%.

Working conditions and the manufacturer's recommendations can also affect the time between maintenance checks:

- In harsh working conditions, time between maintenance checks must be reduced by 5%.
- The vehicle's manufacturer recommends increasing the time between maintenance checks by an additional 3% with the combined use of the additives.

Assuming that the vehicle is treated with all three additives in consecutive order and works under harsh conditions, by what percent is the time between maintenance checks increased in total, taking into account the manufacturer's recommendation?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Among the numbers $1, 2, 3, \cdots, 50$, if 10 consecutive numbers are selected, what is the probability that exactly 3 of them are prime numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知抛物线 \Omega:y=x^2，动线段 AB 在直线 y=\sqrt{3}x-3 上（B 在 A 右侧），且 |AB|=2\sqrt{3}. 过 A 作 \Omega 的切线，取左边的切点为 M. 过 B 作 \Omega 的切线，取右边的切点为 N. 当 MN\parallel AB  时，求点 A 的横坐标.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the plane rectangular coordinate system $xOy$, it is known that $MN$ is a chord of the circle $C: (x-2)^{2} + (y-4)^{2} = 2$, and satisfies $CM\perp CN$. Point $P$ is the midpoint of $MN$. As the chord $MN$ moves on the circle $C$, there exist two points $A$ and $B$ on the line $2x-y-3=0$, such that $\angle APB \geq \frac{\pi}{2}$ always holds. Find the minimum value of the length of segment $AB$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Passing through the origin of the coordinate plane are $180$ lines, including the coordinate axes,
which form $1$ degree angles with one another at the origin. Determine the sum of the x-coordinates
of the points of intersection of these lines with the line $y = 100-x$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given circle $C$: $(x-5)^{2}+(y+1)^{2}=m(m > 0)$, there is exactly one point on the circle that has a distance of $1$ to the line $4x+3y-2=0$. Determine the value of the real number $m$. Express your answer as a single real number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
某个国家有2023个岛和2022座桥，任意一座桥连接两个不同的岛，任意两个岛之间至多有一座桥相连，且可以从任意一个岛通过若干座桥到达其他任何岛。若三个岛中的某个岛与另两个岛都有桥连接，则称这三个岛组成“岛群”.已知任两个“岛群”中都有相同的岛，那么恰有一座桥的岛最少有多少个？
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of integers $n$ with $1 \leq n \leq 2017$ so that $(n-2)(n-0)(n-1)(n-7)$ is an integer multiple of 1001.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A positive integer is *bold* iff it has  $8$  positive divisors that sum up to  $3240$ . For example,  $2006$  is bold because its  $8$  positive divisors,  $1$ ,  $2$ ,  $17$ ,  $34$ ,  $59$ ,  $118$ ,  $1003$  and  $2006$ , sum up to  $3240$ . Find the smallest positive bold number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
求最大的正整数 m，使得存在集合 \{1,2,\cdots,100\} 的 m 个子集 A_1, A_2, \cdots, A_m，满足对任意 1\leq i<j\leq m，都有|A_i\cup A_j|-|A_i\cap A_j|=2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Part of an \(n\)-pointed regular star is shown. It is a simple closed polygon in which all \(2n\) edges are congruent, angles \(A_1,A_2,\cdots,A_n\) are congruent, and angles \(B_1,B_2,\cdots,B_n\) are congruent. If the acute angle at \(A_1\) is \(10^\circ\) less than the acute angle at \(B_1\), then \(n=\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
4.1. A train of length $L=600$ m, moving by inertia, enters a hill with an angle of inclination $\alpha=30^{\circ}$ and stops when exactly a quarter of the train is on the hill. What was the initial speed of the train $V$ (in km/h)? Provide the nearest whole number to the calculated speed. Neglect friction and assume the acceleration due to gravity is $g=10 \mathrm{m} /$ sec $^{2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If \( a \) and \( b \) are distinct prime numbers and \( a^2 - aQ + R = 0 \) and \( b^2 - bQ + R = 0 \), find the value of \( R \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using "学" and "习" to represent two different digits, the product of the four-digit numbers "学学学学" and "习习习习" is a seven-digit number, and its units and millions digits are the same as the digit represented by "学". How many possible two-digit numbers can "学习" represent?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a $7 \times 7$ grid, some squares are painted black and the rest are painted white, with at least one square of each color. A pattern is called "symmetric" if it looks the same when rotated $90^{\circ}$ and if it can be folded along either diagonal such that the two halves match. Find the total number of symmetric patterns. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using the concept of indicator random variables, find the expected value of the product of the number of red and black balls drawn from a box containing 15 green, 10 red, and 10 black balls, when 12 balls are drawn with replacement.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the expected number of times the pattern "21" appears in a sequence of iid random variables $(X_i)_{i\geq 1}$ with $X_i\sim X$, where $X\sim p_1\delta_1+p_2\delta_2+p_3\delta_3$, before seeing 3 for the first time.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of lattice points (including boundary points) inside a square with vertices at \( A(5,0) \), \( B(0,5) \), \( C(-5,0) \), and \( D(0,-5) \) in the Cartesian coordinate plane \( Oxy \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tim plans a weeklong prank to repeatedly steal Nathan's fork during lunch. He involves different people each day:
