Context:
不等式$({{m}^{2}}-2m-3){{x2}}-(m-3)x-1<0$的解集为$R$，求实数$m$的取值范围
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\boxed{$\left( -\frac{1}{5} \right.,\left. 3 \right]$}


Context:
Given the line $x-my-1=0$ intersects the circle $(x-1)^{2}+(y-2)^{2}=4$ at points $A$ and $B$, and the length of chord $AB$ is $2\sqrt{3}$, find the value of the real number $m$.
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Context:
10.301. Find the ratio of the area of a square inscribed in a segment with an arc of $180^{\circ}$ to the area of a square inscribed in a segment of the same circle with an arc of $90^{\circ}$.
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Context:
Find the probability that the minimum of three independent exponential random variables with the same rate is less than or equal to the sum of the other two.
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Context:
For any $x \in \mathbb{R}$, the inequality $|x+5| \geq m+2$ always holds. Find the range of values for the real number $m$ that satisfy this inequality. Express your answer as an inequality involving $m$.
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Context:
Cards numbered 1 and 2 must be placed into the same envelope, and three cards are left to be placed into two remaining envelopes, find the total number of different methods.
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Context:
Let \( A = \{1, 2, \ldots, 2002\} \) and \( M = \{1001, 2003, 3005\} \). For any non-empty subset \( B \) of \( A \), \( B \) is called an \( M \)-free set if the sum of any two elements in \( B \) does not belong to \( M \). If \( A = A_1 \cup A_2 \), \( A_1 \cap A_2 = \varnothing \), and both \( A_1 \) and \( A_2 \) are \( M \)-free sets, then the ordered pair \(\left(A_{1}, A_{2}\right)\) is called an \( M \)-partition of \( A \). Find the number of all \( M \)-partitions of \( A \).
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Context:
A parallelogram $ABCD$ is inscribed in an ellipse $\frac{x^2}{4}+\frac{y^2}{2}=1$. The slope of line $AB$ is $k_1=1$. Calculate the slope of line $AD$.
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Context:
The bases \( AB \) and \( CD \) of trapezoid \( ABCD \) are 65 and 31, respectively, and its diagonals are mutually perpendicular. Find the dot product of vectors \( \overrightarrow{AD} \) and \( \overrightarrow{BC} \).
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Context:
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors and satisfy $\overrightarrow{a} \cdot \overrightarrow{b} = 0$, find the maximum value of $(\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}) \cdot (\overrightarrow{a} + \overrightarrow{c})$.
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Context:
In triangle $ABC$, the angle bisectors $AA_{1}$ and $BB_{1}$ intersect at point $O$. Find the ratio $AA_{1} : OA_{1}$ given $AB=6, BC=5$, and $CA=4$.
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Context:
Sheena is sewing the bridesmaid's dresses for her sister's wedding. She can sew one dress in 15 hours. There are 7 bridesmaids in the wedding. Each bridesmaid's dress takes a different number of hours to sew due to different styles and sizes. The hours needed to sew the bridesmaid's dresses are as follows: 15 hours, 18 hours, 20 hours, 22 hours, 24 hours, 26 hours, and 28 hours. If Sheena sews the dresses for various hours each week as follows: 5 hours in the first week, 3 hours in the second week, 6 hours in the third week, and 4 hours in the fourth week, and then continues by repeating this cycle. Additionally, she needs to spend 10 hours to finalize the dresses by adding accessories and decorations. How many weeks will it take her to complete all the bridesmaid's dresses, including the additional tasks?
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Context:
Find an asymptotic approximation for the number of ways to partition a set of $2N$ distinct elements into $N$ pairs.
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Context:
Five children (Tian Tian, Shi Tou, Kimi, Cindy, Angela) need to switch fathers, with each child able to choose any one of the other four fathers except their own. Find the total number of different combinations of choices for the five children. Express your answer as a single integer.
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Context:
Define $K(n,0)=\varnothing $ and, for all nonnegative integers m and n, $K(n,m+1)=\left\{ \left. k \right|\text{ }1\le k\le n\text{ and }K(k,m)\cap K(n-k,m)=\varnothing  \right\}$.  Find the number of elements of $K(2004,2004)$.
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Context:
设函数 \varphi_1(x),\varphi_2(x) 满足\varphi_i''(x)-\ln(x+1)\varphi_i'(x)+e^x\varphi_i(x)=0,i=1,2,x>-1.定义 w(x)=\varphi_1'(x)\varphi_2(x)-\varphi_1(x)\varphi_2'(x)，若 w(0)=1，则 [w(2)]=__________.
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Context:
5. The volume of tetrahedron $ABCD$ is $\frac{1}{6}$, and it satisfies $\angle ACB=45^{\circ}, AD+BC+\frac{AC}{\sqrt{2}}=3$, then $CD=$
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Context:
Let \( N \) be the positive integers. The function \( f : N \to N \) satisfies \( f(1) = 5 \), \( f(f(n)) = 4n + 9 \), and \( f(2n) = 2n+1 + 3 \) for all \( n \). Find \( f(1789) \).
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Context:
How many ways are there to fill in a $3 \times 3$ grid of cells with $0$'s and $2$'s, one number in each cell, such that each $2 \times 2$ contiguous subgrid contains exactly three $2$'s and one $0$?
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Context:
There are 99 children standing in a circle, each initially holding a ball. Every minute, each child with a ball throws their ball to one of their two neighbors. If a child receives two balls, one of the balls is irrevocably lost. What is the minimum amount of time after which only one ball can remain with the children?
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Context:
The principal of a certain school decided to take a photo of the graduating class of 2008. He arranged the students in parallel rows, all with the same number of students, but this arrangement was too wide for the field of view of his camera. To solve this problem, the principal decided to take one student from each row and place them in a new row. This arrangement displeased the principal because the new row had four students fewer than the other rows. He then decided to take one more student from each of the original rows and place them in the newly created row, and noticed that now all the rows had the same number of students, and finally took his photo. How many students appeared in the photo?
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Context:
Given the set $I = \{x \mid 1 < x < 5, x \in \mathbb{N}\}$ and set $A = \{2, 3\}$, calculate $C_{I} \setminus A$.
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Context:
■ Example 2 For non-negative real numbers $x_{1}$, $x_{2}, \cdots, x_{n}$ satisfying $x_{1}+x_{2}+\cdots+x_{n}=1$, find the maximum value of $\sum_{j=1}^{n}\left(x_{j}^{4}-x_{j}^{5}\right)$. (40th IMO China National Team Selection Exam Question)
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Context:
There is a fence that consists of  $n$  planks arranged in a line. Each plank is painted with one of the available  $100$  colors. Suppose that for any two distinct colors  $i$  and  $j$ , there is a plank with color  $i$  located to the left of a (not necessarily adjacent) plank with color  $j$ . Determine the minimum possible value of  $n$ .
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Context:
Jackson, Brandon, and Meagan all invest in the stock market in different currencies. Jackson invests 500 USD, Brandon invests 600 CAD, and Meagan invests 400,000 JPY. Jackson's investment quadruples in value, Brandon's is reduced to 20% of its initial value, and Meagan's investment increases by 50%.

During this time, the exchange rate changes: 1 USD becomes equivalent to 1.2 CAD, and 1 USD becomes equivalent to 110 JPY. Considering these changes in exchange rates as well, how much more money in USD does Jackson have than Brandon and Meagan combined now?
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Context:
Triangle $ABC$ has $\angle A = 90^\circ$, $\angle C = 30^\circ$, and $AC = 12$. Let the circumcircle of this triangle be $W$. Define $D$ to be the point on arc $BC$ not containing $A$ so that $\angle CAD = 60^\circ$. Define points $E$ and $F$ to be the foots of the perpendiculars from $D$ to lines $AB$ and $AC$, respectively.

Let $J$ be the intersection of line $EF$ with $W$, where $J$ is on the minor arc $AC$. The line $DF$ intersects $W$ at $H$ other than $D$. The area of the triangle $FHJ$ is in the form $\frac{a}{b}(\sqrt{c} - \sqrt{d})$ for positive integers $a, b, c, d$, where $a, b$ are relatively prime, and the sum of $a, b, c, d$ is minimal.

Find $a + b + c + d$.
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Context:
(11) (15 points) Divide each side of the equilateral $\triangle A B C$ with side length 3 into three equal parts, and draw lines parallel to the other two sides through each division point. The 10 points formed by the intersections of the sides of $\triangle A B C$ and these parallel lines are called grid points. If $n$ grid points are chosen from these 10 grid points, there will definitely be three grid points that can form an isosceles triangle (including equilateral triangles). Find the minimum value of $n$.
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Context:
Given real numbers $b_0, b_1, \dots, b_{2019}$ with $b_{2019} \neq 0$, let $z_1,z_2,\dots,z_{2019}$ be the roots in the complex plane of the polynomial \[ P(z) = \sum_{k=0}^{2019} b_k z^k. \] Let $\mu = (|z_1| + \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\dots,z_{2019}$ to the origin. Determine the largest constant $M$ such that $\mu \geq M$ for all choices of $b_0,b_1,\dots, b_{2019}$ that satisfy \[ 1 \leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019. \]
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Context:
What is the radical conjugate of $2\sqrt{7} - 1$?
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Context:
Michelle has a word with $2^{n}$ letters, where a word can consist of letters from any alphabet. Michelle performs a switcheroo on the word as follows: for each $k=0,1, \ldots, n-1$, she switches the first $2^{k}$ letters of the word with the next $2^{k}$ letters of the word. In terms of $n$, what is the minimum positive integer $m$ such that after Michelle performs the switcheroo operation $m$ times on any word of length $2^{n}$, she will receive her original word?
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Context:
Two vertical chords are drawn in a circle, dividing the circle into 3 distinct regions. Two horizontal chords are added in such a way that there are now 9 regions in the circle. A fifth chord is added that does not lie on top of one of the previous four chords. The maximum possible number of resulting regions is \( M \) and the minimum possible number of resulting regions is \( m \). What is \( M^{2} + m^{2} \)?
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Context:
Find all real values of $x$ that satisfy \(\frac{1}{x(x+1)} - \frac{1}{(x+1)(x+2)} < \frac{1}{4}\). (Give your answer in interval notation.)
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Context:
Xiaoming bought 4 apples, 6 pears, and 8 peaches, while Xiaohong bought 5 apples, 7 pears, and 6 peaches. Over the next 18 days, each of them eats one fruit per day. On three days, both of them eat apples; on two days, both eat pears; and on three days, one eats an apple while the other eats a pear. How many days do both of them eat peaches?
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Context:
Given a pair of standard $8$-sided dice is rolled once. The sum of the numbers rolled, if it is a prime number, determines the diameter of a circle. Find the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference.
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Context:
Two distinct natural numbers end in 5 zeros and have exactly 42 divisors. Find their sum.
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Context:
How many real solutions are there to the equation
\[ |||| x|-2|-2|-2|=|||| x|-3|-3|-3| ? \]
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Context:
Let $V$ be a finite-dimensional vector space over the complex numbers $\mathbb{C}$, and let $T: V \rightarrow V$ be a linear transformation with a minimal polynomial of $x^{13}(x-1)$. What is the minimal polynomial of $T^4$?
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Context:
In a linear regression model given by \( Y = XB + U \), where \( U \) is the vector of error terms normally distributed with mean 0 and variance \( \sigma^2 \), and all classical assumptions are satisfied, determine the variance of \( Y \).
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Context:
An integer has $2018$ digits and is divisible by $7$. The first digit is $d$, while all the other digits are $2$. What is the value of $d$?
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Context:
The figure is created by arranging four squares, each with a side length of 5 units, on each side of a central square with a side length of 20 units. Calculate the perimeter of the entire figure.
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Context:
# Problem 5.

