Context:
Compute the flux of water through the parabolic cylinder $S: y = x^2$, from $0 \le x \le 2$ and $0 \le z \le 3$, given that the velocity vector field is $F(x,y,z) = 3z^2\mathbf{i} + 6\mathbf{j} + 6xz\mathbf{k}$. Determine the flow rate of the fluid through this surface.
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Context:
In triangle $ABC$, point $D$ divides side $AB$ in half, and point $E$ lies on side $BC$, where segment $BE$ is 3 times smaller than side $BC$. Segments $AE$ and $CD$ intersect at point $O$. Find $AB$ if it is known that $AE = 5$, $OC = 4$, and $\angle AOC = 120^\circ$.
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Context:
设 x_1,x_2,\cdots,x_{60}\in[-1,1]，求\ds{i=1}{60}x_i^2(x_{i+1}-x_{i-1})的最大值，其中脚标按模 60 理解.
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Context:
Consider a sequence \( x_{n} \) such that \( x_{1} = x_{2} = 1 \), and \( x_{3} = \frac{2}{3} \). Suppose that \( x_{n} = \frac{x_{n-1}^{2} x_{n-2}}{2 x_{n-2}^{2} - x_{n-1} x_{n-3}} \) for all \( n \geq 4 \). Find the least \( n \) such that \( x_{n} \leq \frac{1}{10^{6}} \).
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Context:
A customer enters a store, selects an item costing 20 rubles, and gives the seller a 100-ruble banknote. The seller notices that he doesn't have any change. He goes to a neighboring department, exchanges the 100 rubles, and gives the customer the item and the change. The customer leaves. Suddenly, the seller from the neighboring department returns with the 100-ruble note, claiming it is fake. The first seller then gives him a real 100-ruble note of his own. How much did the first seller lose in total?
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Context:
Please write down a rational number that is smaller than $-\frac{1}{2}$: ______. (Write down one number)
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Context:
Let $AXYZB$ be a convex pentagon inscribed in a semicircle with diameter $AB$. Suppose that $AZ - AX = 6$, $BX - BZ = 9$, $AY = 12$, and $BY = 5$. Find the greatest integer not exceeding the perimeter of quadrilateral $OXYZ$, where $O$ is the midpoint of $AB$.
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Context:
A cube is assembled with $27$ white cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the cube is now completely black on the outside? Give an approximation of the size of your answer.
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Context:
On the surface of a sphere with center O, a point A draws three chords AB, AC, and AD of equal length, and the angle between any two of them is 60°. If the radius of the sphere is 3, then the length of chord AB is ______.
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Context:
Today, Ivan the Confessor prefers continuous functions  $f:[0,1]\to\mathbb{R}$  satisfying  $f(x)+f(y)\geq |x-y|$  for all pairs  $x,y\in [0,1]$ . Find the minimum of  $\int_0^1 f$  over all preferred functions.

(Proposed by Fedor Petrov, St. Petersburg State University)
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Context:
从 $0,1,2,3,4,5,6,7,8,9$ 这 10 个数中取 3 个数, 使其和为不小于 10 的偶数, 则不同的取法有 $\qquad$种。
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Context:
Dikembe wants to teach everyone at school about the importance of flossing, so he plans to bring enough packets of floss for everyone. There are 20 students in his class. Each student needs 1.5 yards of floss. He sees in the store that each packet of floss contains a certain amount of yards. If he buys the least amount necessary, he has 5 yards of floss left over. How many yards of floss does each packet contain?
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Context:
Each week Carina puts 20 more seashells in a jar than she did the week before. If there are 50 seashells in the jar this week, how many will there be in a month?
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Context:
Given \( f(x) = \max \left| x^3 - a x^2 - b x - c \right| \) for \( 1 \leq x \leq 3 \), find the minimum value of \( f(x) \) as \( a, b, \) and \( c \) range over all real numbers.
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Context:
Let the number of $2 pairs of socks be x, the number of $3 pairs of socks be y, and the number of $5 pairs of socks be z. Given that x + y + z = 15 and 2x + 3y + 5z = 35, determine the value of x.
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Context:
Find the probability that the chord $\overline{AB}$ does not intersect with chord $\overline{CD}$ when four distinct points, $A$, $B$, $C$, and $D$, are selected from 2000 points evenly spaced around a circle.
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Context:
对非负实数 $x$ 令 $\pi(x)$ 为不超过 $x$ 的素数个数，如 $\pi(1)=0, \pi(2)=1$ ，再令 $f(x)$为第 $\left[\frac{x}{4}\right]+1$ 个素数, 求 $\int_{0}^{100}\left(\pi(x)+\frac{1}{4} f(x)\right) \mathrm{d} x$ 的值。
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Context:
已知 $x^{2}=4 y, M(2,2)$ ，过 M 点的直线交抛物线于 $A, B$ 两点，过 $A, B$ 两点作抛物线的切线交与点 P ，求 $	riangle A B P$ 的最小面积。请给出最小面积的数值。
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Context:
The base of the pyramid \( T A B C D \) is a rectangle \( A B C D \). The height of the pyramid coincides with the lateral edge \( T A \), and the lateral edge \( T C \) is inclined to the plane of the base at an angle of \( 30^{\circ} \). A plane passing through the edge \( T C \) and parallel to the diagonal of the base \( B D \) forms an angle of \( 45^{\circ} \) with the plane of the base, and the distance between this plane and the diagonal \( B D \) is \( a \). What is the minimum area that a cross-section of the pyramid through the diagonal of the base \( A C \) can have?
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Context:
At a bus stop near Absent-Minded Scientist's house, two bus routes stop: #152 and #251. Both go to the subway station. The interval between bus #152 is exactly 5 minutes, and the interval between bus #251 is exactly 7 minutes. The intervals are strictly observed, but these two routes are not coordinated with each other and their schedules do not depend on each other. At a completely random moment, the Absent-Minded Scientist arrives at the stop and gets on the first bus that arrives, in order to go to the subway. What is the probability that the Scientist will get on bus #251?
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Context:
A flock of 155 birds sit down on a circle \( C \). Two birds \( P_{i}, P_{j} \) are mutually visible if \( m(P_{i} P_{j}) \leq 10^\circ \). Find the smallest number of mutually visible pairs of birds. (One assumes that a position (point) on \( C \) can be occupied simultaneously by several birds.)
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Context:
Find the number of integer solutions to the system of equations:
$$ac+5bd=19,$$
$$ad+bc=0.$$
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Context:
Given the complex number z = (x^2 - 1) + (x + 1)i, where z is purely imaginary, find the value of the real number x that satisfies this condition. Provide your answer as a single value or clearly state if multiple values satisfy the condition.
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Context:
Given the function $y = \frac{kx^2}{k^2 + k - 2}$, if its graph is a hyperbola located in the second and fourth quadrants, find the value of $k$ that satisfies these conditions. Express your answer as a single numerical value.
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Context:
The average age of 12 men is increased by years when two of them whose ages are 21 years and some years are replaced by two new men. The average age of the two new men is 34. What was the age of the second man who was replaced?
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Context:
In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$.  Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively.  What is the minimum possible value of $BE+DE+CD$?
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Context:
The quadrilateral \(ABCD\) is inscribed in a circle. \(I\) is the incenter of triangle \(ABD\). Find the minimum value of \(BD\) given that \(AI = BC = CD = 2\).
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Context:
一个 $10 \times 10$ 的数表在初始状态下每个格内均填有数字1，现在对数表进行如下操作：选择某一行或某一列，并将该行（列）中所有格子内的数变为其相反数。对于非负整数 $N$ ，如果可经过有限次操作，使得最终数表中 -1 的个数恰为 $N$ ，则称 $N$ 是 "好的"，则好的 $N$ 的个数为 $\qquad$.
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Context:
Let $T$ be a $30$-$60$-$90$ triangle with a hypotenuse of length $20$. Three circles, each externally tangent to the other two, have centers at the three vertices of $T$. The area of the union of the circles intersected with $T$ is $(m + n \sqrt{3}) \pi$ for rational numbers $m$ and $n$. Find $m + n$.
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Context:
From the beginning to the end of a certain year, the price of a stock rose by a certain percent. In 2008, it dropped 25 percent. In 2009, it rose 30 percent. The price of the stock at the end of 2009 was 117 percent of the stock's starting price in that certain year. By what percent did the stock price rise in that year?
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Context:
In an NGO, the daily average wages of 35 illiterate employees is decreased from Rs. 40 to Rs. 15, thus the average salary of all the literate and illiterate employees is decreased by Rs. 15 per day. Find the number of educated (literate) employees working in the NGO.
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Context:
Antonia received 400 emails in her mail. 1/4 were spam emails, while some of the remaining emails were promotional messages. There were 180 important emails in her inbox. What is the ratio of promotional messages to the remaining emails after removing spam emails?
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Context:
There exist constants $p$ and $q$ so that for any vectors $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c},$ the vectors $\mathbf{a} - 2 \mathbf{b} + 3 \mathbf{c},$ $2 \mathbf{a} + 3 \mathbf{b} - 4 \mathbf{c},$ and $p \mathbf{b} + q \mathbf{c}$ are always collinear.  Enter the ordered pair $(p,q).$
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Context:
There are 1200 medical professionals at a hospital consisting of doctors, nurses, administrative staff, and medical technicians. If the ratio of doctors to nurses is 3 to 4, the ratio of nurses to administrative staff is 2 to 1, and the ratio of administrative staff to medical technicians is 3 to 5, how many nurses are at the hospital?
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Context:
Given that $F\_1$ and $F\_2$ are two foci of the hyperbola $x^2-y^2=1$, and $P$ is a point on the hyperbola such that $\angle F\_1PF\_2=60^{\circ}$, determine the area of $\triangle F\_1PF\_2$.
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Context:
设 $a \in \mathbf{R}$, 且对任意实数 $b$ 均有 $\max _{x \in[0,1]}\left|x^{2}+a x+b\right| \geq 1$, 求 $a$ 的取值范围.
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Context:
Through the vertices $A, B$ and $C$ of the parallelogram $A B C D$ with sides $A B=3$ and $B C=5$, a circle is drawn, intersecting the line $B D$ at point $E$, and $B E=9$. Find the diagonal $B D$.