- On Monday, he convinces Joe to do it.
- On Tuesday, either Betty or John could undertake the prank.
- On Wednesday, there are only three friends from whom he can seek help, as Joe, Betty, and John are not available.
- On Thursday, neither those involved earlier in the week nor Wednesday's helpers are willing to participate, but four new individuals are ready to help.
- On Friday, Tim decides he could either do it himself or get help from one previous assistant who has volunteered again.
How many different combinations of people could be involved in the prank over the week?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $\triangle ABC$, where $3\sin^{2}B + 7\sin^{2}C = 2\sin A\sin B\sin C + 2\sin^{2}A$, find the value of $\sin(A + \frac{\pi}{4})$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\triangle ABC,\triangle BCD,$ and $\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$? Express your answer in the form $a + b\sqrt{3}$, where $a$ and $b$ are integers. Please provide the value of a + b.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $a$, $b$, $c$, $d$, $e$, and $f$ are integers such that $8x^3 + 64 = (ax^2 + bx + c)(dx^2 + ex + f)$ for all $x$, then what is $a^2 + b^2 + c^2 + d^2 + e^2 + f^2$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Petya and Vasya participated in the elections for the position of president of the chess club. By noon, Petya had 25% of the votes, and Vasya had 45%. After noon, only Petya's friends came to vote (and accordingly, only voted for him). As a result, Vasya ended up with only 27% of the votes. What percentage of the votes did Petya receive?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a set of data: $2$, $6$, $5$, $4$, $7$, $9$, $8$, $10$, what is the $80\%$ quantile of this data set?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Laurie loves multiplying numbers in her head. One day, she decides to multiply two 2-digit numbers $x$ and $y$ such that $x \leq y$ and the two numbers collectively have at least three distinct digits. Unfortunately, she accidentally remembers the digits of each number in the opposite order (for example, instead of remembering $51$ she remembers $15$). Surprisingly, the product of the two numbers after flipping the digits is the same as the product of the two original numbers. How many possible pairs of numbers could Laurie have tried to multiply?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The coefficients of the polynomial  $P(x)$  are nonnegative integers, each less than 100. Given that  $P(10) = 331633$  and  $P(-10) = 273373$ , compute  $P(1)$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square is drawn on a plane with its sides horizontal and vertical. Several line segments parallel to its sides are drawn inside the square, such that no two segments lie on the same line and do not intersect at a point that is internal for both segments. The segments divide the square into rectangles, with each vertical line crossing the square and not containing any of the segments intersecting exactly $k$ rectangles, and each horizontal line crossing the square and not containing any of the segments intersecting exactly $l$ rectangles. What could be the number of rectangles in the partition?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $0 \leq \alpha \leq \pi$. Consider the sequence $u_n$ where $u_n = \cos(n\alpha)$. Denote by $V_n(\alpha)$ the count of sign changes in this sequence. Determine the limit of the ratio $\frac{V_n(\alpha)}{n}$ as $n$ approaches infinity, given that the sign changes in $u_n$ are irregular.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tim buys a book of esoteric words. He decides to learn 10 words a day, but only on weekdays, as he takes weekends off for leisure. In 2 years, with occasional vacations (30 days in total) during which he doesn't learn new words, the book has increased the number of words he knows by 50%. Additionally, Tim forgets 2 words per week. How many words did he know originally?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a $k$-element subset $T$ of the set $\{1,2,\cdots,242\}$, every pair of elements (which may be the same) in $T$ has a sum that is not an integer power of 3. Find the maximum value of $k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An isosceles triangle with a base \( a \) and a base angle \( \alpha \) is inscribed in a circle. Additionally, a second circle is constructed, which is tangent to both of the triangle's legs and the first circle. Find the radius of the second circle.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\frac{a}{2 \sin \alpha (1 + \cos \alpha)}}