The segments of two lines, enclosed between two parallel planes, are in the ratio $5: 9$, and the acute angles between these lines and one of the planes are in the ratio $2: 1$. Find the cosine of the smaller angle.
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Context:
Using 2 colors to paint the 4 small squares on a $2 \times 2$ chessboard, how many different coloring schemes are there?
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Context:
Find the sum of the infinite series: \[ 2^2 + \frac{3^2}{2!} + \frac{4^2}{3!} + \cdots \]
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Context:
A hot air balloon with 200 balloons is blowing up. After about half an hour, 1/5 of the total number of balloons in the hot air balloon have blown up. The wind speed then increases, creating a variation in the rate at which balloons blow up. In the next hour, 30% of the remaining balloons blow up. Due to different manufacturing quality, 10% of the balloons still intact have double durability and are less likely to blow up in the same conditions. At the end of the second hour, twice the number of balloons that had already blown up, excluding the more durable ones, also blow up. How many balloons in the hot air balloon remain intact?
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Context:
Given an integer \( n \geq 2 \), find the smallest positive real number \( c \) such that for any complex numbers \( z_1, z_2, \ldots, z_n \), the following inequality holds:
$$
\left|\sum_{i=1}^{n} z_{i}\right| + c \sum_{1 \leq i < j \leq n} \left|z_{i} - z_{j}\right| \geq \sum_{i=1}^{n} \left|z_{i}\right|.
$$
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Context:
What is the minimum possible cardinality of a finite set $X$ such that there are precisely seven open sets in a topology on $X$?
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Context:
What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1? The original answer is in the format $\frac{\sqrt{m}}{n}$, please provide the value of m + n.
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Context:
In the diagram, $\triangle QRS$ is an isosceles right-angled triangle with $QR=SR$ and $\angle QRS=90^{\circ}$. Line segment $PT$ intersects $SQ$ at $U$ and $SR$ at $V$. If $\angle PUQ=\angle RVT=y^{\circ}$, calculate the value of $y$ in degrees, rounding to one decimal place if necessary.
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Context:
It rained for 5 days out of a week (which has 7 days). In how many ways can it rain on exactly 3 consecutive days?
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Context:
Determine the range of $a-b$ if the equation $x^2+ax+b-2=0$ has one root in the interval $(-\infty, -1)$ and another root in the interval $(-1, 0)$.
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Context:
Example 25 (2004 National Women's Mathematical Olympiad Problem) A deck of tri-color cards, totaling 32 cards, includes 10 cards of each of the red, yellow, and blue colors, numbered $1, 2, \cdots, 10$; there are also a big joker and a small joker, each numbered 0. From this deck, several cards are drawn, and then the score is calculated according to the following rule: each card numbered $k$ is counted as $2^{k}$ points; if their total score is 2004, then these cards are called a "good card group." Try to find the number of "good card groups."
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Context:
Find the conditional probability that a standard Gaussian random variable $x$ is greater than $0$ given that $x$ is greater than another independent standard Gaussian random variable $y$. Express your answer in terms of standard Gaussian distribution properties.
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Context:
The circle constructed on the larger lateral side $AB$ of the right trapezoid $ABCD$ as its diameter intersects the base $AD$ at its midpoint. It is known that $AB=10, CD=6$. Find the midline of the trapezoid.
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Context:
Determine the number of ways to distribute the elements of the set \([n]\) into four subsets \(A_1, A_2, A_3, A_4\) such that \(A_1 \cap A_2 \cap A_3 \cap A_4 = \emptyset\) and \(A_1 \cup A_2 \cup A_3 \cup A_4 = [n]\).
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Context:
Calculate the following limit involving the Gamma function:

$$ \lim_{r \to \infty}\sqrt{\frac{r}{2}} \cdot \frac{\Gamma\left(\frac{r - 1}{2}\right)}{\Gamma\left(\frac{r}{2}\right)} $$
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Context:
Evaluate the integral \( \int_{0}^{\infty}\frac{\sin (7\sqrt{x})}{\sqrt{x}}dx \).
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Context:
Mark and William are playing a game with a stored value. On his turn, a player may either multiply the stored value by 2 and add 1 or he may multiply the stored value by 4 and add 3. The first player to make the stored value exceed \(2^{100}\) wins. The stored value starts at 1 and Mark goes first. Assuming both players play optimally, what is the maximum number of times that William can make a move?

(Optimal play means that on any turn the player selects the move which leads to the best possible outcome given that the opponent is also playing optimally. If both moves lead to the same outcome, the player selects one of them arbitrarily.)
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Context:
Alex runs a successful family bakery and is planning to expand the operation by opening two new branches. Each new branch requires an initial investment of $15,000. To fund the expansion, Alex decides to reinvest 40% of the bakery's monthly profit of $10,000 into the new branches. Additionally, Alex has saved $5,000 specifically for this expansion. After covering the initial investments for both branches, how much money will Alex have left from the reinvested profits and savings?
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Context:
Determine the number of distinct prime divisors of the integer \(5685858885855807765856785858569666876865656567858576786786785^{22}\) given that it has 6,436,343 divisors.
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Context:
A $1.4 \mathrm{~m}$ long rod has $3 \mathrm{~kg}$ masses at its ends. Where should the rod be pivoted so that, when released from a horizontal position, the mass on the left side passes under the pivot at a speed of $1.6 \mathrm{~m} / \mathrm{s}$?
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Context:
A train of a certain length moving at a speed of 72 kmph takes 79.99360051195904 seconds to cross another train of length 500 m, moving at 36 kmph in the same direction. What is the length of the first train?
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Context:
Given the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ with an eccentricity of $\frac{\sqrt{2}}{2}$, a line $MA$ is drawn through a point $M$ on the ellipse, intersecting the ellipse at points $A$ and $B$ with slopes $k_{1}$ and $k_{2}$, respectively. If points $A$ and $B$ are symmetric about the origin, find the value of $k_{1} \cdot k_{2}$.
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Context:
Jake has 84 more apples and some fewer peaches than Steven. Steven has 13 peaches and 52 apples. How many fewer peaches does Jake have compared to Steven?
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Context:
Three equally spaced parallel lines intersect with a circle, resulting in three chords of lengths 38, 38, 34 units. Calculate the distance between two adjacent parallel chords. Express your answer in units.
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Context:
In an arithmetic sequence $\{a_n\}$ with a non-zero common difference, if $2a_3-a_7^2+2a=0$, and $\{b_n\}$ is a geometric sequence with $b_7=a_7$, then $b_6b_8=$ ______.
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Context:
The numbers $a, b, c, d$ belong to the interval $[-8.5,8.5]$. Find the maximum value of the expression $a + 2b + c + 2d - ab - bc - cd - da$.
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Context:
In how many ways can the set \( \{a_{1}, a_{2}, \ldots, a_{9}, b_{1}, b_{2}, \ldots, b_{9}, c_{1}, c_{2}, \ldots, c_{9}\} \) be partitioned into 9 subsets of the form \( \{ a_{i}, b_{j}, c_{k} \} \)?
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Context:
Given a sequence $\{a_{n}\}$ that satisfies $\frac{a_1}{2}+\frac{a_2}{2^2}+⋯+\frac{a_n}{2^n}=n(n≤N^*)$, and $b_n=λ(a_n-1)-n^2+4n$. If the sequence $\{b_{n}\}$ is a monotonically increasing sequence, then the range of $\lambda$ is ______.
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Context:
Problem 11.5. In a chess tournament, a team of schoolchildren and a team of students, each consisting of 15 people, are competing against each other. During the tournament, each schoolchild must play against each student exactly once, and each person must play no more than one game per day. The number of games played on different days may vary.