#
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Context:
设 P(z)=z^4-(6i+6)z^3+24iz^2-(18i-18)z-13. 求 P(z)=0 的四个根在复平面上对应的四个点所围成凸四边形的面积.
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Context:
Find the value of $y$ that satisfies the equation:

$$\sqrt{y+\sqrt{y+\sqrt{y+\sqrt{\cdots}}}}=\frac{y+\sqrt{y^2+4y}}{2}$$
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Context:
Let $n=p_1^{e_1}p_2^{e_2}\dots p_k^{e_k}=\prod_{i=1}^k p_i^{e_i}$, where $p_1<p_2<\dots<p_k$ are primes and $e_1, e_2, \dots, e_k$ are positive integers. Let $f(n) = \prod_{i=1}^k e_i^{p_i}$. Find the number of integers $n$ such that $2 \le n \le 2023$ and $f(n)=128$.
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Context:
设 A 是 5\times5 的方阵，A 的第一行是 (1,1,1,1,1)，第二行是 (1,-1,1,-1,1). 已知 \text{tr}(AA^T)=28，求 [\det A] 的最大可能值.
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Context:
If $\int_{1}^{m} (2x-1)dx=6$ where $m > 1$, then the constant term of the expansion of the polynomial $(x^{2}+ \frac{1}{x^{2}}-2)^{m}$ is ______.
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Context:
A large gathering of people stand in a triangular array with $2020$ rows, such that the first row has $1$ person, the second row has $2$ people, and so on. Every day, the people in each row infect all of the people adjacent to them in their own row. Additionally, the people at the ends of each row infect the people at the ends of the rows in front of and behind them that are at the same side of the row as they are. Given that two people are chosen at random to be infected with COVID at the beginning of day 1, what is the earliest possible day that the last uninfected person will be infected with COVID?
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Context:
If a line forms a $72^{\circ}$ angle with a plane, then the maximum angle formed by this line with a line in the plane passing through the foot of the perpendicular is $\_\_\_\_\_.$
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Context:
There are 300 non-zero integers written in a circle such that each number is greater than the product of the three numbers that follow it in the clockwise direction. What is the maximum number of positive numbers that can be among these 300 written numbers?
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Context:
Anne-Marie has a deck of 16 cards, each with a distinct positive factor of 2002 written on it. She shuffles the deck and begins to draw cards from the deck without replacement. She stops when there exists a nonempty subset of the cards in her hand whose numbers multiply to a perfect square. What is the expected number of cards in her hand when she stops?
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Context:
Through the point \((1,2)\), two tangent lines to the circle \(x^2 + y^2 = 1\) are drawn. What is the area of the quadrilateral formed by these two tangent lines and the \(x\)-axis and \(y\)-axis?
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Context:
Two linear functions \( f(x) \) and \( g(x) \) satisfy the properties that for all \( x \),
- \( f(x) + g(x) = 2 \)
- \( f(f(x)) = g(g(x)) \)
and \( f(0) = 2022 \). Compute \( f(1) \).
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Context:
Find four distinct positive integers $a, b, c, d$ less than $15$ which are invertible modulo $15$. Calculate the remainder when $(abc + abd + acd + bcd)(abcd)^{-1}$ is divided by $15$.
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Context:
What is the units (i.e., rightmost) digit of \[ \left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor ? \]
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Context:
The increasing sequence of positive integers $a_1, a_2, a_3, \dots$ is defined by the rule
\[a_{n + 2} = a_{n + 1} + a_n\]
for all $n \ge 1.$ If $a_7 = 210$, then find $a_8.$
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Context:
Problem 3. In a certain company, there are 100 shareholders, and any 66 of them collectively own at least $50\%$ of the company's shares. What is the largest percentage of all shares that one shareholder can own? (The percentage of shares in the company owned by a shareholder can be non-integer.)
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Context:
If there exist two different points on the graph of the function $y=f(x)$ such that the tangents to the function's graph at these two points are perpendicular to each other, then the function $y=f(x)$ is said to have property $T$. If the function $g(x)=ax-\frac{c}{2}+bsinxcosx+c{cos^2}x$ has property $T$, where $a$, $b$, and $c$ are real numbers and satisfy $b^{2}+c^{2}=1$, then the range of the real number $a+b+c$ is ______.
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Context:
Given a set of linearly independent vectors in $\mathbb{R}^3$: $\left\{a, b, c\right\}$. Determine the value of $\lambda$ such that the dimension of the subspace generated by the vectors $2a - 3b, (\lambda - 1)b - 2c, 3c - a, \lambda c - b$ is 2. In other words, find $\lambda$ for which there are only two linearly independent vectors in the span of the given vectors.
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Context:
Given that $f(x)$ is an even function defined on $\mathbb{R}$ and is differentiable, its derivative is $f′(x)$. When $x < 0$, we have $2f(x)+xf′(x) > x^{2}$. Find the solution set of the inequality $(x+2014)^{2}f(x+2014)-4f(-2) < 0$. Express your answer as an interval.
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Context:
Two triangles intersect to form seven finite disjoint regions, six of which are triangles with area 1. The last region is a hexagon with area \(A\). Compute the minimum possible value of \(A\).
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Context:
Given the parabola y^2 = 4x whose directrix intersects the x-axis at point P, draw line l through point P with the slope k (k > 0), intersecting the parabola at points A and B. Let F be the focus of the parabola. If |FB| = 2|FA|, calculate the length of segment AB. Express your answer as a single value.
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Context:
For any set \( S \), let \( |S| \) represent the number of elements in set \( S \) and let \( n(S) \) represent the number of subsets of set \( S \). If \( A \), \( B \), and \( C \) are three finite sets such that:

(1) \( |A|=|B|=2016 \);
(2) \( n(A) + n(B) + n(C) = n(A \cup B \cup C) \),

then the maximum value of \( |A \cap B \cap C| \) is ________.
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Context:
记 $Q$ 为 $1,2, \cdots, 100$ 的若干个排列组成的集合，且满足对于任意的 $1 \leqslant a 、 b \leqslant 100$ ， $a \neq b$ ，至多存在一个 $\sigma \in Q$ ，使得在 $\sigma$ 中 $a$的下一个数恰为 $b$ 。求集合 $Q$ 的元素个数的最大值。
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Context:
How many times are the hands of a clock at right angle in 2 days?
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Context:
Two particles are placed at opposite vertices of the same face of a cube. If they are on the same edge at some time t, then they remain in the same position at time t+1. If not, one of the particles is chosen with equal probability and at any time t+1, it goes to one of the three neighboring vertices with probability 1/3. What is the expected value of the minimum time they are in the same edge?
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Context:
Among the 100 natural numbers 1, 3, 5, 7, …, 199, select several numbers so that none of the selected numbers is an integer multiple of another. The maximum number of such numbers that can be selected is     .
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Context:
Paco had 26 salty cookies and 17 sweet cookies. He ate some sweet cookies and 9 salty cookies. Paco had 17 salty cookies left. How many sweet cookies did Paco eat?
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Context:
A trainee cook took two buckets of unpeeled potatoes and peeled everything in an hour. Meanwhile, $25\%$ of the potatoes went into peels. How much time did it take for him to collect exactly one bucket of peeled potatoes?
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Context:
In how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$? For example, $2+10$, $10+2$, and $3+2+2+5$ are three such ways.
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Context:
设f是定义在｛0,1，···，2005｝上的函数，取值于非负整数，对任意满足定义域的变量x，都有f(2x+1)=f(2x),f(3x+1)=f(3x),f(5x+1)=f(5x).问：这个函数最多能取到多少个函数值？
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Context:
For constant $k>1$, 2 points $X,\ Y$ move on the part of the first quadrant of the line, which passes through $A(1,\ 0)$ and is perpendicular to the $x$ axis, satisfying $AY=kAX$. Let a circle with radius 1 centered on the origin $O(0,\ 0)$ intersect with line segments $OX,\ OY$ at $P,\ Q$ respectively. Express the maximum area of $\triangle{OPQ}$ in terms of $k$.

[i]2011 Tokyo Institute of Technology entrance exam, Problem 3[/i]
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\boxed{\frac{1}{2} \left( \frac{k-1}{k+1} \right)}


Context:
In how many ways can we change the sign $*$ with $+$ or $-$, such that the following equation is true?
\[ 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 = 29 \]
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Context:
Points $E$ and $F$ are chosen on sides $BC$ and $CD$ respectively of rhombus $ABCD$ such that $AB=AE=AF=EF$, and $FC,DF,BE,EC>0$. Compute the measure of $\angle ABC$.
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Context:
ABCD is a trapezoid with AB parallel to CD. M is the midpoint of AD, ∠MCB = 150°, BC = x, and MC = y. Find the area of trapezoid ABCD in terms of x and y.
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Context:
In the acute-angled triangle $ABC$, the angle $\angle BAC = 60^\circ$. A circle constructed with $BC$ as the diameter intersects side $AB$ at point $D$ and side $AC$ at point $E$. Determine the ratio of the area of quadrilateral $BDEC$ to the area of triangle $ABC$.
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Context:
在正方体中随机取 3 条棱, 它们两两异面的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Given the number $500 \ldots 005$ (with 80 zeros in between). We need to replace any two zeros with non-zero digits so that the resulting number is divisible by 165. In how many ways can this be done?
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Context:
For a modified toothpick pattern, the first stage is constructed using 5 toothpicks. If each subsequent stage is formed by adding three more toothpicks than the previous stage, what is the total number of toothpicks needed for the $15^{th}$ stage?
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Context:
已知正数 $a, b, c$, 满足 $(a+b-c)\left(\frac{1}{a}+\right.$ $\left.\frac{1}{b}-\frac{1}{c}\right)=3$, 求 $f=\left(a^{4}+b^{4}+c^{4}\right)\left(\frac{1}{a^{4}}+\frac{1}{b^{4}}+\frac{1}{c^{4}}\right)-240\sqrt{3}$ 的最小值.
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Context:
Using Cauchy's integral formula, evaluate the contour integral $$\oint_{\Gamma} \frac{\cos(z)+i\sin(z)}{(z^2+36)(z+2)}dz$$ where $\Gamma$ is the circle centered at the origin with radius $3$ traced counterclockwise. Note that the denominator cannot be factored further for partial fraction decomposition due to non-interior points.
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Context:
The quadrilateral $ABCD$ satisfies the following conditions: \( \angle ABC = \angle BCD = 150^{\circ} \). Additionally, \( AB = 18 \) and \( BC = 24 \). Equilateral triangles \( \triangle APB \), \( \triangle BQC \), and \( \triangle CRD \) are constructed outside the quadrilateral. If \( P(X) \) represents the perimeter of polygon \( X \), then the equation \( P(APQRD) = P(ABCD) + 32 \) holds. Determine the length of side \( CD \).
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Context:
Let \( P_{1} \) and \( P_{2} \) be any two different points on the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\), and let \( P \) be a variable point on the circle with diameter \( P_{1} P_{2} \). Find the maximum area of the circle with radius \( OP \).
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Context:
Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with left and right foci denoted as $F_{1}$ and $F_{2}$, respectively. Draw a line $l$ passing through the right focus that intersects the ellipse at points $P$ and $Q$. What is the maximum area of the inscribed circle of triangle $\triangle F_{1} P Q$?
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Context:
A conveyor system produces on average 85% of first-class products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the frequency of first-class products from 0.85 in absolute magnitude does not exceed 0.01?
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Context:
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?