Context:
The least common multiple of a positive integer $n$ and $18$ is $180$, and the greatest common divisor of $n$ and $45$ is $15$. What is the sum of the digits of $n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $$f(x)=\sin(x)+\sqrt3 \cos(x)$$ Find the real number $m$ such that $$f'(x)+f''(x)=\ln(m^2-4m+5)$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Which quadrant does the line \( l: ax + by + c = 0 \) not pass through when \( ab < 0 \) and \( bc < 0 \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the symmetry center of the function $y=\cos (2x-\frac{\pi }{4})$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the remainder when the value of $m$ is divided by 1000 in the number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_6 \le 1500$ such that $a_i-i$ is odd for $1\le i \le 6$. The total number of sequences can be expressed as ${m \choose n}$ for some integers $m>n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
During a unique three-team soccer game, in the first half, Team A scored 8 points, Team B scored half as many as Team A, and Team C scored twice as many as Team B. Additionally, Team A missed one penalty kick, and Team C missed two penalty kicks. In the second half, Team A scored the same number of goals as Team C did in the first half, Team B scored as many goals as Team A in the first half, and Team C scored three more goals than Team B in the second half. Team A scored one successful penalty kick, while Team B scored two successful penalty kicks. How many goals did all three teams score during the whole match, including the penalty kicks?
Please reason step by step, and put your final answer within \boxed{}.


Context:
实数$x _ { 1 } , x _ { 2 } , \cdots , x _ { 2 0 0 1 }$满足$\sum _ { k = 1 } ^ { 2 0 0 0 } \vert x _ { k } - x _ { k + 1 } \vert = 2 0 0 1 ,$，令$y_ { k }= \frac { 1 } { k } ( x _ { 1 } + x _ { 2 } + \cdots + x _ { k } ) , k = 1 ,$,2,⋯,2001,，求$\sum _ { k = 1 } ^ { 2 0 0 0 } \vert y _ { k } - y _ { k + 1 } \vert$的最大可能值．
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the decomposition rate $v$ of a certain type of garbage approximately satisfies the relationship $v=a\cdot b^{t}$, where $a$ and $b$ are positive constants, and the decomposition rate is $5\%$ after $6$ months and $10\%$ after $12$ months, calculate the time it takes for this type of garbage to completely decompose.
Please reason step by step, and put your final answer within \boxed{}.


Context:
8. For the cubic function $y=x^{3}+a x^{2}+b x+c$, its graph intersects the $x$-axis at points $A, T$, and $B$ in sequence. The lines $A P$ and $B Q$ are tangent to the cubic curve at points $P$ and $Q$ (where $P$ does not coincide with $A$, and $Q$ does not coincide with $B$). Then the ratio of the projections of $\overrightarrow{A B}$ and $\overrightarrow{P Q}$ on the $x$-axis is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cylinder with a volume of 9 is inscribed in a cone. The plane of the top base of this cylinder cuts off a frustum from the original cone, with a volume of 63. Find the volume of the original cone.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Dacid obtained 72, 45, 72, 77, and 75 marks (out of 100) in English, Mathematics, Physics, Chemistry, and Biology, respectively. Given that each subject has a different credit hour value - English: 3 credit hours, Mathematics: 4 credit hours, Physics: 4 credit hours, Chemistry: 3 credit hours, and Biology: 2 credit hours - calculate his weighted average marks.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the decimal digits of \[ \left\lfloor \frac{51525354555657\dots979899}{50} \right\rfloor. \] Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let
$$
S=\sum_{r=0}^{n}\binom{3 n+r}{r}
$$

Evaluate \( S \div (23 \times 38 \times 41 \times 43 \times 47) \) when \( n = 12 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the function $f(x)$ is an even function, and shifting its graph one unit to the right results in an odd function, and $f(2)=-1$, calculate the sum of $f(1)+f(2)+...+f(2017)$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
if the function q is defined by the formula q = 5w^2 / (4d^2 (z^3 + x^2)), by what factor will q be multiplied if w is quadrupled, d is doubled, z is tripled, and x is halved?
Please reason step by step, and put your final answer within \boxed{}.