At some point in the tournament, the organizer noticed that the schedule for the next day can be arranged in exactly 1 way with 15 games, and in $N$ ways with 1 game (the order of the games in the schedule does not matter, only who plays against whom). Find the maximum possible value of $N$.
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Context:
Josh had some money at first. He spent $1.75 on a drink, and then spent another amount. After spending, he had $6 left. How much money did Josh spend after buying the drink?
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Context:
Let  $n$  be a positive integer. For a permutation  $a_1, a_2, \dots, a_n$  of the numbers  $1, 2, \dots, n$  we define  $$ b_k = \min_{1 \leq i \leq k} a_i + \max_{1 \leq j \leq k} a_j $$  We say that the permutation  $a_1, a_2, \dots, a_n$  is *guadiana* if the sequence  $b_1, b_2, \dots, b_n$  does not contain two consecutive equal terms. How many guadiana permutations exist?
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Context:
Evaluate the limit of the double integral: \[ \lim_{n\rightarrow \infty}\int_{-n}^{n}\int_{-n}^{n}\sin(x^2+y^2)\,dx\,dy \]
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Context:
Given there are 2, 1, 3, and 4 paths leading to the top of the mountain from the east, west, south, and north sides, respectively, calculate the maximum number of ways to ascend from one side and descend from any other side.
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Context:
Find the number of solutions to the equation \(x_1 + x_2 + x_3 + \ldots + x_{10} = 27\) where each \(x_i\) is an integer satisfying \(1 \leq x_i \leq 4\).
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Context:
Given the function $f(x)=({x}^{3}-x)(\frac{1}{2}-\frac{1}{{e}^{x}+1})$, the solution set of the inequality $f\left(x\right)\ln \left(x+2\right)\leqslant 0$ is ______.
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Context:
Point \( A \) lies on the line \( y = \frac{15}{8} x - 4 \), and point \( B \) on the parabola \( y = x^{2} \). What is the minimum length of segment \( AB \)?
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Context:
Find the largest six-digit number in which all digits are distinct, and each digit, except for the extreme ones, is equal either to the sum or the difference of its neighboring digits.
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Context:
In the triangular pyramid \( PABC \), the sums of the three plane angles at each of the vertices \( A \) and \( B \) are equal to \( 180^\circ \), and \( PC = AB \). Inside the pyramid, a certain point \( D \) is taken such that the sum of the distances from \( D \) to the three lateral faces \( PAB \), \( PAC \), and \( PBC \) is 7. Find the distance from the center of the circumscribed sphere to the face \( PAB \), given that the volumes of the pyramids \( PABC \) and \( BABC \) are in the ratio \( 8: 1 \).
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Context:
Calculate the maximum number of distinct real roots that a polynomial $P$ can have if it satisfies the following property: the product of any two distinct roots of $P$ is also a root of $P$.
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Context:
Given the integers 2, 2, 5, 5, 8, and 9 on six cards, determine the number of integers from 1 to 31 that cannot be obtained as sums by choosing any number of these cards. Express your answer as a whole number.
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Context:
Find the number of solutions to:
\[\sin x = \left( \frac{1}{3} \right)^x\]
on the interval $(0,50 \pi)$.
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Context:
Given that \( A \) is an antisymmetric matrix and \( B \) is a symmetric matrix, find the condition under which \((A - B)^2\) is antisymmetric.
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Context:
Evaluate the limit of the integral over the semicircle as $R \to \infty$:

$$\lim_{R\to\infty}\int_{\Gamma_R}\frac{e^{izt}}{\pi\cos(iz)}dz$$

where $t > 0$ and $\Gamma_R$ is the upper semicircle of radius $R$ in the complex plane.
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Context:
There are $2023$ distinct points on a plane, which are colored either white or red. For each white point, there are exactly two red points such that the distance from the white point to each of these red points is $2023$. Find the minimum number of red points.
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Context:
Sets $A$, $B$, and $C$, depicted in the Venn diagram, are such that the total number of elements in set $A$ is three times the total number of elements in set $B$. Their intersection has 1200 elements, and altogether, there are 4200 elements in the union of $A$, $B$, and $C$. If set $C$ intersects only with set $A$ adding 300 more elements to the union, how many elements are in set $A$?

[asy]
label("$A$", (2,67));
label("$B$", (80,67));
label("$C$", (41,10));
draw(Circle((30,45), 22));
draw(Circle((58, 45), 22));
draw(Circle((44, 27), 22));
label("1200", (44, 45));
label("300", (44, 27));
[/asy]
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Context:
Let $n$ be a positive integer. Find as many zeros as possible for the last digits of the following expression: $1^n + 2^n + 3^n + 4^n$.
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Context:
Malcolm has 240 followers on Instagram and 500 followers on Facebook. The number of followers he has on Twitter is half the number of followers he has on Instagram and Facebook combined. Meanwhile, the number of followers he has on TikTok is 3 times the number of followers he has on Twitter, and he has 510 more followers on Youtube than he has on TikTok. Additionally, he has 120 followers on Pinterest which is double the number of his Snapchat followers.

Over the next month, he gains 15% more followers on Instagram, 20% more followers on Facebook, loses 12 followers on Twitter, gains 10% more followers on TikTok, and loses 5% followers on Snapchat. During the same period, Malcolm gains 8% more followers on Youtube and a fixed number of 20 new followers on Pinterest. After this one month period, what is the total number of followers Malcolm has on all his social media platforms?
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Context:
The captain's assistant, who had been overseeing the loading of the ship, was smoking one pipe after another from the very start of the process. When $2 / 3$ of the number of loaded containers became equal to $4 / 9$ of the number of unloaded containers, and noon struck, the old sea wolf started smoking his next pipe. When he finished this pipe, the ratio of the number of loaded containers to the number of unloaded containers reversed the ratio that existed before he began smoking this pipe. How many pipes did the second assistant smoke during the loading period (assuming the loading rate and the smoking rate remained constant throughout)?
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Context:
Example 2 There are $r$ people participating in a chess tournament, every 2 people play one game, the winner of each game gets 2 points, the loser gets 0 points, and in the case of a draw, each gets 1 point. After the tournament, there is exactly one person who has the fewest wins, and only he has the highest score. Find the minimum value of $r$. (16th All-Russian Mathematical Olympiad problem)
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Context:
Let $\triangle ABC$ be a triangle with $\angle ABC = 76^\circ$ and $\angle ACB = 72^\circ$. Points $P$ and $Q$ lie on the sides $AB$ and $AC$, respectively, such that $\angle ABQ = 22^\circ$ and $\angle ACP = 44^\circ$. Find the measure of $\angle APQ$.
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Context:
在锐角三角形 $A B C$ 中, $A B$ 上的高 $C E$ 与 $A C$ 上的高 $B D$ 相交于点 $H$, 以 $D E$ 为直径的圆分别交 $A B 、 A C$ 于 $F 、 G$ 两点, $F G$ 与 $A H$ 相交于点 $K$, 已知 $B C=25, B D=20, B E=7$, 求 $25A K$ 的长。
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Context:
Determine the number of homomorphisms from the symmetric group $S_4$ to the dihedral group $D_4$. Provide your answer as a single integer.
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Context:
Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half dollar. What is the probability that at least 25 cents worth of coins come up heads?
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Context:
In the tetrahedron P-ABC, PB=6, AC=3, and G is the centroid of triangle PAC. A section is made through point G such that the section is parallel to line PB and AC. The perimeter of the section is ______.
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Context:
$P$ is a point inside triangle $A B C$, and lines $A P, B P, C P$ intersect the opposite sides $B C, C A, A B$ in points $D, E, F$, respectively. It is given that $\angle A P B=90^{\circ}$, and that $A C=B C$ and $A B=B D$. We also know that $B F=1$, and that $B C=999$. Find $A F$.
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Context:
5. In several packages, there are 20 candies, and there are no two packages with the same number of candies and no empty packages. Some packages may be inside other packages (then it is considered that a candy lying in the inner package also lies in the outer package). However, it is forbidden to place a package inside another package that already contains a package. What is the maximum possible number of packages?
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Context:
If $X$ is a random variable with a chi-squared distribution with $k$ degrees of freedom, and $Y=\sqrt{X}$, determine the distribution of $Y$. \( \text{Express your answer in terms of known distributions.} \)
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Context:
Determine the Galois group of the polynomial \(x^4 - 10\) over \(\mathbb{F}_7\).
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Context:
Compute the degree of the zero cycle in the Fano variety of lines on a smooth quadric $Q \subset \mathbb{P}_k^3$, where $F_{osc} = \{[l] \in F(Q), \exists P\text{ plane such that } 2l \subseteq P \cap Q\}$.
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Context:
Compute the limit: \[ \lim_{n\to\infty}\int_{0}^{\infty}\frac{\sin\left(\frac{x}{n}\right)}{\left(1+\frac{x}{n}\right)^{n}}\,dx. \]
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Context:
Evaluate the expression \( \sum_{n=1}^{1009} \gcd(2n, 2020) - \sum_{n=0}^{1008} \gcd(2n+1, 2019) \).
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Context:
In a universe $U$ shaped like $\mathbb{R}^3$, points are randomly distributed in a Poisson fashion with density parameter $\lambda$, such that in any measurable set $S$ of volume $V$, the expected number of points in $S$ is $V$. Consider a random line $L$ that intersects the origin. What is the probability that $L$ intersects at least one point in this distribution?
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Context:
Determine the supremum of the set \(\{(\omega+2)^k \mid k \in \omega\}\).
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Context:
In a square room, there is a bulb on each wall that can shine one of the seven colors of the rainbow. There are no bulbs in the room that shine the same color. With each move, a person can change the color of one bulb to a color that no bulb is currently shining, while also changing the colors of the two other bulbs to the remaining two unused colors. (After the move, there still won't be two bulbs with the same color in the room). What is the minimum number of moves required so that eventually, each bulb has shone each of the seven colors?
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Context:
Evaluate the double integral \( \int_{0}^{2}\int_{0}^{\sqrt{2x-x^2}} \sqrt{x^2 + y^2} \,dy\,dx \) using the change of variables \( x = \cos\theta \) and \( y=\sin\theta \).
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Context:
For each positive integer $n$, let $r_n$ be the smallest positive root of the equation $x^n = 7x - 4$. There are positive real numbers $a$, $b$, and $c$ such that \[\lim_{n \to \infty} a^n (r_n - b) = c.\] If $100a + 10b + c = \frac{p}{7}$ for some integer $p$, find $p$.
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Context:
Find the number of positive integers \( j \leq 3^{2013} \) such that
\[ j = \sum_{k=0}^{m} \left((-1)^k \cdot 3^{a_k}\right) \]
for some strictly increasing sequence of nonnegative integers \( \{a_k\} \). For example, we may write \( 3 = 3^1 \) and \( 55 = 3^0 - 3^3 + 3^4 \), but 4 cannot be written in this form.
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Context:
Consider two distinct quadratic polynomials, $f(x)$ and $g(x)$. If $|f(x)| \geq |g(x)|$ for all $x \in \mathbb{R}$ and $f(x)$ has real roots, determine the number of distinct roots of the equation $h(x)h''(x) + (h'(x))^2 = 0$, where $h(x) = f(x)g(x)$. Provide your answer as an integer.
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Context:
Determine the radius of convergence of the function \( f(x) = \int_{0}^{x} \ln{(t+\sqrt{t^2+1})} \, dt \) using its Maclaurin series.
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Context:
Evaluate the factorial of \(-\frac{1}{2}\) using the Gamma function.
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Context:
Determine the quotient group \( \mathbb{C}/P \), where \( \mathbb{C} \) is the multiplicative group of non-zero complex numbers and \( P \) is the subgroup of positive real numbers.
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Context:
Determine the value of \( k \) such that the number of monotonically increasing functions from the set \( \{1,2,3,4,5,6\} \) to itself, with the property that \( f(x) \ge x \) for all \( x \), is equal to \( \frac{2}{k} \cdot \binom{11}{5} \).
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Context:
The least positive angle \( \alpha \) for which
 \( \left( {\frac{3}{4} - {\sin }^{2}\left( \alpha \right) }\right) \left( {\frac{3}{4} - {\sin }^{2}\left( {3\alpha }\right) }\right) \left( {\frac{3}{4} - {\sin }^{2}\left( {{3}^{2}\alpha }\right) }\right) \left( {\frac{3}{4} - {\sin }^{2}\left( {{3}^{3}\alpha }\right) }\right)  = \frac{1}{256} \)
 has a degree measure of \( \frac{m}{n} \) ,where \( m \) and \( n \) are relatively prime positive integers. Find
 \( m + n \) .
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Context:
Siva has the following expression, which is missing operations:\n$$\\frac12 \\,\\, \\_  \\,\\,\\frac14 \\,\\, \\_ \\,\\, \\frac18 \\,\\, \\_  \\,\\,\\frac{1}{16} \\,\\, \\_  \\,\\,\\frac{1}{32}.$$ For each blank, he flips a fair coin: if it comes up heads, he fills it with a plus, and if it comes up tails, he fills it with a minus. Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coinflips and operations. If the probability that the positive difference between his computed values is greater than $\\frac12$ can be expressed as $\\frac{a}{b}$ for relatively prime positive integers $a$, $b$, then find $a + b$.
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Context:
Determine the number of different 6-digit codes that can be formed using the digits 4, 6, and 9, with the following conditions: the code must use all the digits, it ends in 4, and 4 and 9 are never consecutive.
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Context:
Find the value of $c$ that minimizes the sum of the areas of two regions bounded by the curve $f(x) = x^3 - x^2$ and the line $y = c + x$. Assume $c$ is such that exactly two of these regions have finite area.
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Context:
Given two fixed points A(-2, 0) and B(1, 0), if a moving point P satisfies $|PA| = \sqrt{3}|PB|$, then the area of the figure enclosed by the trajectory of point P is equal to \_\_\_\_\_\_.
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Context:
Compute the factor group: $\mathbb{Z} \times \mathbb{Z} \big / \langle (1,2) \rangle$.
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Context:
Find all positive integers \( n \) such that \( n \) is equal to 100 times the number of positive divisors of \( n \).
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Context:
If the eight digits \(1, 2, 3, 4, 5, 6, 7, 8\) are randomly permuted, what is the probability that the resulting eight-digit number is divisible by 8?
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Context:
用 3 个 1,2 个 2,5 个 3 这十个数字能构成多少个四位数?
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Context:
Given a convex quadrilateral \(ABCD\) where \(AB=AD=1\), \(\angle A=80^\circ\), and \(\angle C=140^\circ\), find the length of the diagonal \(AC\).
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Context:
$x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } , x _ { 5 } , x _ { 6 } \in \{ 1 , 2 , 3 , 4 , 5 , 6 \}$，且$x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } , x _ { 5 } , x _ { 6 }$各不相同，