*(A. Golovanov)*
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Context:
What is the minimal polynomial of $\sqrt{3} + \sqrt{5}$ over $\mathbb{Q}(\sqrt{5})$?
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Context:
Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex on the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies?
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Context:
In the first year, Grandma Molly created 4 statues and placed them on her front lawn. Each year after, she began creating sets of statues that were inspired by mathematical sequences. In the second year, she created two sets. The first set contained statues based on the Fibonacci sequence, placing 1, 1, and 2 statues on her lawn. The second set she placed was inspired by triangular numbers, adding 3 statues. However, that year a hail storm broke 2 statues from each set, which she threw away. In the third year, Molly added two more sets. One set contained 3 statues as the next number in the Fibonacci sequence and the other set contained 6 statues as the next triangular number. Unfortunately, another hail storm broke statues again. This time, 3 statues from the Fibonacci set and 4 from the triangular set were destroyed. For the fourth year, keeping to her tradition, she was planning to add two more sets again. The first set was supposed to have 5 statues, as the next Fibonacci number, and the other 10 statues, as the next triangular number, but a forecasted snow storm made her reconsider. So she decided to add a number of statues equal to the subtraction of the sum of the broken statues in previous years from the total number she was planning to add. After four years of such intricate work and adjustments for weather, how many turtle statues were left on her front lawn?
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Context:
For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean of $x$ and $y$ equal to $12^{10}$?
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Context:
The focus of the parabola \( y^{2}=2px \) \( (p>0) \) is point \( F \), and the directrix is line \( l \). Points \( A \) and \( B \) are on the parabola and satisfy \( \angle AFB=\frac{\pi}{3} \). Let \( M \) be the midpoint of segment \( AB \), and let \( N \) be the projection of \( M \) onto \( l \). Find the maximum value of \( \frac{|MN|}{|AB|} \).
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Context:
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?
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Context:
If the angle formed by two skew lines is 60°, then this pair of skew lines is called a "Golden Skew Line Pair." Among all the lines connecting the vertices of a cube, there are _______ pairs of "Golden Skew Line Pairs."
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Context:
Let the integer $n \geq 3$, and let $P = \{1, 2, \ldots, n\}$. $A$ and $B$ are two non-empty subsets of $P$. The number of pairs of sets $(A, B)$ such that the maximum number in $A$ is less than the minimum number in $B$ is: ______.
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Context:
Natural numbers $1,2, \ldots, 64$ are written in the cells of an $8 \times 8$ table so that for all $k=1,2,3, \ldots, 63$, the numbers $k$ and $k+1$ are in adjacent cells along one side. What is the maximum possible sum of the numbers on the main diagonal?
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Context:
If a semicircle is used to form the lateral surface of a cone, what is the maximum angle between any two generatrices?
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Context:
Suppose that for some positive integer $n$, the first two digits of $5^n$ and $2^n$ are identical. Suppose the first two digits are $a$ and $b$ in this order. Find the two-digit number $\overline{ab}$.
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Context:
Given a sphere resting on a flat surface and a 1.5 m tall post, the shadow of the sphere is 15 m and the shadow of the post is 3 m, determine the radius of the sphere.
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Context:
Peyton has 3 children - John, Samantha, and Heather. John gets 2 juice boxes and Samantha gets 1 juice box in their lunch, 5 days a week. Heather gets 3 juice boxes on Monday and Wednesday and 2 juice boxes on Tuesday and Thursday while on Friday, she gets just 1 juice box. The school year is 18 weeks long, but Heather has a one-week break, making her school year 17 weeks. Samantha attends after-school activities, which shorten her school year to 16 weeks. Additionally, during the last 2 weeks of the school year, there is a special holiday break where none of the children need packed lunches. How many juice boxes will Peyton need for the entire school year for all of her children, considering all these factors?
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Context:
How many such five-digit Shenma numbers exist, where the middle digit is the smallest, the digits increase as they move away from the middle, and all the digits are different?
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Context:
In $\triangle ABC$ with $AC>AB$, let $D$ be the foot of the altitude from $A$ to side $\overline{BC}$, and let $M$ be the midpoint of side $\overline{AC}$. Let lines $AB$ and $DM$ intersect at a point $E$. If $AC=8$, $AE=5$, and $EM=6$, find the square of the area of $\triangle ABC$.
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Context:
Given two circles $O\_1$: $(x-2)^2 + y^2 = 16$ and $O\_2$: $x^2 + y^2 = r^2 (0 < r < 2)$, a moving circle $M$ is tangent to both circles $O\_1$ and $O\_2$. The locus of the center of the moving circle $M$ forms two ellipses, with eccentricities $e\_1$ and $e\_2 (e\_1 > e\_2)$. Find the minimum value of $e\_1 + 2e\_2$.
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Context:
Four pairs of siblings, in each pair one is a boy and the other is a girl. We divide them into three groups so that each group has at least two people, but siblings do not end up in the same group. In how many ways can this be done?
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Context:
In the plane Cartesian coordinate system $xOy$, there is a circle $C:(x+2)^{2}+(y-m)^{2}=3$. If circle $C$ has a chord $AB$ with a midpoint $G$, and the length of $AB$ is twice the distance from $G$ to the center $O$ of circle $C$, i.e., $|AB|=2|GO|$, then the range of the real number $m$ is ______________.
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Context:
In a hockey league, there are 15 teams. The number of times each team faces the other teams varies based on their current ranking. The top 5 ranked teams face off against the other top 5 teams 12 times each, and the remaining 10 teams 8 times each. Rank 6 to 10 teams play against each other 10 times each and against the remaining 5 teams 6 times each. Finally, the bottom 5 ranked teams face off against each other 8 times each. How many games are played in the season?
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Context:
Find the angle $\angle XYZ$ where $X, Y, Z$ are points in the complex plane such that $$\frac{X-Y}{X-Z}+\frac{\overline{X-Y}}{\overline{X-Z}}=2.$$
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Context:
现有边长为3、4、5的三角形两个，边长分别为4、5、41的三角形四个，边长分别为$\frac { 5 } { 6 }\sqrt{2} $、4、5的三角形六个．用上述三角形为面，可以拼成__个四面体.
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Context:
We define $N$ as the set of natural numbers $n < 10^6$ with the following property: There exists an integer exponent $k$ with $1 \leq k \leq 43$, such that $2012 \mid n^k - 1$. Find $|N|$.
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Context:
if 4^x + y = 34, 2^x - y = 20, for integers of x and y, y ^ 2 = ?
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Context:
For any nonnegative integer $n$, let $S(n)$ be the sum of the digits of $n$. Let $K$ be the number of nonnegative integers $n \leq 10^{10}$ that satisfy the equation \[ S(n) = (S(S(n)))^2. \] Find the remainder when $K$ is divided by $1000$.
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Context:
A fan can create an airflow of 10 liters per second. If the fan works for a certain amount of time each day, the amount of airflow it will create in one week is 42000 liters. How long does the fan work each day?
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Context:
已知实数 x 和 y 满足 $5 x^{2}+6 x y+5 y^{2}-8 x-8 y=-3$. 记 $x+y-x^{2}-y^{2}$ 的最大值为 I ，则 $[100 I]=$？
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Context:
In 1854, geologist W.K. Loftus excavated two clay tablets in Senkereh (a location in ancient Babylon) on which one of the tablets recorded:
$9^2 = 81 = 60 + 21 = 1•21$
$10^2 = 100 = 60 + 40 = 1•40$
$11^2 = 121 = 2 \times 60 + 1 = 2•1$
$12^2 = 144 = 2 \times 60 + 24 = 2•24$
...
Following this pattern, $58^2 = \_\_\_\_\_\_$．(Write in the form of “a•b”)
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Context:
Given a geometric sequence with an even number of terms, where the sum of the even-numbered terms is twice the sum of the odd-numbered terms, the first term is 1, and the sum of the two middle terms is 24, find the total number of terms in the sequence. Express your answer as a whole number.
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Context:
已知三棱锥$P-ABC$的四个顶点均在某球面上，$PC$为该球的直径，$\vartriangle ABC$是边长为4的等边三角形，三棱锥$P-ABC$的体积为$\frac{16}{3}$，则此三棱锥的外接球的表面积为______.
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Context:
A round robin tournament is held with $2016$ participants. Each player plays each other player once, and no games result in ties. We say a pair of players $A$ and $B$ is a dominant pair if all other players either defeat both $A$ and $B$ or are defeated by both $A$ and $B$. Find the maximum number of dominant pairs.
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Context:
Evaluate the contour integral $$\oint_{|z|=1}\frac{dz}{1+2z+2/z}$$ using Cauchy's residue theorem.
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Context:
Let triangle $ABC$ have $\angle BAC = 45^{\circ}$ and circumcircle $\Gamma$. Let $M$ be the intersection of the angle bisector of $\angle BAC$ with $\Gamma$. Let $\Omega$ be the circle tangent to segments $\overline{AB}$ and $\overline{AC}$ and internally tangent to $\Gamma$ at point $T$. Given that $\angle TMA = 45^{\circ}$ and that $TM = \sqrt{100 - 50 \sqrt{2}}$, the length of $BC$ can be written as $a \sqrt{b}$, where $b$ is not divisible by the square of any prime. Find $a + b$.
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Context:
Given that function $f(x)$ is an odd function defined on $\mathbb{R}$ with its derivative ${f}'(x)$, when $x < 0$, $2f(x) + x{f}'(x) < 0$ always holds. Determine the relationship among $f(1)$, $2016f(\sqrt{2016})$, and $2017f(\sqrt{2017})$, expressing your answer using inequality signs.
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Context:
We are given some three-element subsets of $\{1,2, \dots ,n\}$ for which any two of them have at most one common element. We call a subset of $\{1,2, \dots ,n\}$ nice if it doesn't include any of the given subsets. If, no matter how the three-element subsets are selected initially, we can add one more element to every 29-element nice subset while keeping it nice, find the minimum value of $n$.
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Context:
Let $0$ $\leq$ $\alpha$ $\leq$ $\pi$. Denote by $V_n$$(\alpha)$ the number of sign changes in the sequence ${u_n}$ where $u_n$ $=$ $\displaystyle \cos n\alpha$. Find the limit of the sequence $\frac{V_n(\alpha)}{n}$ as $n$ $\to$ $\infty$.
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Context:
For a positive integer  $n$ , let  $f(n)$  be the sum of the positive integers that divide at least one of the nonzero base  $10$  digits of  $n$ . For example,  $f(96)=1+2+3+6+9=21$ . Find the largest positive integer  $n$  such that for all positive integers  $k$ , there is some positive integer  $a$  such that  $f^k(a)=n$ , where  $f^k(a)$  denotes  $f$  applied  $k$  times to  $a$ .