Context:
集合$\{ x \vert - 1 \leq \log _ { \frac { 1 } { x } } 1 0 < - \frac { 1 } { 2 } , x \in N _ { + } \}$的真子集的个数是$\underline{\hspace{1cm}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 7 balls of each of the three colors: red, blue, and yellow. Each set of 7 balls of the same color is numbered from 1 to 7. When randomly selecting 3 balls with different numbers, find the number of ways to pick such that the 3 balls are of different colors and their numbers are not consecutive.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many distinct functions $f: \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \to \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ are there such that $(f \circ f)(1) = 2$ and $f$ is onto (surjective)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can two distinct squares be chosen from an $8 \times 8$ chessboard such that the midpoint of the line segment connecting their centers is also the center of a square on the board?
Please reason step by step, and put your final answer within \boxed{}.


Context:
1021. Among all rectangular parallelepipeds having a given volume $\mathrm{V}$, find the one with the smallest total surface area.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $s(m)$ denote the sum of the digits of the positive integer $m$. Find the largest positive integer that has no digits equal to zero and satisfies the equation \[2^{s(n)} = s(n^2).\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. Find the value of $AR+BR+CR+DR$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There is a $6 \times 6$ grid of lights. There is a switch at the top of each column and on the left of each row. A light will only turn on if the switches corresponding to both its column and its row are in the "on" position. Compute the number of different configurations of lights.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let f(x) be a non-decreasing function on the interval [0, 1], meaning that for any 0 ≤ x1 ≤ x2 ≤ 1, we have f(x1) ≤ f(x2). The function satisfies the following conditions:
1. f(0) = 0;
2. f(x/3) = (1/2)f(x);
3. f(1-x) = 1 - f(x).

Find the value of f(1/2016). Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
若存在正整数 a,b，使得 \df{k}{2023}(a+b)=[a,b]，则整数 k 的个数为__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For each positive integer $n$, let $H_n = \frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n}$. If \[ \sum_{n=4}^{\infty} \frac{1}{nH_nH_{n-1}} = \frac{M}{N} \] for relatively prime positive integers $M$ and $N$, compute $100M+N$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an isosceles triangle \( ABC \) with \(\angle B = \arctan \frac{8}{15}\), a circle with radius 1 is inscribed in angle \( C \) and touches side \( CB \) at point \( M \) and cuts off a segment \( KE \) from the base. It is known that \( MB = \frac{15}{8} \). Find the area of triangle \( KMB \), given that points \( A, K, E, B \) follow in the mentioned order along the base \( AB \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $a,$ $b,$ $x,$ and $y$ are real numbers such that $ax+by=3,$ $ax^2+by^2=7,$ $ax^3+by^3=16,$ and $ax^4+by^4=42,$ find $ax^5+by^5.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
liam is pulled over for speeding just as he is arriving at work . he explains to the police officer that he could not afford to be late today , and has arrived at work only four minutes before he is to start . the officer explains that if liam had driven 5 mph slower for his whole commute , he would have arrived at work exactly on time . if liam ' s commute is 40 miles long , how fast was he actually driving ? ( assume that liam drove at a constant speed for the duration of his commute . )
Please reason step by step, and put your final answer within \boxed{}.


Context:
A triangular array of numbers has a first row consisting of the odd integers $1,3,5,\cdots,99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it. How many entries in the array are multiples of $67$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a group of 500 students, each student is registered for at least one of 5 classes - History, Math, English, Science, and Geography. 120 students are registered for History, 105 students are registered for Math, 145 students are registered for English, 133 students are registered for Science, and 107 students are registered for Geography. If 15 students are registered for all 5 classes, 40 students are registered for both History and Math, 35 students are registered for both English and Science, and 25 students are registered for both Math and Geography, how many students are registered for exactly two classes?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the minimum degree of the Lagrange polynomial required to approximate $f(x) = \sin(x)$ in the interval $[0, \pi]$ such that the error, $| \sin(x) - L_n(x) |$, is less than or equal to $0.001$? In other words, given the Lagrange interpolation error formula:
$$| \sin(x) - L_n(x) | = \left| \frac{\sin^{(n+1)}(c)}{(n+1)!} \right| (x - x_0)(x - x_1) \cdots (x - x_n) \leq 0.001, \quad 0 \leq c \leq \pi$$
How do you determine $n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
At a party, there are $2011$ people, each with a glass of fruit juice, sitting around a circular table. Every second, they clink glasses according to the following two rules:
1. They do not clink glasses crosswise.
2. At each point in time, everyone can clink glasses with at most one other person.