则满足$x _ { 1 } - 5 x _ { 2 } + 1 0 x _ { 3 } - 1 0 x _ { 4 } + 5 x _ { 5 } - x _ { 6 } = 0$的组数为多少？
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Context:
(JBMO 2000 - Shortlist, Problem G6), treated in class

Let $A B C D$ be a quadrilateral such that $\widehat{D A B}=60^{\circ}, \widehat{A B C}=90^{\circ}$ and $\widehat{B C D}=120^{\circ}$. Let $M$ be the intersection point of the diagonals $[A C]$ and $[B D]$. Given that $B M=1$ and $D M=2$, calculate the area of the quadrilateral $A B C D$.
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Context:
Given \( S = \frac{2}{1 \times 3} + \frac{2^2}{3 \times 5} + \frac{2^3}{5 \times 7} + \cdots + \frac{2^{49}}{97 \times 99} \) and \( T = \frac{1}{3} + \frac{2}{5} + \frac{2^2}{7} + \cdots + \frac{2^{48}}{99} \), find \( S - T \).
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Context:
A permutation of \{1,2, \ldots, 7\} is chosen uniformly at random. A partition of the permutation into contiguous blocks is correct if, when each block is sorted independently, the entire permutation becomes sorted. For example, the permutation $(3,4,2,1,6,5,7)$ can be partitioned correctly into the blocks $[3,4,2,1]$ and $[6,5,7]$, since when these blocks are sorted, the permutation becomes $(1,2,3,4,5,6,7)$. Find the expected value of the maximum number of blocks into which the permutation can be partitioned correctly.
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Context:
If $x < 0$ and $0 < y < 1$, then the order from smallest to largest of $\frac{y^2}{x}$, $\frac{y}{x}$, $\frac{1}{x}$ is __________.
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Context:
A plane is drawn through the midpoint of the diagonal of a cube and is perpendicular to this diagonal. Find the area of the resulting cross-section if the edge length of the cube is \(a\).
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Context:
Evaluate \(\lim_{n \to \infty} \frac{1}{n^5} \sum (5r^4 - 18r^2s^2 + 5s^4)\), where the sum is over all \(r, s\) satisfying \(0 < r, s \leq n\).
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Context:
In a regular truncated quadrilateral pyramid with the height of the lateral face \( a \), a sphere can be inscribed. Find the area of its lateral surface.
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Context:
Find the largest positive integer value of $n < 1000$ such that $\phi(36n) = \phi(25n) + \phi(16n)$, where $\phi(n)$ denotes the number of positive integers less than $n$ that are relatively prime to $n$.
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Context:
a, b, and c started a business with different capitals. At the end of the year, the profit share of b is Rs. 1800, and the difference between the profit shares of a and c is Rs. 720. If b invested Rs. 10000 and c invested Rs. 12000, how much did a invest?
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Context:
Let \( a < b < c < d < e \) be real numbers. We calculate all the possible sums of two distinct numbers among these five numbers. The three smallest sums are 32, 36, and 37, and the two largest sums are 48 and 51. Find all possible values of \( e \).
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Context:
3. In the equation $A \times B \times \overline{B B C}=\overline{A D A A}$, the same letters represent the same digits, different letters represent different digits, and $D$ is the smallest natural number, $A$ is the smallest prime number. Using the digits $A$, $B$, and $C$ to form natural numbers without repeated digits, the number of prime numbers formed is $\qquad$.
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Context:
Given circle $O$: $x^{2}+y^{2}=4$ and curve $C$: $y=3|x-t|$, there are two points $A(m,n)$ and $B(s,p)$ on curve $C$ ($m$, $n$, $s$, $p$ are all positive integers). The ratio of the distance from any point on circle $O$ to point $A$ and point $B$ is a fixed value $k(k > 1)$. Find $m^{s}-n^{p}=$ _____.
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Context:
One hundred friends, including Petya and Vasya, live in several cities. Petya found out the distance from his city to the city of each of the remaining 99 friends and summed these 99 numbers. Vasya did the same. Petya obtained 1000 km. What is the largest number Vasya might have obtained? (Consider the cities as points on a plane; if two people live in the same city, the distance between their cities is considered to be zero.)
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Context:
A turtle crawled out of its home and crawled in a straight line at a constant speed of 5 m/hour. After an hour, it turned $90^{\circ}$ (either right or left) and continued crawling, then crawled for another hour, then turned again $90^{\circ}$ (either right or left), and so on. It crawled for a total of 11 hours, turning $90^{\circ}$ at the end of each hour. What is the minimum distance it could be from its home?
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Context:
Given three sequences $\{F_n\}$, $\{k_n\}$, $\{r_n\}$ satisfying: $F_1=F_2=1$, $F_{n+2}=F_{n+1}+F_n$ ($n\in\mathbb{N}^*$), $r_n=F_n-3k_n$, $k_n\in\mathbb{N}$, $0\leq r_n<3$, calculate the sum $r_1+r_3+r_5+\ldots+r_{2011}$.
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Context:
A regular pentagon with a unit side, by drawing two diagonals from one vertex, results in the net of a pyramid with a regular triangle base. What is the volume of the pyramid?
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Context:
已知p为质数，$\sqrt{p}$的小数部分为x$, \frac { 1 } { x }$的小数部分为$\frac { \sqrt { p } - 3 1 } { 7 5 }$.求所有满足条件的质数p的值.
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Context:
Calculate the double integral of the function \( f(x,y) = (x+y)^9(x-y)^9 \) over the region \( R = \{(x,y) \mid 0 \leq x \leq 1/2, x \leq y \leq 1-x\} \).
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Context:
In a particular game, each of $4$ players rolls a standard $6{ }$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo's first roll was a $5,$ given that he won the game?
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Context:
Evaluate the double integral: \[ J = \int\limits_0^1 \int\limits_0^1 \dfrac{x-y}{(x+y)^3} \, dy \, dx. \] Determine the correct value of \( J \).
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Context:
Given the ellipse $\Gamma_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ and $c=\sqrt{a^{2}-b^{2}}$, the parabola $\Gamma_{2}: y^{2}=2c\left(x-\frac{c}{2}\right)$ intersects with $\Gamma_{1}$ at point $A$. Let $O$ be the origin. The line $l: x=\frac{a^{2}}{c}$ intersects the x-axis and the line $OA$ at points $D$ and $E$, respectively. A point $B$ on the ellipse $\Gamma_{1}$ satisfies $\frac{S_{\triangle ABE}}{S_{\triangle ABD}}=\frac{a}{c}$, with points $A$ and $B$ on opposite sides of the y-axis. Find the slope of line $AB$.
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Context:
A factory produces rattles shaped in the form of a ring with 3 red and 7 blue spherical beads on it. Two rattles are said to be of the same type if one can be obtained from the other one by moving a bead along the ring or by flipping the ring over in space. How many different types of rattles can be manufactured?
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Context:
已知正四棱雉 $\Gamma$ 的高 $h=3$, 侧面与底面所成角 $\theta=\frac{\pi}{3}$ 。先在正四棱雉 $\Gamma$ 内放人一个内切球 $O_{1}$, 然后依次放人球 $O_{2}, O_{3}, \cdots$, 使得后放入的各球均与前一个球及正四棱雉 $\Gamma$的四个侧面都相切。则放入所有球的体积之和为$\frac{m \pi}{n}$，求$m+n$的值。
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Context:
Given the hyperbola x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0), one of its asymptotes intersects the ellipse x^2/4 + y^2 = 1 and the length of the chord obtained is 4*sqrt(3)/3. Find the eccentricity of this hyperbola.
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Context:
Alex drives from point \( P \) to point \( Q \), then immediately returns to \( P \). Bob drives from point \( Q \) to point \( P \), then immediately returns to \( Q \). They travel on the same road, start at the same time, and each travels at a constant speed. Alex's speed is three times Bob's speed. They meet each other for the first time 15 minutes after the start. How long after the start will they meet each other for the second time? Express your answer in minutes as a whole number.
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Context:
A man saves 25% of his income during a year and spends the remaining portion on his personal expenses. Next year his income increases by 25%, but his savings increase by some percentage. His total expenditure in 2 years is double his expenditure in the 1st year. By what percentage did his savings increase in the second year?
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Context:
A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in 2007. A set of plates in which each possible sequence appears exactly once contains $N$ license plates. Find $\frac{N}{10}$.
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Context:
Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number $1$, then Todd must say the next two numbers ($2$ and $3$), then Tucker must say the next three numbers ($4$, $5$, $6$), then Tadd must say the next four numbers ($7$, $8$, $9$, $10$), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number $10,000$ is reached. Find the $2019$th number said by Tadd.