*2021 CCA Math Bonanza Lightning Round #4.3*
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Context:
Given that $f(x)$ is an even function defined on $\mathbb{R}$, when $x < 0$, $xf'(x) - f(x) > 0$, and $f(1) = 0$, then the solution set of the inequality $\frac{f(x)}{x} < 0$ is ______.
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Context:
The integers 2, 2, 5, 5, 8, and 9 are written on six cards. Any number of the six cards is chosen, and the sum of the integers on these cards is determined. Note that the integers 1 and 30 cannot be obtained as sums in this way. How many of the integers from 1 to 31 cannot be obtained as sums?
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Context:
对正整数 n，考虑含有 n 阶元的有限非交换群，记 a_n 为这些群的阶的最小可能值，则 a_1+a_2+a_3+a_4 的值为__________.
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Context:
Given a quadrilateral $ABCD$ where the internal angles form an arithmetic progression, the angles are such that $\angle ADB$ is double the $\angle DBA$. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Furthermore, the angles within triangle $ABD$ form an arithmetic progression as well. Find the sum of the two smallest angles in $ABCD$.
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Context:
Consider the polynomial \( p(x) = x^n + n x^{n-1} + a_2 x^{n-2} + \cdots + a_n \) having all real roots. If \( r_1^{16} + r_2^{16} + \cdots + r_n^{16} = n \), where the \( r_j \) are the roots of \( p(x) \), find all such roots.
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Context:
An up-right path between two lattice points $P$ and $Q$ is a path from $P$ to $Q$ that takes steps of length 1 unit either up or to the right. How many up-right paths from $(0,0)$ to $(7,7)$, when drawn in the plane with the line $y=x-2.021$, enclose exactly one bounded region below that line?
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Context:
Three sets of parallel lines and another set of five parallel lines intersect on a plane. Calculate the number of parallelograms that can be formed. Express your answer as a single integer.
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Context:
We write the following equation: \((x-1) \ldots (x-2020) = (x-1) \ldots (x-2020)\). What is the minimal number of factors that need to be erased so that there are no real solutions?
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Context:
In the given diagram, in the cube \( A B C D - A_{1} B_{1} C_{1} D_{1} \), what is the measure of the dihedral angle \( B - A_{1} C - D \)?
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Context:
Given the sequence $\{a_n\}$ where $a_n=n^2-kn$ ($n\in\mathbb{N}^*$), and $\{a_n\}$ is monotonically increasing, find the range of values for $k$. Express your answer as an interval.
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Context:
Triangle $ABC$ has side lengths $AB = 11, BC=24$, and $CA = 20$. The bisector of $\angle{BAC}$ intersects $\overline{BC}$ in point $D$, and intersects the circumcircle of $\triangle{ABC}$ in point $E \ne A$. The circumcircle of $\triangle{BED}$ intersects the line $AB$ in points $B$ and $F \ne B$. What is the length of $CF$?
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Context:
A couple with four children, ages 6, 10, 13, and 8 years old, decided to go to an amusement park. The regular ticket costs $129, but children below 12 years old have a $15 discount. If they gave the cashier $800, how much change will they receive?
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Context:
The sequence \( x_{0}, x_{1}, x_{2}, \ldots \) has its first two terms positive, and it holds that \( x_{n+2} = \frac{x_{n+1} + 1}{x_{n}} \). Express the 2003rd term of the sequence in terms of \( x_{0} \) and \( x_{1} \).
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Context:
Let \( A \) be an \( n \)-element subset of the set \( \{1, 2, \cdots, 2021\} \), such that any two numbers in \( A \) are neither coprime nor do they have a divisibility relationship. What is the maximum value of \( n \)?
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Context:
Find the minimum number  $n$  such that for any coloring of the integers from  $1$  to  $n$  into two colors, one can find monochromatic  $a$ ,  $b$ ,  $c$ , and  $d$  (not necessarily distinct) such that  $a+b+c=d$ .
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Context:
Let ABCD be a quadrilateral inscribed in a circle with diameter AC, and let E be the foot of the perpendicular from D onto AB. Given that AD = DC and the area of quadrilateral ABCD is 24 cm^2, calculate the length of DE in cm.
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Context:
13.5.3 * Given: The hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>b>0)$ has an eccentricity $e=2+\sqrt{6}-\sqrt{3}-\sqrt{2}$, and a line $e$ passing through its right focus $F_{2}$ and perpendicular to the $x$-axis intersects the hyperbola at points $A$ and $B$. Find the value of $\angle A F_{1} F_{2}$.
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Context:
记集合 \(T=\{0,1,2,3,4,5,6\}, M=\left\{\left.\frac{a_{1}}{7}+\frac{a_{2}}{7^{2}}+\frac{a_{3}}{7^{3}}+\frac{a_{4}}{7^{4}} \right\rvert\, a_{i} \in T, i=1,2,3,4\right\}\)。将 \(M\) 中的元素按从大到小的顺序排列，求第 2005 个数。答案的形式为 \(\frac{k}{7}+\frac{m}{7^{2}}+\frac{n}{7^{3}}+\frac{p}{7^{4}}\)，请给出k+m+n+p的值。
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Context:
Lindsey owns 4 more cars than Cathy. Susan owns 2 fewer cars than Carol. Carol owns twice as many cars as Cathy. Erica owns 25% more cars than Lindsey. If Cathy owns 5 cars, how many total cars do they all own put together?
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Context:
In a country with 15 cities, some of them are connected by air routes belonging to three airlines. It is known that even if any one of the airlines stops flying, it will still be possible to travel from any city to any other city (possibly with transfers), using the flights of the remaining two airlines. What is the minimum number of air routes that can exist in the country?
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Context:
设函数 $f(x)$ 为奇函数, 当 $x \geqslant 0$ 时, $f(x)=x^{2}$ 。则不等式 $$ f(f(x))+f(x-1)<0 $$ 的解集的上限为一个涉及 \sqrt{} 的表达式，请给出该表达式的整数近似值。
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Context:
How many different ways can we write digits in place of the letters such that a valid subtraction is obtained? (Naturally, different letters correspond to different digits, and zero is not used as the starting digit of a number.)

$$
\overline{F T C} - \overline{E T O} = \overline{K K}
$$
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Context:
To prevent the spread of the novel coronavirus, individuals need to maintain a safe distance of at least one meter between each other. In a certain meeting room with four rows and four columns of seats, the distance between adjacent seats is more than one meter. During the epidemic, for added safety, it is stipulated that when holding a meeting in this room, there should not be three people seated consecutively in any row or column. For example, the situation shown in the first column of the table below does not meet the condition (where "$\surd $" indicates a seated person). According to this rule, the maximum number of participants that can be accommodated in this meeting room is ____.