How many seconds are needed, at a minimum, for everyone to have clinked glasses with everyone else?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest positive integer $n$ such that the following statement holds:

If $a_1, a_2, a_3, a_4, a_5, a_6$ are six distinct positive integers less than or equal to $n$, then there exist three distinct positive integers from these six, say $a$, $b$, and $c$, such that $ab > c$, $bc > a$, and $ca > b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
John Smith buys 3 cakes for $12 each and a pack of cookies for $5. He has a 10% discount coupon off the total cost and there is a 5% sales tax applied. After applying the discount and tax, he splits the cost with his brother. How much did each brother pay?
Please reason step by step, and put your final answer within \boxed{}.


Context:
6. Ms. Olga Ivanovna, the class teacher of 5B, is staging a "Mathematical Ballet." She wants to arrange the boys and girls so that exactly 2 boys are 5 meters away from each girl. What is the maximum number of girls that can participate in the ballet if it is known that 5 boys are participating?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit, if it exists, of the expression $$\lim_{x\to 0}\left[\left[\log \left(1+\frac{1}{2x}\right)\right]^2-\left[\log \left(1+\frac{1}{x}\right)\right]^2\right].$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the inverse function of the given function $f: \mathbb{R} \to (0,2)$ defined by $f(x) = \frac{e^x-e^{-x}}{e^x+e^{-x}} + 1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a park, there are 2 maple trees and 5 poplar trees currently. Today, park workers will plant 3 times the current number of poplar trees as maple trees. If there are also 4 oak trees in the park, how many maple, poplar, and oak trees will the park have in total when the workers are finished planting today?
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Context:
For which natural numbers \( n \) can a board of size \( n \times n \) be divided along the grid lines into dominos of size \( 1 \times 2 \) so that the number of vertical and horizontal dominos is equal?
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Context:
Given two sequences $\{a_n\}$ and $\{b_n\}$ satisfying $a_1= \frac{1}{2}$, $a_n+b_n=1$, $b_{n+1}= \frac{b_n}{1-a_n^2}$, for $n\in\mathbb{N}_+$, find the value of $b_{2016}$.
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Context:
72 vertices of a regular 3600-gon are painted red in such a way that the painted vertices form the vertices of a regular 72-gon. In how many ways can 40 vertices of this 3600-gon be chosen so that they form the vertices of a regular 40-gon and none of them are red?
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Context:
Given real numbers \(x\) and \(y\) satisfying \(x^{2}+(y-2)^{2} \leq 1\), determine the range of values for \(\frac{x+\sqrt{3} y}{\sqrt{x^{2}+y^{2}}}\).
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Context:
There are 20 points on a plane, and each pair of points is connected by a segment (edges) - each point with every other. Then, 35 edges are randomly selected and removed. Find the probability that the resulting graph is connected (i.e., it is possible to reach any point from any other point via edges).
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Context:
A function $f:\{1,2, \ldots, n\} \to \{1, \ldots, m\}$ is [i]multiplication-preserving[/i] if $f(i)f(j) = f(ij)$ for all $1 \le i \le j \le ij \le n$, and [i]injective[/i] if $f(i) = f(j)$ only when $i = j$. For $n = 9, m = 88$, the number of injective, multiplication-preserving functions is $N$. Find the sum of the prime factors of $N$, including multiplicity. (For example, if $N = 12$, the answer would be $2 + 2 + 3 = 7$.)
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Context:
Given the sets \( A = \{(x, y) \mid |x| + |y| = a, a > 0\} \) and \( B = \{(x, y) \mid |xy| + 1 = |x| + |y| \} \), if the intersection \( A \cap B \) is the set of vertices of a regular octagon in the plane, determine the value of \( a \).
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Context:
In $\triangle ABC$, $\angle A = 42^{\circ}$. The trisectors of $\angle B$ and $\angle C$ meet at points $D$ and $E$ respectively. Find the measure of $\angle BDC$. Express your answer in degrees as a whole number.
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Context:
Example 7 $n$ people greet each other by phone during a holiday, it is known that each person has called at most 3 of their friends; any 2 people have had at most 1 phone call; and among any 3 people, at least 2 of them, one has called the other. Find the maximum value of $n$. (Original problem)
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Context:
把某公司的 $9$ 男 $3$ 女共 $12$ 名员工平均分配到甲、乙、丙三个部门。求甲、乙、丙三个部门各有 $1$ 名女员工的概率.用分数表示，写出分子加分母的值。
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Context:
Given that the even function $f(x)$ satisfies $f(x-2) = -f(x)$ for all $x \in \mathbb{R}$, and when $x \in [-1, 0]$, $f(x) = 2^x$. Find the value of $f(2015)$.
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Context:
Dima needed to arrive at the station at 18:00, where his father was supposed to pick him up by car. However, Dima caught an earlier train and arrived at the station at 17:05. Instead of waiting for his father, he started walking towards him. They met on the way, Dima got into the car, and they arrived home 10 minutes earlier than the calculated time. What was Dima's walking speed before he met his father, if the car's speed was $60 \ \text{km/h}$?
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Context:
Given that the function f(x) = 4x^2 - kx - 8 has monotonicity on the interval (∞, 5], determine the range of values for the real number k.
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Context:
A number x is 12 times another number y, and the third number z is 20% greater than x. Find the percentage that y is less than z.
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Context:
There is a basket of apples. After dividing them into three equal parts, 2 apples remain. Taking out two of these parts, and dividing them into three equal parts again, 2 apples remain. After taking out two of these parts again and dividing them into three equal parts, 2 apples remain. How many apples are in the basket, at minimum?