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Context:
Compute the tensor product \( \mathbb{Z} \otimes_{\mathbb{Z}[x]} \mathbb{Z} \), where the left \( \mathbb{Z} \) is endowed with a \( \mathbb{Z}[x] \)-module structure via the ring homomorphism \( \phi_n: \mathbb{Z}[x] \to \mathbb{Z} \) defined by \( \phi_n(f) = f(n) \), and the right \( \mathbb{Z} \) uses the structure from \( \phi_m \). The answer should be expressed in terms of the integers \( n \) and \( m \).
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Context:
There are 4 willow trees and 4 poplar trees planted in a row. How many ways can they be planted alternately?
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Context:
In the symmetric group $S_5$, how many equivalence classes of subgroups isomorphic to the non-cyclic group of order 4 are there?
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Context:
$已知椭圆E：x^2/a^2 + y^2/b^2 = 1 (a > b > 0) 的一个顶点为 A(0，1)，离心率 e = \sqrt{6}/3.$
$过点P(-\sqrt{3}，1)作斜率为k的直线与椭圆E交于不同的两点B，C，直线AB，AC分别与x轴交于点M，N.设椭圆的左顶点为D，求\frac{|MD|}{|MN|}的值.$
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Context:
Identify the quotient group $G/H$ where $G = GL_n(\mathbb{R})$ and $H$ is the subgroup of $G$ consisting of all matrices with positive determinant.
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Context:
Provide an example of a function \( f(x) \) such that the improper integral \( \int_{1}^{\infty} f(x) \, dx \) converges, but \( \lim_{x \to \infty} f(x) \) does not exist.
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Context:
Mrs. Toad has a class of 2017 students, with unhappiness levels $1,2, \ldots, 2017$ respectively. Today in class, there is a group project and Mrs. Toad wants to split the class in exactly 15 groups. The unhappiness level of a group is the average unhappiness of its members, and the unhappiness of the class is the sum of the unhappiness of all 15 groups. What's the minimum unhappiness of the class Mrs. Toad can achieve by splitting the class into 15 groups?
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Context:
Let $S$ be a subset of the natural numbers such that $0 \in S$, and for all $n \in \mathbb{N}$, if $n$ is in $S$, then both $2n+1$ and $3n+2$ are in $S$. What is the smallest number of elements $S$ can have in the range $\{0,1,\ldots, 2019\}$?
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Context:
7. In a race with six runners, $A$ finished between $B$ and $C, B$ finished between $C$ and $D$, and $D$ finished between $E$ and $F$. If each sequence of winners in the race is equally likely to occur, what is the probability that $F$ placed last?
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Context:
The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y|\leq 1000$?
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Context:
A figure in the plane has exactly two axes of symmetry. Find the angle between these axes.
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Context:
In a running competition, Jungkook came in the 7th and Minyoung came in the 5th. Taehyung did worse than Minyoung but better than Jungkook. In what place did Taehyung come in?
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Context:
Determine the number of irreducible polynomials in $GF(2)[x]$ of degree 8 whose roots generate a subfield of $GF(2^8)$ rather than the whole field.
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Context:
Determine the asymptotic growth rate of the function defined by the recurrence relation: \[ F(n) = \begin{cases} \sqrt{3F\left(\frac n3\right)} + n & \mbox{if }\ n>3\\ 2 & \mbox{if }\ n\leq3 \end{cases} \]
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Context:
Determine the total number of automorphisms of the symmetric group $S_6$. Express your answer as a single integer.
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Context:
Assuming a Poisson distribution with an unknown rate parameter \(\lambda\) for software failures, where two failures occurred in one week, determine the waiting time without any additional failures needed to declare with 95% confidence that a workaround is successful. A successful workaround will permanently eliminate the problem. Calculate this waiting time.
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Context:
Consider the Burgers equation \(u_t + uu_x = 0\) with the initial condition given by:
\[
u(x,0) = \phi(x) = \begin{cases} 2, & x \leq \pi/2 \\ \sin x + 1, & \pi/2 \leq x \leq 3\pi/2 \\ 0, & x \geq 3\pi/2 \end{cases}.\]
Determine the breaking time \(t_B\).
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Context:
Evaluate the limit: $$\lim_{x \rightarrow \infty} \sum_{n=0}^\infty (-1)^{n} \frac{x \lvert x \rvert^{2n}}{2n+1}$$
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Context:
Evaluate the limit:
\[ \lim\limits_{n\rightarrow\infty}\sum\limits_{k=2}^n\frac{1-e^{-(b_k-a_{k})}}{1+e^{-(b_k-a_{k})}} \]
where \(0 \leq a_k \leq b_k \leq 1\) and \(0 \leq b_k - a_k \leq \epsilon\).
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Context:
Determine the range of values for $c$ such that the equation $x(x^2-7) = 2x+c$ has exactly two solutions. Express your answer in the form $c_1 < c < c_2$, where $c_1$ and $c_2$ are integers.
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Context:
Determine the number of possible values for $a$ such that the equation $ax^2 + xy + y^2 + bx + cy + d = 0$ represents a pair of straight lines with slopes $m$ and $m^2$. How many such values of $a$ exist?
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Context:
Let $K$ be a number field and $G_K = \text{Gal}(\overline{K}/K)$ be its absolute Galois group. Consider a discrete $G_K$-module $M$ with a trivial $G_K$-action. For each place $v$ of $K$, let $K_v$ be the completion of $K$ at $v$. The restriction to a decomposition group $G_v$ at $v$ defines a homomorphism $H^1(K, M) \to H^1(K_v, M)$. Define $$F^1(K, M) := \text{Ker}\left(H^1(K, M) \to \prod_v H^1(K_v, M)\right).$$ Determine the structure of $F^1(K, M)$ when the $G_K$-action on $M$ is trivial.
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Context:
Compute the residue of the function \( \frac{z}{e^z - 1} \) at its poles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of values for \( r \) such that the curves \( C_1 = \arg(z) = \frac{\pi}{6} \) and \( C_2 = |z - i2\sqrt{3}| = r \) intersect at exactly two points.
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Context:
Calculate the flux of the vector field \( \mathbf{v} = \mathrm{curl}\, \mathbf{u} \) through the surface \( \Sigma \), where \( \Sigma \) is the portion of the paraboloid \( z = 1 - x^2 - y^2 \) with \( x \geq 0 \) and \( z \geq 0 \). The vector field \( \mathbf{u} \) is given by \( \mathbf{u} = (xy, xz^2, x^2y) \). Use Stokes' theorem to compute the flux.
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Context:
Find the linearization of the differential equation \( y''(t) + y(t) e^{0.02y} = 0 \) near \( y = 0 \).
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Context:
Let \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) be a \( 2 \times 2 \) matrix over a field \( k \) such that \( A^2 = 0 \). Consider the polynomials \( a^2 + bc, \ c(a+d), \ b(a+d), \ d^2 + bc \) in \( a, b, c, d \). Let \( I \) be the ideal generated by these polynomials. Determine the smallest integer \( r \) such that \( (a+d)^r \in I \).
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Context:
Is 6 the minimum number of bishops needed to guarantee that Player A can always checkmate Player B's knight on an infinite chessboard, assuming both players play optimally?
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Context:
Find a basis for the field extension $\mathbb{R}(x^2)(x)$ over $\mathbb{R}(x^2)$. Provide your answer as a set of elements that form a basis.
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Context:
On an infinite tape, numbers are written in a row. The first number is one, and each subsequent number is obtained by adding the smallest non-zero digit of its decimal representation to the previous number. How many digits are in the decimal representation of the number that is in the $9 \cdot 1000^{1000}$-th place in this sequence?
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Context:
Given that \(a, b, c, d\) are positive real numbers such that \(a+b+c+d = 2\), determine the range of values that the expression \(\lambda = (1+a+b)(1+c+d)\) can take.
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Context:
Let $\mathcal{O}$ be a regular octahedron. How many lines are there such that a rotation of at most $180^{\circ}$ around these lines maps $\mathcal{O}$ onto itself?
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Context:
Three non-collinear lattice points  $A,B,C$  lie on the plane  $1+3x+5y+7z=0$ . The minimal possible area of triangle  $ABC$  can be expressed as  $\frac{\sqrt{m}}{n}$  where  $m,n$  are positive integers such that there does not exists a prime  $p$  dividing  $n$  with  $p^2$  dividing  $m$ . Compute  $100m+n$ .