|       |       |       |       |
|-------|-------|-------|-------|
| $\surd $ |       |       |       |
| $\surd $ |       |       |       |
| $\surd $ |       |       |       |
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Context:
James decides to start making his own coffee. He buys a coffee machine for $200 and gets a $20 discount. He figures it will cost him $3 a day to make his coffee. He previously bought 2 coffees a day for a certain price each. The machine pays for itself in 36 days. How much did he pay for each coffee he bought before?
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Context:
Let $P_1, P_2, \dots, P_n$ be points on the plane. There is an edge between distinct points $P_x, P_y$ if and only if $x \mid y$. Find the largest $n$, such that the graph can be drawn with no crossing edges.
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Context:
A trapezium and a triangle are joined along one of their sides forming a complex 5-sided shape. The trapezium has parallel sides of 20 cm and 18 cm, and the distance between them is 5 cm. The triangle is a right-angled triangle, with its right angle adjacent to the shorter parallel side of the trapezium and one leg along the longer parallel side. If the longer leg of the right-angled triangle has a length of 7 cm, find the area of the combined shape.
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Context:
A rectangular tea bag $PART$ has a logo in its interior at the point $Y$. The distances from $Y$ to $PT$ and $PA$ are $12$ and $9$ respectively, and triangles $\triangle PYT$ and $\triangle AYR$ have areas $84$ and $42$ respectively. Find the perimeter of pentagon $PARTY$.
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Context:
A gives B as many cents as B has and C as many cents as C has. Similarly, B then gives A and C as many cents as each then has. C, similarly, then gives A and B as many cents as each then has. After this, each person gives half of what they have to each other person. If each finally has 24 cents, calculate the number of cents A starts with.
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Context:
Michael is celebrating his fifteenth birthday today. Calculate the total number of Sundays that have occurred in his lifetime.
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Context:
Fresh grapes contain a certain percentage of water by weight, while dried grapes contain 20% water by weight. The weight of dry grapes available from 25 kg of fresh grapes is 3.125 kg. What is the percentage of water in fresh grapes by weight?
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Context:
In the given sequence \(1, 4, 8, 10, 16, 19, 21, 25, 30, 43\), how many groups of consecutive numbers have a sum that is divisible by 11?
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Context:
5 设$x _ { 1 }$1$x _ { 2 } 、 x _ { 3 } 、 x _ { 4 }$都是正实数，且$x _ { 1 } + x _ { 2 } + x _ { 3 } + x _ { 4 } = \pi$，求$A = ( 2 \sin ^ { 2 } x _ { 1 } + \frac { 1 } { \sin ^ { 2 } x _ { 1 } } ) ( 2 \sin ^ { 2 } x _ { 2 } + \frac { 1 } { \sin ^ { 2 } x _ { 2 } } ) ( 2 \sin ^ { 2 } x _ { 3 } + \frac { 1 } { \sin ^ { 2 } x _ { 3 } } ) ( 2 \sin ^ { 2 }$ $\frac { 1 } { \sin ^ { 2 } x _ { 4 } } )$ 的最小值．（1991年中国国家集训队测试题）
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Context:
Given the function $f(x) = |x+1| + |x+2| + ... + |x+2018| + |x-1| + |x-2| + ... + |x-2018|$ where $x \in \mathbb{R}$, and $f(a^2 - 3a + 2) = f(a - 1)$, find the sum of all distinct integer values of $a$.
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Context:
设 \left[\df{10^{2022}}{10^{100}-9}\right] 能被 10^n 整除，则非负整数 n 的最大可能值为__________.
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Context:
Jana had to calculate the product of two six-digit numbers for her homework. When copying from the board, she omitted one digit from one of the numbers, and instead of a six-digit number, she wrote only 85522. When she got home, she realized her mistake. However, she remembered that the number she had copied incorrectly was divisible by three. She decided to try to determine what the original number could have been.

Determine how many such six-digit numbers exist.

(M. Dillingerová)
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Context:
The lateral surface of a right circular cone is developed into a circle of radius \( R \) with a removed sector corresponding to a central angle of \( (2-\sqrt{3}) \pi \). Find the maximum area of a plane section of the cone that passes through its apex.
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Context:
In tetrahedron S-ABC, if $SA=CB=\sqrt{5}$, $SB=AC=\sqrt{10}$, and $SC=AB=\sqrt{13}$, then the surface area of the circumscribed sphere of this tetrahedron is ________.
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Context:
Given a hyperbola $\frac{x^2}{4} - y^2 = 1$ and a straight line $l$ passing through the right focus of the hyperbola intersecting it at points A and B, if there exist exactly three such lines for which the length $|AB|=a$, then find the range of values for the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the derivative.

\[ y = -\frac{1}{4} \arcsin \frac{5+3 \cosh x}{3+5 \cosh x} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the arc length of the curve $$x(t) = \frac{\cos(3t)}{t^3}; \ y(t)=\frac{\sin(3t)}{t^3};\ t\ge \frac{1}{\sqrt3} $$
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Context:
A man uses a piece of canvas having an area of $$ 551\;\mathrm m^2, $$ to make a conical tent of base radius $$ 7\;\mathrm m $$. Assuming that all the stitching margins and wastage incurred while cutting, amount to approximately $$ 1\;\mathrm m^2 $$, find the volume of the tent that can be made with it.
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Context:
What is the sum of the values of $n$ with $2 \leq n \leq 10$ for which a pizza cut into 10 pieces, with two pieces each $\frac{1}{24}$ of the whole pizza, four pieces each $\frac{1}{12}$, two pieces each $\frac{1}{8}$, and two pieces each $\frac{1}{6}$, cannot be distributed equally among $n$ friends?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a dumpling restaurant, dumplings can be ordered in portions of 6, 9, and 20 pieces. Therefore, not every number of dumplings can be ordered using these sets; for example, it is not possible to buy 1, 2, 3, 4, 5, 7, or 8 dumplings. What is the largest number of dumplings that cannot be ordered at the restaurant?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A certain lottery has tickets labeled with the numbers $1,2,3, \ldots, 1000$. The lottery is run as follows: First, a ticket is drawn at random. If the number on the ticket is odd, the drawing ends; if it is even, another ticket is randomly drawn (without replacement). If this new ticket has an odd number, the drawing ends; if it is even, another ticket is randomly drawn (again without replacement), and so forth, until an odd number is drawn. Then, every person whose ticket number was drawn (at any point in the process) wins a prize. You have ticket number 1000. What is the probability that you get a prize?
Please reason step by step, and put your final answer within \boxed{}.


Context:
It is known that the numbers \( x, y, z \) form an arithmetic progression in the given order with a common difference \( \alpha = \arccos \left(-\frac{3}{7}\right) \), and the numbers \( \frac{1}{\cos x}, \frac{7}{\cos y}, \frac{1}{\cos z} \) also form an arithmetic progression in the given order. Find \( \cos^{2} y \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x)=x^{4}+ax^{3}+2x^{2}+b$, where $a, b \in \mathbb{R}$. If the function $f(x)$ has an extreme value only at $x=0$, find the range of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{-\frac{8}{3} \leq a \leq \frac{8}{3}}