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Context:
Evaluate the integral $$\int_0^\infty e^{-ay^{2}-\frac{b}{y^2}}dy $$ where a and b are real and positive.
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Context:
已知曲线 $C_{1}: y=\sqrt{-x^{2}+10 x-9}$ 及点 $A(1,0)$. 若曲线 $C_{1}$ 上存在相异的两点 $B 、 C$,其到直线 $l: 3 x+1=0$ 的距离分别为 $|A B|$ 、 $|A C|$, 则 $|A B|+|A C|=$
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Context:
A total of  $n$  people compete in a mathematical match which contains  $15$  problems where  $n>12$ . For each problem,  $1$  point is given for a right answer and  $0$  is given for a wrong answer. Analysing each possible situation, we find that if the sum of points every group of  $12$  people get is no less than  $36$ , then there are at least  $3$  people that got the right answer of a certain problem, among the  $n$  people. Find the least possible  $n$ .
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Context:
若 T=\ds{k=1}{100}2^{99+k}\cdot3^{101-k}，则 T 的末尾数字 0 的个数为__________.
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Context:
Determine the maximum number of distinct positive integer solutions for the equation $x(x+1)(x+2)=ay^b$, where $a,b$ are positive integers and $b\ge3$.
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Context:
There are 20 students in the class. Each boy has made a list of all the girls in the class they find attractive (the list may be empty). It is known that there are no three boys whose lists contain the same number of girls. What is the minimum number of girls that could be in the class?
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Context:
A bamboo pole of length 24 meters is leaning against a wall with one end on the ground. If the vertical distance from a certain point on the pole to the wall and the vertical distance from that point to the ground are both 7 meters, find the vertical distance from the end of the pole touching the wall to the ground in meters, or $\qquad$ meters.
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Context:
The origin coincides with the pole, and the positive $x$-axis coincides with the polar axis. What are the polar coordinates of the point $(-5-5\sqrt{3})$?
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Context:
A canister contains $2\frac{1}{2}$ cups of flour. Greg and Sally have a brownie recipe that calls for $1\frac{1}{3}$ cups of flour. They want to make $1\frac{1}{2}$ recipes of brownies. To the nearest whole percent, what percent of the flour in the canister would they use?
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Context:
If the following limit $$\lim_{x\to 0}\frac{\cos^2x-\cos x-e^x\cos x+e^x-\frac{x^3}{2}}{x^n}$$ is a non zero finite number, find $n$ where $n\in\mathbb{N}$
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Context:
Let $ABC$ be an equilateral triangle. Let $P$ and $S$ be points on $AB$ and $AC$, respectively, and let $Q$ and $R$ be points on $BC$ such that $PQRS$ is a rectangle. If $PQ = \sqrt{3} \cdot PS$ and the area of $PQRS$ is $28\sqrt{3}$, what is the length of $PC$?
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Context:
In the rectangular parallelepiped \(ABCD A_1 B_1 C_1 D_1\), two diagonals of the side faces \(A_1 C_1\) and \(C_1 D\) are drawn. Find the angle between them, given that the diagonal \(A_1 C_1\) is equal in length to one of the edges of the parallelepiped, and the diagonal \(C_1 D\) forms a 30-degree angle with the same edge.
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Context:
Two identical equilateral triangles can form a rhombus. If the side length of the equilateral triangle is 10, what is the area of the largest square inside this rhombus?
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Context:
Evaluate the limit $$\lim_{n \rightarrow \infty } n \sqrt[n]{n^{-n+1}+n^{-n+2}+\cdots+1} \cdot \left(1-\sin{\frac{2}{n}}\right).$$
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Context:
Let $a_0 = 2,$ $b_0 = 3,$ and
\[a_{n + 1} = \frac{a_n^2}{b_n} \quad \text{and} \quad b_{n + 1} = \frac{b_n^2}{a_n}\]for all $n \ge 0.$  Then $b_8 = \frac{3^m}{2^n}$ for some integers $m$ and $n.$  Enter the ordered pair $(m,n).$
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Context:
$P$为边长为$a$的正三角形$ABC$所在平面外一点且$PA=PB=PC=a$，则$P$到$AB$的距离为_____.
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Context:
Let $p$ be a prime number such that $p \equiv 3 \pmod{4}$. Determine the remainder when the sum $\sum_{i=1}^{p-1}\left(\frac{i^4+1}{p}\right)$ is divided by $p$, where $\left(\frac{i^4+1}{p}\right)$ denotes the Legendre symbol. Use the fact that if $p-1 \nmid k$, then $\sum_{i=1}^{p-1}i^k \equiv 0 \pmod{p}$.
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Context:
设函数 f(x)=(x^2+ax+1)\ln x，过点 (1,0) 作其切线 y=g(x). 若对任意 x\geq 1，均有 f(x)\geq g(x)，则实数 a 的最小可能值为__________.
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Context:
Given the group $G = \mathbb{Z}_{100} \times \mathbb{Z}_{500}$, how many subgroups $H$ of $G$ are isomorphic to $\mathbb{Z}_{25} \times \mathbb{Z}_{25}$?
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Context:
Alice and Bob are each secretly given a real number between 0 and 1 uniformly at random. Alice states, “My number is probably greater than yours.” Bob repudiates, saying, “No, my number is probably greater than yours!” Alice concedes, muttering, “Fine, your number is probably greater than mine.” If Bob and Alice are perfectly reasonable and logical, what is the probability that Bob’s number is actually greater than Alice’s?
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Context:
Find the minimum of \( \sum\limits_{k=0}^{40} \left(x+\frac{k}{2}\right)^2 \) where \( x \) is a real number.
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Context:
We draw circles over the legs of an isosceles right-angled triangle, which intersect at point $D$ on the hypotenuse. What is the common area determined by the circles if each of the legs is $a$?
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Context:
The line segments connecting an internal point of a convex, non-equilateral $n$-gon with its vertices divide the $n$-gon into $n$ equal triangles.