*Proposed by Yannick Yao*
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Context:
Given $f_{1}(x)=\sin x+\cos x$, let $f_{2}(x)=f'_{1}(x)$, ..., $f_{n+1}(x)=f'_{n}(x)$, ..., determine the value of $f_{1}\left( \frac{\pi}{3}\right)+f_{2}\left( \frac{\pi}{3}\right)+f_{3}\left( \frac{\pi}{3}\right)+\ldots+f_{2017}\left( \frac{\pi}{3}\right)$.
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Context:
The Evil League of Evil is plotting to poison the city's water supply. They plan to set out from their headquarters at $(5,1)$ and put poison in two pipes, one along the line $y=x$ and one along the line $x=7$. However, they need to get the job done quickly before Captain Hammer catches them. What's the shortest distance they can travel to visit both pipes and then return to their headquarters?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence for the power series $\displaystyle\sum_{n = 0} \dfrac{(n+1)!}{n^2} x^{2n}$.
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Context:
The digits 1, 2, 3, 4 can be arranged to form twenty-four different four-digit numbers. If these twenty-four numbers are then listed from smallest to largest, what is the position of 3142? Express your answer as an ordinal position (e.g., 1st, 2nd, 3rd, etc.).
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Context:
Find the sum of all the positive solutions of
\[2\cos2x \left(\cos2x - \cos{\left( \frac{2014\pi^2}{x} \right) } \right) = \cos4x - 1\]
If the answer is \(x\pi\), what is the value of \(x\)?
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Context:
Given positive integers \( r \) and \( n \), find the smallest positive integer \( m \) such that for any partition of the set \( S = \{1, 2, \ldots, m\} \) into \( r \) pairwise disjoint subsets \( A_1, A_2, \ldots, A_r \), there exist two positive integers \( a \) and \( b \) in the same subset \( A_i \) (\(1 \leq i \leq r\)) such that \( b < a \leq \frac{n+1}{n} b \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sales tax on an article was reduced to 3 1/3%. Due to this change, a person who purchases an article with a market price of Rs. 10,800 saves Rs. 18. What was the original sales tax percentage?
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Context:
A boat travels upstream from port $A$ and encounters a drifting raft at a distance of 4 kilometers from port $A$. The boat continues its journey to port $B$ upstream and immediately turns around, arriving at port $A$ simultaneously with the raft. Given that the current speed is 2 meters/second, and the boat's speed downstream is 8 meters/second, what is the distance between ports $A$ and $B$ in kilometers?
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Context:
First, select $n$ numbers from the set $ \{1, 2, \cdots, 2020\} $. Then, from these $n$ numbers, choose any two numbers $a$ and $b$, such that $a$ does not divide $b$. Find the maximum value of $n$.
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Context:
A point is chosen uniformly at random from the interior of a unit square. Let $p$ be the probability that any circle centered at the point that intersects a diagonal of the square must also intersect a side of the square. Given that $p^2$ can be written as $m-\sqrt{n}$ for positive integers $m$ and $n$, what is $m+n$?
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Context:
88. Xiaoming and his sister Xiaomei go to two different schools, which are located to the east and west of their home, respectively. Xiaoming leaves home 6 minutes before his sister Xiaomei starts for her school. Two minutes after Xiaomei leaves, their father, who is at home, realizes that both Xiaoming and Xiaomei have forgotten their homework. So, he sets out on a bike to catch up with Xiaoming and Xiaomei to deliver their homework. It is known that the father's speed is three times that of Xiaoming and twice that of Xiaomei. Throughout the process of delivering the homework, Xiaomei and Xiaoming continue moving forward and do not reach their schools. How many minutes at least will the father need from the time he sets out until he returns home? $\qquad$
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Context:
Petya wants to place 99 coins on a \(2 \times 100\) board so that no two coins are in adjacent cells (sharing an edge) and each cell contains at most one coin. How many ways are there to place the coins?
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Context:
A point $P$ moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection.Let $N( t)$ be the number of starting directions from a fixed interior point $P_0$ for which $P$ escapes within $t$ units of time. Find the least constant $a$ for which constants $b$ and $c$ exist such that
$$N(t) \leq at^2 +bt+c$$
for all $t\gt 0$ and all initial points $P_0 .$
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Context:
What is the maximum possible number of sides of smaller convex polygons formed by the diagonals of a convex heptagon?
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Context:
Evaluate the integral \( \int _{ -\infty }^{ +\infty }{ { e }^{ { (ix-b) }^{ 2 } } } \, dx \) for real constants \( x \) and \( b \).
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Context:
The $600$ students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of three lunch groups. Find the probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
Determine the function \( f(x) \) represented by the Riemann sum: \[ \lim\limits_{n \to \infty} \sum\limits_{i=1}^{n} \left(\frac{i^3}{n^4}-\frac{i}{n}\right). \]
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Context:
Evaluate the Cauchy Principal Value of the integral: \[ \int_{-\infty}^\infty \frac{\cos kx}{x-a} \, dx \] where \( a \) is a real number and \( k > 0 \).
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Context:
Given the differential equation $$0 = -gt - \frac{F}{\alpha}\cdot \log\left( 1 - \frac{\alpha t}{M+m} \right),$$ with $M=5~kg$, $m=10~kg$, $\alpha = 1 kg/s$, and $F = 0.9(M +m)g$, find the second solution for $t$ when $\alpha t \ll M+m$, given that one solution is $t = 0$. What is the second, more important solution?
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Context:
Line l is tangent to circle O at point A. Let D be the foot of the perpendicular from A to chord BC. The perpendiculars from points B and C to line l are drawn with feet at E and F, respectively. Given that CF = 12 and AD = 9, find the length of BE.
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Context:
Determine the values of \(\beta\) for which the inequality \(\left(\sum_{i=1}^{N}{x_i}\right)^{\beta} \geq \sum_{i=1}^{N}{x_i^{\beta}}\) holds for all positive real numbers \(\{x_i\}_{i=1}^{N}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the decimal digits of the number
 \( 5\mathop{\sum }\limits_{{k = 1}}^{{99}}k\left( {k + 1}\right) \left( {{k}^{2} + k + 1}\right) . \)
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Context:
Evaluate the integral $$\int_{0}^{+\infty }\frac{x \log(x)}{(x^{2}+1)^{2}}dx$$ and determine whether it converges or diverges.
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Context:
Natural numbers \( x, y, z \) are such that \( \operatorname{GCD}(\operatorname{LCM}(x, y), z) \cdot \operatorname{LCM}(\operatorname{GCD}(x, y), z) = 1400 \).

What is the maximum value that \( \operatorname{GCD}(\operatorname{LCM}(x, y), z) \) can take?
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Context:
Consider the differential equation \((1+e^x)y'+ye^x=1\). Given that \(y(0)=2\), determine the interval(s) where the solution \(y(x)\) has critical points.
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Context:
In the figure, quarter arc $AD$ of the circle $x^2+y^2=16$ is given. Points $B$ and $C$ are arbitrary on the arc $AD$ such that $C$ is between $D$ and $B$. Determine the maximum possible area of the pentagon $OABCD$, where $O$ is the origin.
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Context:
Given that $\phi = \frac{1 + \sqrt{5}}{2}$, determine the value of $n$ defined as follows:

$$
n = \frac{1}{1} + \frac{1}{1 + \phi} + \frac{1}{1 + \phi + \phi^2} + \frac{1}{1 + \phi + \phi^2 + \phi^3} + \dots
$$