Context:
One day in the 20th century (1900-1999), a younger brother said to his older brother: "Brother, look, if you add up the four digits of the year you were born, it gives my age." The elder brother responded: "Dear brother, you are right! The same applies to me, if I add up the four digits of the year you were born, it gives my age. Additionally, if we swap the two digits of our respective ages, we can get each other's age." Given that the two brothers were born in different years, in which year did this conversation take place?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, if for any real number $x$, the inequality $|{x\overrightarrow{a}+\overrightarrow{b}}|≥\frac{{\sqrt{3}}}{2}$ always holds, find the range of values for the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. Express your answer as an interval of angles in radians, using standard mathematical notation for intervals.
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Context:
At time 0, an ant is at \((1,0)\) and a spider is at \((-1,0)\). The ant starts walking counterclockwise along the unit circle, and the spider starts creeping to the right along the \(x\)-axis. It so happens that the ant's horizontal speed is always half the spider's. What will the shortest distance ever between the ant and the spider be?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $P(x)$ be a polynomial with integer coefficients, leading coefficient 1, and $P(0) = 3$. If the polynomial $P(x)^2 + 1$ can be factored as a product of two non-constant polynomials with integer coefficients, and the degree of $P$ is as small as possible, compute the largest possible value of $P(10)$.
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Context:
Find the number of different quadruples \((a, b, c, d)\) of positive integers such that \(ab = cd = a + b + c + d - 3\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sequence $\{a_{n}\}$ satisfies ${a_{n+1}}=(2|\sin \frac{n\, !!\, \pi\, !!}{2}|-1){a_{n}}+2n$. Find the sum of the first 100 terms of the sequence $\{a_{n}\}$.
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Context:
Yves and his seven siblings ordered pizza and asked to have it cut into 24 slices. During dinner time, they only ate one-third of it. The next day, Yves ate one-fifth of the remaining pizza. Then his three oldest siblings ate 3 slices each, and the remaining siblings ate 2 slices each. How many slices of pizza were left?
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Context:
If point P $(a, b)$ is in the second quadrant, then the point symmetric to $P' (a-1, -b)$ with respect to the y-axis is in the     quadrant.
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Context:
Given $A = \{a + 2, (a + 1)^2, a^2 + 3a + 3\}$ and $1 \in A$, find the value of the real number $a$.
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Context:
将正九边形的每个顶点等概率地涂上红、蓝两种颜色之一, 则存在三个同色的顶点构成锐角三角形的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Let $X(t)$ be a stationary random variable with expected value $E[X(t)] = m$ and covariance function $r_X(\tau) = 2e^{-|\tau|}$. Calculate the variance of $\int_0^1X(t)dt$, $$V[\int_0^1X(t)dt].$$
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Context:
Example 5 Let $S=\{1,2,3, \cdots, 98,99,100\}$, if a three-element subset $A=\{a, b, c\}$ of $S$ satisfies $a+b=3 c$, then $A$ is said to have property $P$. Find the number of all three-element subsets of $S$ that have property $P$.
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Context:
A sphere inscribed in a pyramid \( SABC \) touches the faces \( SAB, SBC, SCA \) at points \( D, E, F \) respectively.

Find all possible values of the sum of the angles \( SDA, SEB \), and \( SFC \).
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Context:
Ancient astronaut theorist Nutter B. Butter claims that the Caloprians from planet Calop, 30 light years away and at rest with respect to the Earth, wiped out the dinosaurs. The iridium layer in the crust, he claims, indicates spaceships with the fuel necessary to travel at 30\% of the speed of light here and back, and that their engines allowed them to instantaneously hop to this speed. He also says that Caloprians can only reproduce on their home planet.

Call the minimum life span, in years, of a Caloprian, assuming some had to reach Earth to wipe out the dinosaurs, $T$. Assume that, once a Caloprian reaches Earth, they instantaneously wipe out the dinosaurs. Then, $T$ can be expressed in the form $m\sqrt{n}$, where $n$ is not divisible by the square of a prime. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On side $AC$ of triangle $ABC$, a circle is constructed such that $AC$ acts as its diameter. This circle intersects sides $AB$ and $BC$ at points $D$ and $E$, respectively. The angle $\angle EDC$ is $30^\circ$, $EC=1$, and the area of triangle $DBE$ is half the area of triangle $ABC$. Find the length of segment $BO$, where $O$ is the intersection point of segments $AE$ and $CD$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Problem 4. A set will be called interesting if its elements are prime numbers and the following condition is met:

no matter how we choose three distinct elements of the set, the sum of the chosen numbers is a prime number.

Determine the maximum number of elements an interesting set can have.
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Context:
11. Given the parabola $y^{2}=2 p x(p>0), A, B$ are two moving points on the parabola different from the vertex $O$. Let $\angle A O B=\theta\left(\theta \neq 90^{\circ}\right)$. If $S_{\triangle A O B}=m \tan \theta$, find the maximum value of $\tan \theta$ when $m$ is minimized.
Please reason step by step, and put your final answer within \boxed{}.