What is the smallest $n$ for which this is possible?
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Context:
Given a circular park 20 feet in diameter with a straight walking path 5 feet wide passing through its center, calculate the area of the park covered by grass after the path is laid.
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Context:
There are two libraries, A and B, involved in an inter-library loan program. Library A has a unique collection of 150 books, all of which were in Library A at the start of the year. Library B also loans out a certain number of books to Library A. Throughout the year, books are loaned out between the libraries. By the end of the year, 60 percent of the books loaned out from Library A's unique collection are returned, and there are only 100 books from the unique collection in Library A. Additionally, it is known that 30 percent of the loaned books from Library A's unique collection were returned within the same year. During the year, Library B had loaned 20 books to Library A, and 50 percent of them were returned at the end of the year. Calculate the total number of books from Library A's unique collection that were loaned out during that year.
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Context:
A sum was put at simple interest at a certain rate for 10 years. Had it been put at a higher rate, it would have fetched Rs. 600 more. The sum was Rs. 1200. What was the higher rate percentage?
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Context:
2016 students are lined up in a row and count off from left to right according to $1,2 \cdots, n$ $(n \geqslant 2)$. If the 2016th student counts to $n$, all students who count to $n$ in this round will receive a New Year's gift. How many students will definitely not receive a New Year's gift, regardless of what $n$ is?
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Context:
9. Find the maximum value of the volume of a regular quadrilateral frustum with the upper base edge and side edge both equal to 1 (accurate to 0.001).