What is the value of $\lfloor2n\rfloor + \lceil2n\rceil$?
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Context:
Given that point $P$ is a common point of circle $O\_1$ and circle $O\_2$, where circle $O\_1$ is defined as $(x-a)^{2}+(y-b)^{2}=b^{2}+1$, and circle $O\_2$ is defined as $(x-c)^{2}+(y-d)^{2}=d^{2}+1$. If $ac=8$, $\frac {a}{b} = \frac {c}{d}$, then the minimum value of the distance between point $P$ and any point $M$ on line $l$: $3x-4y-25=0$ is _____.
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Context:
Given that $21^{-1} \equiv 17 \pmod{53}$, find $32^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.)
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Context:
Given the function $$f(x)= \begin{cases} \overset{2^{x},x\geq 3,}{f(x+1),x<3}\end{cases}$$, find the value of $f(\log_2 6)$.
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Context:
A point A moves uniformly with velocity $v$ so that the vector $v$ is continually "aimed" at point B which in its turn moves rectilinearly and uniformly with velocity $u<v$. Initially, $v⊥u$, and the points are separated by a distance $l$. What is the time at which the points converge?
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Context:
9.3. Three cyclists started simultaneously from one point in the same direction on a circular track. The first one completes the entire track in 5 minutes, the second in 7 minutes, and the third in 9 minutes. What is the least amount of time after which all the cyclists will be at the same point on the track again? The speeds of all cyclists are constant.
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Context:
There are relatively prime positive integers $s$ and $t$ such that
\[
\sum_{n=2}^{100}\left(\frac{n}{n^2-1}- \frac{1}{n}\right)=\frac{s}{t}
\]
Find $s + t$.
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Context:
For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in a row. What is the remainder when $S(2015)$ is divided by $12$?
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Context:
Determine the values of \( p > 0 \) for which the sequence of functions \( f_n(x) = \frac{nx}{1+n^2x^p} \) converges uniformly to the pointwise limit \( f(x) = 0 \) on the interval \([0,1]\).
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Context:
Two trains of length 100 m and another length are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. From the moment they meet, they will be clear of each other in 18.998480121590273 seconds. What is the length of the second train?
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Context:
Given the function $f(x)=2x+\sin x$, the solution set of the inequality $f(m^{2})+f(2m-3) < 0$ (where $m\in\mathbb{R}$) is ______.
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Context:
A customer makes three deposits towards the purchase of a certain product. The first deposit is 10% of the total product cost, and it amounts to $130. The second deposit is twice the amount of the first deposit and covers an additional 20% of the total product cost. There is also an ongoing promotional discount of 5% on the total product cost, which the customer hasn't availed yet. If the customer decides to pay off the remaining balance including the discount in a final payment which also has an interest rate of 2%, determine how much more the customer needs to pay for the product.
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Context:
When one ounce of water is added to a mixture of acid and water, the new mixture is $20\%$ acid. When one ounce of acid is added to the new mixture, the result is $33\frac13\%$ acid. The percentage of acid in the original mixture is
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Context:
Find the area bounded by the inverse of the function $f(x) = x^3 + 3x + 2$, the x-axis, and the vertical lines $x = -2$ and $x = 6$. You do not need to explicitly find the inverse function to solve this problem.
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Context:
A man buys three items - a cycle for Rs. 1,400, a helmet for Rs. 400, and two safety lights costing Rs. 200 each. The store offers a 10% discount on the cycle, 5% discount on the helmet, and the safety lights are sold at the listed price. After purchasing the items, the man pays an additional 5% tax on the total cost. After a few months, he decides to sell the cycle at a 12% loss, the helmet at a 25% profit, and the safety lights at a no profit no loss condition. He also buys a new lock for Rs. 300 and includes it in the sale. If a 3% transaction fee is applied to the total selling price and he decides to round the final selling price down to the nearest whole number, what is the final selling price of all the items combined?
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Context:
Three circles $C_{1}$, $C_{2}$, and $C_{3}$ with radii 1, 2, and 3, respectively, touch each other externally. The centers of $C_{1}$ and $C_{2}$ are on the x-axis, and $C_{3}$ touches them from above. Determine the radius of the circle that is enclosed by $C_{1}$, $C_{2}$, and $C_{3}$ and is tangent to all three circles.
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Context:
Evaluate \( \lim_{n\to \infty} \int_{n^2+1}^{n^2+n+1}\frac{\sin^{-1}x}{\sqrt{x}}\,dx \) without performing the integration.
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Context:
Given that the sequence $\{a\_n\}$ is an arithmetic progression, and the sequence $\{b\_n\}$ satisfies $b\_n=a\_n a_{n+1} \cdot a_{n+2} (n \in \mathbb{N}^*)$, let $S\_n$ be the sum of the first $n$ terms of $\{b\_n\}$. If $a_{12}=\frac{3}{8} a_{5} > 0$, find the value of $n$ when $S\_n$ reaches its maximum.
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Context:
A system consists of 7 components with independent, identically distributed lifetimes. The system will operate until any one of the components fails. If the lifetime \( X \) of each component has a probability density function given by:

\[f(x) = \begin{cases} \frac{3}{x^4}, & \text{for } x > 1\\ 0, & \text{otherwise} \end{cases}\]

Calculate the expected lifetime until failure of the system.
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Context:
Find the minimum distance from the point \( P = (2, 5, 1) \) to the ellipsoid \( 2x^2 + 4y^2 + z^2 = 9 \), restricted to the first octant.
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Context:
Calculate \(\int_{S}\omega\), where \(\omega\) is the 2-form on \(\mathbb{R}^3-\{(1,0,0),(-1,0,0)\}\) given by:
\[
\omega=((x-1)^2+y^2+z^2)^{-3/2}((x-1)dy\wedge dz+ydz\wedge dx+zdx \wedge dy)+
  ((x+1)^2+y^2+z^2)^{-3/2}((x+1)dy\wedge dz+ydz\wedge dx+zdx \wedge dy)
\]
and \(S=\{(x,y,z)\in \mathbb{R}^3: x^2+y^2+z^2=5 \}\). The orientation of \(S\) is the natural orientation induced by \(D=\{(x,y,z)\in \mathbb{R}^3: x^2+y^2+z^2 \leq 5 \}\).
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Context:
In a $33 \times 33$ grid, each cell is colored with one of three different colors such that the number of cells of each color is equal. Determine the minimum number of separating edges, where a separating edge is defined as an edge between two adjacent cells of different colors.
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Context:
Determine the values of $q$ for which the integral \( \int_1^\infty \frac{x^q}{\ln x}\,dx \) converges.
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Context:
Evaluate the limit \( \lim_{R \to \infty} \int_{0}^{\frac{\pi}{2}} Re^{-R\sin\theta} \, d\theta \).
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Context:
Compute the covariance \( \text{Cov}[X,Y] \) for the joint mass density function given by: \[ f_{X,Y}(x,y) = \frac{{e}^{-(x + \frac{y}{x})}}{x}, \ y > 0, \ x > 0. \] Assume that \( X \) follows an exponential distribution with parameter 1.
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Context:
Determine the number of distinct sequences of rational numbers that converge to 1.
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Context:
Let \( f = \sum_{i \ge m} a_i t^i \) be a formal Laurent series, where \( m \in \mathbb{Z} \). Consider the formal differential \( df \). What is the coefficient of degree \(-1\) in the differential form \( \frac{df}{f} \)? Provide an explicit expression for this coefficient.
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Context:
Determine the order of the singularity at \( z_0 \) for the function \( \frac{f(z) + g(z)}{r(z) + h(z)} \), where \( f(z) \), \( g(z) \), \( r(z) \), and \( h(z) \) are analytic near \( z_0 \). At \( z_0 \), \( f(z) \) has a pole of order 2, \( g(z) \) has a zero of order 3, \( r(z) \) has a zero of order 2, and \( h(z) \) has a zero of order 1.
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Context:
Hong and Song each have a shuffled deck of eight cards, consisting of four red and four black cards. Each turn, they play the top two cards from their decks, which can be either two black cards, two red cards, or one of each color. What is the probability that Hong and Song play exactly the same pairs as each other for all four turns? If this probability is expressed as $\frac{m}{n}$, where $m$ and $n$ are coprime positive integers, compute $100m+n$. \(\text{(Proposed by Sean Li)}\)
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Context:
Calculate the norm \( \|\Lambda\| \) of the linear functional \( \Lambda(f) = \int_{0}^{1} (9t^4 - 18t^3 + 11t^2 - 2t) f(t) \, dt \) in the Banach space \( C([0, 1]) \).
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Context:
Petya has a deck of 36 cards, consisting of 4 suits with 9 cards each. He selects 18 cards to give to Vasya, keeping the remaining 18 for himself. The players take turns placing one card on the table, starting with Petya. If Vasya can place a card of the same suit or rank in response to Petya's move, he earns 1 point. What is the maximum number of points that Vasya can guarantee?
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Context:
Evaluate the limit: $$\lim_{n\to \infty} \frac{1}{n}\cdot \left((m+1)(m+2) \ldots(m+n)\right)^{\frac{1}{n}}$$ where $m$ is a fixed positive integer.
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Context:
Find the matrix that corresponds to a dilation centered at the origin with scale factor $-3.$
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\boxed{\begin{pmatrix} -3 & 0 \\ 0 & -3 \end{pmatrix}}


Context:
On February 1, it was 16.2°C outside Jacinta's house at 3:00 p.m. On February 2, it was -3.6°C outside Jacinta's house at 2:00 a.m. If the temperature changed at a constant rate between these times, calculate the rate at which the temperature decreased. Express your answer in °C per hour.
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Context:
Evaluate the double integral \( \iint_{\Omega} (x+y^2) \, dx \, dy \) where \( \Omega \) is the region bounded by the lines \( x+y=0 \), \( x+y=1 \), \( 2x-y=0 \), and \( 2x-y=3 \).
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Context:
Given seven points on a plane, where the distances between them are denoted by $a_1, a_2, \ldots, a_{21}$, determine the maximum number of times the same distance value can occur among these 21 distances.
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Context:
Find the minimal distance between the origin and the intersection of the surfaces given by the equations \(x^2 = 2yz\) and \(x^2 + 3y^2 + 2z^2 = 30\).
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Context:
Determine the order of the numerical scheme given by \(y_{n+1} = y_n + \frac{h}{2}\left(f(x_n, y_n) + f(x_{n+1}, y_{n+1})\right)\) by inserting the exact solution into the scheme.
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Context:
Consider a sequence of independent two-dimensional vectors of random variables \((A_n, B_n)_{n=1}^{\infty}\), where each vector is uniformly distributed on the square \([-2,2] \times [-2,2]\). Define \(V_n=(S_n, T_n) = (\sum_{i=1}^n A_i, \sum_{i=1}^n B_i)\) and \(|V_n| = \sqrt{(S_n)^2+(T_n)^2}\). Determine the constant \(c\) such that \(\lim_{n \to \infty} P(|V_n|<c\sqrt{n})=0.95\).
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Context:
Consider the parabola  $ C: y \equal{} x^2$  in the  $ x \minus{} y$  plane. Let  $ l_1$  be the tangent line of  $ C$  with slope  $ \sqrt {3}$  and  $ l_2$  be the tangent line of  $ C$  with negative slope which makes angle  $ 45^\circ$  with  $ l_1$ . Find the area bounded by  $ C,\ l_1$  and  $ l_2$ .
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Context:
A jar contains 97 marbles that are either red, green, or blue. Neil draws two marbles from the jar without replacement and notes that the probability that they would be the same color is $\frac{5}{12}$. After Neil puts his marbles back, Jerry draws two marbles from the jar with replacement. Compute the probability that the marbles that Jerry draws are the same color.
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Context:
A carton contains milk that is \(2\%\) fat, an amount that is \(40\%\) less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk? Provide your answer as a fraction in the form \(\frac{k}{m}\), where the fraction is in its simplest form. Please give the value of k + m.
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Context:
Given vectors OA = a, OB = b, OC = c, and the condition AC = 3BC, express c in terms of a and b. Provide your answer as a vector equation, including all necessary operations and coefficients.
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Context:
Abie initially had 20 bags of chips. She decided to share 2/5 of the bags with her friends and bought 18 bags in the store which only cost her half the price of the initial 20 bags. On her way home, Abie found a coupon to buy 4 more bags of chips for 3/4 of the original price per bag. How much did Abie spend in total if each bag originally cost her $2?
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Context:
Let \( f \) be an arbitrary, twice differentiable function for which \( f'' \neq 0 \). The function \( u(x,y) = f(x^2 + axy + y^2) \) satisfies the equation \( U_{xx} - U_{yy} = 0 \). Determine the value of the constant \( a \).
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Context:
Determine the cofinality of the ordinal expression \( \omega_3^{\omega_2} \).
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Context:
Twenty-five adults went to a soccer match with thirty-two children. Adult tickets are $12 each and the total bill was $450. How many dollars is one child's ticket?
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Context:
Given two independent standard normal random variables, $X$ and $Z$, find the probability $P(X+Z<0, Z>0)$.
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Context:
Find the sum of all three-digit natural numbers that contain at least one odd digit and at least one even digit.
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Context:
Determine the number of subgroups of order 125 in the group $G = \mathbb{Z}_{360} \oplus \mathbb{Z}_{150} \oplus \mathbb{Z}_{75} \oplus \mathbb{Z}_{3}$.
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Context:
Given $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $\Gamma$: $\frac{{x}^{2}}{{a}^{2}}+y^{2}=1\left(a \gt 1\right)$, $A$ is the top point of $\Gamma$, and the line $l$ passes through point $F_{1}$ and intersects $\Gamma$ at points $B$ and $C$. If $l$ vertically bisects the line segment $AF_{2}$, then the perimeter of $\triangle ABC$ is ______.
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Context:
Define the sequence \(\{a_n\}\) where \(a_n = n^3 + 4\) for \(n \in \mathbf{N}_+\). Let \(d_n = \gcd(a_n, a_{n+1})\), which is the greatest common divisor of \(a_n\) and \(a_{n+1}\). Find the maximum value of \(d_n\).
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Context:
Given a $4 \times 4$ squares table, how many ways can we fill the table with $\{0,1\}$ such that any two neighboring squares (sharing one common side) have a product equal to $0$?
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Context:
What is the smallest number of regular hexagons of side length 1 needed to completely cover a disc of radius 1 ?
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Context:
Given that \( F_{1} \) and \( F_{2} \) are the left and right foci of the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\), and \( A \) is a moving point on the ellipse. Circle \( C \) is tangent to the extended line of \( F_{1}A \), the extended line of \( F_{1}F_{2} \), and segment \( AF_{2} \). If \( M(t, 0) \) is a point of tangency, then \( t=\_\_\_\_\_ \).
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Context:
In the "Happy Dictionary," there is a question that presents a sequence of numbers and asks you to fill in the next few numbers based on the pattern. Here is the sequence given: $$- \frac {1}{2}, \frac {1}{2}, - \frac {3}{8}, \frac {1}{4}, \ldots, - \frac {5}{32}, \frac {3}{32}, \ldots$$ The 8th number in the sequence can be \_\_\_\_\_\_.
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Context:
A river flows at a constant speed. Piers A and B are located upstream and downstream respectively, with a distance of 200 kilometers between them. Two boats, A and B, depart simultaneously from piers A and B, traveling towards each other. After meeting, they continue to their respective destinations, immediately return, and meet again for the second time. If the time interval between the two meetings is 4 hours, and the still water speeds of boats A and B are 36 km/h and 64 km/h respectively, what is the speed of the current in km/h?
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Context:
Let $a_n$ be the product of the complex roots of $x^{2n} = 1$ that are in the first quadrant of the complex plane. That is, roots of the form $a + bi$ where $a, b > 0$. Let $r = a_1 \cdot a_2 \cdot \ldots \cdot a_{10}$. Find the smallest integer $k$ such that $r$ is a root of $x^k = 1$.
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Context:
What is the probability that among the five drawn lottery numbers there are consecutive numbers?
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Context:
In an examination, a student scores 5 marks for every correct answer and loses 2 marks for every wrong answer. If he attempts all 150 questions and secures 370 marks, the number of questions he attempts correctly is:
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Context:
Josh had 16 marbles in his collection. He decided to triple his collection and then added 10 more marbles. After that, he lost 25% of his marbles and found 5 marbles the next day. What is the square root of the total number of marbles Josh has now?
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Context:
Find the smallest positive integer  $k$  such that, for any subset  $A$  of  $S=\{1,2,\ldots,2012\}$  with  $|A|=k$ , there exist three elements  $x,y,z$  in  $A$  such that  $x=a+b$ ,  $y=b+c$ ,  $z=c+a$ , where  $a,b,c$  are in  $S$  and are distinct integers.
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Context:
The integers  $1, 2, \dots, 64$  are written in the squares of a  $8 \times 8$  chess board, such that for each  $1 \le i < 64$ , the numbers  $i$  and  $i+1$  are in squares that share an edge. What is the largest possible sum that can appear along one of the diagonals?
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Context:
Three of the edges of a [[cube]] are $\overline{AB}, \overline{BC},$ and $\overline{CD},$ and $\overline{AD}$ is an interior [[diagonal]].  Points $P, Q,$ and $R$ are on $\overline{AB}, \overline{BC},$ and $\overline{CD},$ respectively, so that $AP = 5, PB = 15, BQ = 15,$ and $CR = 10.$  What is the [[area]] of the [[polygon]] that is the [[intersection]] of [[plane]] $PQR$ and the cube?
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Context:
What is the minimum number of straight cuts required to divide a $5 \times 5$ square into unit squares, given that after each cut the resulting pieces can be rearranged arbitrarily before making the next cut, allowing multiple pieces to be cut simultaneously?
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Context:
What is the expected number of random integers in the range [1, N] that need to be generated before encountering a duplicate integer, assuming the generator is ideal?
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Context:
Sixteen wooden Cs are placed in a 4-by-4 grid, all with the same orientation, and each is to be colored either red or blue. A quadrant operation on the grid consists of choosing one of the four two-by-two subgrids of Cs found at the corners of the grid and moving each C in the subgrid to the adjacent square in the subgrid that is 90 degrees away in the clockwise direction, without changing the orientation of the C. Given that two colorings are the considered same if and only if one can be obtained from the other by a series of quadrant operations, determine the number of distinct colorings of the Cs.
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Context:
$A_1, A_2, ..., A_n$  are the subsets of  $|S|=2019$  such that union of any three of them gives  $S$  but if we combine two of subsets it doesn't give us  $S$ . Find the maximum value of  $n$ .
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Context:
A, B, C, and D rent a pasture. If A puts 10 oxen for 7 months, B puts 12 oxen for 5 months, C puts 15 oxen for 3 months, and D puts 20 oxen for 6 months for grazing, and the rent of the pasture is 250 USD. Given the currency conversion rate of 1 USD = 50 local currency units (LCU), how much amount should C pay as his share of rent in LCU?
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Context:
Given the function $f(x) = \log_a(8-ax)$ ($a > 0$ and $a \neq 1$), if $f(x) > 1$ holds true in the interval $[1,2]$, then the range of the real number $a$ is ______.
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Context:
In trapezoid $PQRS$ with $PQ$ parallel to $RS$, the diagonals $PR$ and $QS$ intersect at $T$. If the area of triangle $PQT$ is 75 square units, and the area of triangle $PST$ is 30 square units, calculate the area of trapezoid $PQRS$.
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Context:
The volume of a certain rectangular solid is $8$ cm3, its total surface area is $32$ cm2, and its three dimensions are in geometric progression. The sums of the lengths in cm of all the edges of this solid is
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Context:
The equation $ ax^3\plus{}bx^2\plus{}cx\plus{}d\equal{}0$ has three distinct solutions. How many distinct solutions does the following equation have:

$ 4(ax^3\plus{}bx^2\plus{}cx\plus{}d)(3ax\plus{}b)\equal{}(3ax^2\plus{}2bx\plus{}c)^2?$
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Context:
Athletes A, B, and C, along with 4 volunteers, are lining up for a group photo. If A and B must be next to each other and C cannot be at either end, find the total number of different ways to arrange them. Express your answer as a numerical value.
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Context:
In the trapezium FGHI, side GF is parallel to HI. The lengths of FG and HI are 50 and 20 respectively. Point J is on side FG such that segment IJ divides the trapezium into two parts of equal area. Find the length of FJ. Express your answer as a whole number.
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Context:
During training target practice, each soldier fired 10 times. One of them was successful and scored 90 points in total. How many times did he score 7 points if he scored 10 points 4 times and the results of the remaining shots were 7, 8, and 9 points? Note that there were no misses.
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Context:
设 $x, y, z \geq 0$, 且至多有一个为 0 , 求

$$
f(x, y, z)=\sqrt{\frac{x^{2}+256 y z}{y^{2}+z^{2}}}+\sqrt{\frac{y^{2}+256 z x}{z^{2}+x^{2}}}+\sqrt{\frac{z^{2}+256 x y}{x^{2}+y^{2}}}
$$

的最小值.
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Context:
Given a circle $C: (x-2)^2+y^2=1$ and a point $P$ on the line $l: x+y+1=0$. If there exists a line $m$ passing through point $P$ intersecting circle $C$ at points $A$ and $B$, with point $A$ being the midpoint of $PB$, then the range of the x-coordinate $x_0$ of point $P$ is \_\_\_\_\_.
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Context:
In a tetrahedron \(A B C D\), edge \(A B\) is perpendicular to edge \(C D\), \(\angle A C B = \angle A D B\), the area of the cross-section passing through edge \(A B\) and the midpoint of edge \(D C\) is \(S\), and \(|D C| = a\). Find the volume of tetrahedron \(A B C D\).
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Context:
A quality inspector sequentially selects products numbered 3, 13, 23,..., 93 from 100 products numbered 1 to 100 for inspection. What sampling method is used in this case? Provide your answer as the name of the specific sampling method.
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Context:
In the quadrilateral \(ABCD\), it is known that \(AB = BC\) and \(\angle ABC = \angle ADC = 90^\circ\). A perpendicular \(BH\) is dropped from vertex \(B\) to side \(AD\). Find the area of quadrilateral \(ABCD\) if it is known that \(BH = h\).
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Context:
In trapezoid \(A B C D\), the bases \(A D\) and \(B C\) are 8 and 18, respectively. It is known that the circumscribed circle of triangle \(A B D\) is tangent to lines \(B C\) and \(C D\). Find the perimeter of the trapezoid.
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