Note: If the upper base area of the regular quadrilateral frustum is $S_{1}$, the lower base area is $S_{2}$, and the height is $h$, then its volume
$$
V=\frac{1}{3} h\left(S_{1}+S_{2}+\sqrt{S_{1} S_{2}}\right) .
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知$x$，$y$，$z$为正数，且满足等式$x y z ( x + y + z ) = 1$，求表达式$( x + y ) ( y + z )$的最小值.
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Context:
定义在 $\mathbf{R}$ 上的函数 $f(x)$ 满足 $f(0)=0, f(x)+f(1-x)=1, f\left(\frac{x}{5}\right)=\frac{1}{2} f(x)$ ，且当 $0 \leqslant x_{1}<x_{2} \leqslant 1$ 时, $f\left(x_{1}\right) \leqslant f\left(x_{2}\right)$. 求 $f\left(\frac{1}{2007}\right)$ 的值。答案的形式为\frac{k}{m}，请给出k+m的值。
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Context:
Li Ming goes home for lunch after school and cooks noodles for himself, which includes the following steps:
① Boil the noodles for 4 minutes;
② Wash vegetables for 5 minutes;
③ Prepare the noodles and condiments for 2 minutes;
④ Boil the water in the pot for 10 minutes;
⑤ Wash the pot and add water for 2 minutes.
Apart from step ④, only one step can be performed at a time. Calculate the minimum number of minutes Li Ming needs to cook the noodles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
12. On the highway between $\mathrm{A}$ and $\mathrm{B}$, the speed limit for vehicles traveling from $\mathrm{A}$ to $\mathrm{B}$ is 50 kilometers per hour, and for vehicles traveling from $\mathrm{B}$ to $\mathrm{A}$ is 60 kilometers per hour. Now, two cars, Car A and Car B, start simultaneously from $\mathrm{A}$ and $\mathrm{B}$ respectively, and travel back and forth between the two places. When Car A reaches $\mathrm{B}$ and is on its way back to $\mathrm{A}$, it stops due to a malfunction, and the stopping point is 30 kilometers from $\mathrm{B}$. The two cars meet for the second time at this location, which is 1 hour and 12 minutes later than the originally scheduled second meeting time. What is the distance between the two places? $\qquad$ kilometers.
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Context:
Find the maximum possible value obtainable by inserting a single set of parentheses into the expression $1 + 2 \times 3 + 4 \times 5 + 6$.
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Context:
Consider the sum   $$ S =\sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|. $$  The value of  $S$  can be written as  $\tan \left( \frac{c\pi}{d} \right)$  for some relatively prime positive integers  $c, d$ , satisfying  $2c < d$ . Find the value of  $c + d$ .
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Context:
3. A student wrote a program for recoloring a pixel into one of 128 different colors. These colors he numbered with natural numbers from 1 to 128, and the primary colors received the following numbers: white color - number 1, red - 5, orange - 13, yellow - 21, green - 45, blue - 75, dark blue - 87, purple - 91, black - 128. If the initial color of the pixel has a number \( n \leq 19 \), then the student's program recolors it to the color with the number \( n+4 \), and if the initial color of the pixel has a number \( n \geq 20 \), then the pixel is recolored to the color with the number \( |129-2n| \). Initially, the pixel was red. The student applied his program 2019 times sequentially. What color did the pixel end up as a result?
Please reason step by step, and put your final answer within \boxed{}.