Context:
We define the ridiculous numbers recursively as follows:
[list=a]
[*]1 is a ridiculous number.
[*]If $a$ is a ridiculous number, then $\sqrt{a}$ and $1+\sqrt{a}$ are also ridiculous numbers.
[/list]
A closed interval $I$ is ``boring'' if
[list]
[*]$I$ contains no ridiculous numbers, and
[*]There exists an interval $[b,c]$ containing $I$ for which $b$ and $c$ are both ridiculous numbers.
[/list]
The smallest non-negative $l$ such that there does not exist a boring interval with length $l$ can be represented in the form $\dfrac{a + b\sqrt{c}}{d}$ where $a, b, c, d$ are integers, $\gcd(a, b, d) = 1$, and no integer square greater than 1 divides $c$. What is $a + b + c + d$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A right triangle and a circle are drawn such that the circle is tangent to the legs of the right triangle. The circle cuts the hypotenuse into three segments of lengths 1,24 , and 3 , and the segment of length 24 is a chord of the circle. Compute the area of the triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with its left focus $F(c, 0)$ and eccentricity $\frac{\sqrt{3}}{3}$. Point $M$ lies on the ellipse in the first quadrant, and the line segment formed by intersecting line $FM$ with the circle $x^{2} + y^{2} = \frac{b^{2}}{4}$ has a length of $c$. Calculate the slope of the line $FM$, expressing your answer as a simplified fraction or decimal.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If one side of square \(ABCD\) lies on the line \(y = 2x - 17\) and two other vertices lie on the parabola \(y = x^2\), then the minimum area of the square is \(\quad\).
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Context:
If there are several heaps of stones on the table, it is said that there are $\textit{many}$ stones on the table, if we can find $50$ piles and number them with the numbers from $1$ to $50$ so that the first pile contains at least one stone, the second - at least two stones,..., the $50$-th has at least $50$ stones. Let the table be initially contain $100$ piles of $100$ stones each. Find the largest $n \leq 10 000$ such that after removing any $n$ stones, there will still be $\textit{many}$ stones left on the table.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two positive integers that only have 1 as a common factor are called coprime numbers. For example, 2 and 7 are coprime, as are 3 and 4. In any permutation of 2, 3, 4, 5, 6, 7, where each pair of adjacent numbers are coprime, there are a total of \_\_\_\_\_\_\_\_ different permutations (answer with a number).
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Context:
After receiving his weekly allowance, a student decided to save 10% of his allowance each day. On the first day, he spent 3/5 of his remaining allowance at the arcade. On the second day, he saved 10% of his remaining allowance and then spent one-third of what was left at the toy store. The next day, again after saving 10%, he spent his last $0.60 at the candy store. What is this student's weekly allowance?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the set $S=\{1,2,3,4,5,6,7,8\}$, for $x\in S$, if $x+1 \notin S$ and $x-1 \notin S$, then $x$ is a "good element" of $S$. Among all sets composed of $3$ elements from $S$, the number of sets that do not contain any "good elements" is ____.
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Context:
Compute
$$
\sum_{k=0}^{100}\left\lfloor\frac{2^{100}}{2^{50}+2^{k}}\right\rfloor .
$$
(Here, if \( x \) is a real number, then \( \lfloor x \rfloor \) denotes the largest integer less than or equal to \( x \).)
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Context:
Calculate the result of the expression $826446281 \times 11 \times 11$. Express your answer as a single integer.
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Context:
Calculate the integral
$$\int_{\mathbb{R}^2}\exp\left({-\frac{(q^2+p^2)}{2}}\right)\exp\left({-\frac{\left(\cosh(s)q+\sinh(s)p\right)^2+\left(\sinh(s)q+\cosh(s)p\right)^2}{2}}\right)dq\;dp$$
where $s\geq 0$ is a constant.
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Context:
A person can swim in still water at 6 km/h. If the speed of water is 2 km/h, how many hours will the person take to swim back against the current for a certain distance, if it takes them 3.5 hours?
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Context:
You roll three fair six-sided dice. Given that the highest number you rolled is $5$, the expected value of the sum of the three dice can be written as $\frac{a}{b}$ in simplest form. Find $a + b$.
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Context:
Consider the rational function $Q(x)$ satisfying $x \cdot Q(x+2018) = (x-2018)Q(x)$ for all $x \neq 2018$ and $0$. Given that $Q(1) = 1$, find the value of $Q(2017)$.
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Context:
Five soccer teams compete, with each pair of teams playing one match: the winner gets 3 points, the loser gets 0 points, and in case of a draw each team gets 1 point. After the matches, the total points of each team do not exceed 9 points, and there are exactly two teams that have the same points. Let the scores of the five teams from highest to lowest be $A, B, C, D, E$ (where there are exactly two equal scores). If $\overline{\mathrm{ABCDE}}$ (the concatenation of the scores) is a multiple of 15, how many draws were there in total?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The integer $k$ is a \textit{good number}, if we can divide a square into $k$ squares. Find how many good numbers not greater than $2006$ there are.
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Context:
Determine all pairs $(a, p)$ of positive integers with $p$ prime such that $p^{a}+a^{4}$ is a perfect square. Please provide the sum of all elements in these pairs.
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Context:
Determine the number of distinct integers among the numbers
$$
\left\lfloor\frac{1^{2}}{2007}\right\rfloor,\left\lfloor\frac{2^{2}}{2007}\right\rfloor, \quad \cdots, \quad \left\lfloor\frac{2007^{2}}{2007}\right\rfloor.
$$
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Context:
In how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$? For example, $2 + 10$, $10 + 2$, and $3 + 2 + 2 + 5$ are three such ways.
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Context:
each of the positive integers a and c is a 4 - digit integer . if each of the digits 0 through 9 appears in one of these 3 integers , what is the maximum possible value of the sum of a and c ?
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Context:
Suppose $a,$ $b,$ and $c$ are real numbers such that
\[\frac{ac}{a + b} + \frac{ba}{b + c} + \frac{cb}{c + a} = -9\]and
\[\frac{bc}{a + b} + \frac{ca}{b + c} + \frac{ab}{c + a} = 10.\]Compute the value of
\[\frac{b}{a + b} + \frac{c}{b + c} + \frac{a}{c + a}.\]
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Context:
A large cube is made up of 27 small cubes. A plane is perpendicular to one of the diagonals of this large cube and bisects the diagonal. How many small cubes are intersected by this plane?
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Context:
The ratio of investments of three partners p, q, and r is 7 : 5 : 4, and the ratio of their profits is 7 : 10 : 8. If p invested the money for 2 months and q invested the money for t months, find for how much time did r invest the money?
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Context:
Given the function $f(x) = \ln x$, its tangent line equation at $x=x_0$ is $y=kx+b$. Determine the range of values for $k+b$.
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Context:
For any positive integer  $a$ , define  $M(a)$  to be the number of positive integers  $b$  for which  $a+b$  divides  $ab$ . Find all integer(s)  $a$  with  $1\le a\le 2013$  such that  $M(a)$  attains the largest possible value in the range of  $a$ .
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Context:
Given unit vectors $\vec{a}$ and $\vec{b}$ with an acute angle between them, for any $(x, y) \in \{(x, y) \mid | x \vec{a} + y \vec{b} | = 1, xy \geq 0 \}$, it holds that $|x + 2y| \leq \frac{8}{\sqrt{15}}$. Find the minimum possible value of $\vec{a} \cdot \vec{b}$.
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Context:
How many positive five-digit integers have the product of their five digits equal to $900$?
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Context:
Find the largest integer $d$ that divides all three numbers $abc$, $bca$, and $cab$, where $a$, $b$, and $c$ are some nonzero and mutually different digits.
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Context:
A circle touches the extensions of two sides $AB$ and $AD$ of square $ABCD$ with a side length of $2-\sqrt{5-\sqrt{5}}$ cm. From point $C$, two tangents are drawn to this circle. Find the radius of the circle, given that the angle between the tangents is $72^{\circ}$ and it is known that $\sin 36^{\circ} = \frac{\sqrt{5-\sqrt{5}}}{2 \sqrt{2}}$.
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Context:
On a game show, a contestant is given 5 keys, each of which opens exactly one of 5 identical boxes. The first box contains $100, the second $500, the third $1,000, the fourth $5,000, and the fifth $10,000. The contestant assigns each key to one of the boxes and wins the total amount of money contained in any boxes that are opened by the keys assigned to them. What is the probability that a contestant will win more than $12,000?
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Context:
In the diagram, a large circle and a rectangle intersect such that the rectangle halves the circle with its diagonal, and $O$ is the center of the circle. The area of the circle is $100\pi$. The top right corner of the rectangle touches the circle while the other corner is at the center of the circle. Determine the total shaded area formed by the parts of the circle not included in the intersection with the rectangle. Assume the intersection forms a sector.
[Diagram not shown: Assume descriptive adequacy for the composition of the circle and the rectangle.]
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Context:
Let $a$ be a positive number. Consider the set $S$ of all points whose rectangular coordinates $(x, y)$ satisfy all of the following conditions:
\begin{enumerate}
    \item $\frac{a}{2} \le x \le 2a$
    \item $\frac{a}{2} \le y \le 2a$
    \item $x+y \ge a$
    \item $x+a \ge y$
    \item $y+a \ge x$
\end{enumerate}
The boundary of set $S$ is a polygon with
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Context:
In three of the six circles of a diagram, the numbers 4, 14, and 6 are written. In how many ways can natural numbers be placed in the remaining three circles so that the product of the triplets of numbers along each of the three sides of the triangular diagram are the same?
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Context:
What is the smallest value of \( n \) needed for the \( n \)th order Taylor polynomial \( T_n(x,0) \) of \( \cos x \) centered at \( 0 \) to guarantee an error \( |\cos x - T_n(x,0)| \) less than \( 10^{-3} \) for \( x \) in the interval \( [-\frac{\pi}{2}, \frac{\pi}{2}] \)?
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Context:
Tessa has a figure created by adding a semicircle of radius 1 on each side of an equilateral triangle with side length 2, with the semicircles oriented outwards. She then marks two points on the boundary of the figure. What is the greatest possible distance between the two points?
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Context:
For the parabola $C$, suppose a line $l$ passes through the focus $F$ of $C$, and the angle between $l$ and the axis of symmetry of $C$ is $\frac{\pi}{4}$. If the chord intercepted by $l$ on $C$ has a length of 4, then the distance from the focus of $C$ to its vertex is ______.
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Context:
解方程$2 \sqrt { x ^ { 2 } - 1 2 1 } + 1 1 \sqrt { x ^ { 2 } - 4 } = 7 \sqrt { 3 } x .$
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Context:
The sum of the first $n$ terms of an infinite sequence $\{a_n\}$ is $S_n$. If for any positive integer $n$, $S_n$ belongs to $\{k_1,k_2,k_3,…,k_{10}\}$, then the maximum number of possible values for $a_{10}$ is __________.
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Context:
Find the limit of the following expression as x approaches 0 from the right:

$$\lim_{x \to 0+}\frac{x^{(\sin x)^x}-(\sin x)^{x^{\sin x}}}{x^3}$$
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Context:
Triangle $ABC$ has $AB = AC$. Point $D$ is on side $\overline{BC}$ so that $AD = CD$ and $\angle BAD = 36^\circ$. Find the degree measure of $\angle BAC$.
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Context:
A circle is circumscribed around the right triangle $ABC$ ($\widehat{C} = 90^\circ$). Let $CD$ be the altitude of the triangle. A circle with center at $D$ passes through the midpoint of the arc $\widetilde{AB}$ and intersects $AB$ at point $M$. Find $|CM|$, if $|AB|=c$.
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Context:
We have an $n$-gon, and each of its vertices is labeled with a number from the set $\{1, \ldots, 10\}$. We know that for any pair of distinct numbers from this set there is at least one side of the polygon whose endpoints have these two numbers. Find the smallest possible value of $n$.
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Context:
Question 78: Given $x, y, z > 0$, find the minimum value of $2 \sqrt{(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)}-\sqrt{\left(1+\frac{x}{y}\right)\left(1+\frac{y}{z}\right)}$.
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Context:
Two 5-digit positive integers are formed using each of the digits from 0 through 9 once. What is the smallest possible positive difference between the two integers?
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Context:
Evaluate the limit:

$$\lim\limits_{n\to\infty}{\sum\limits_{k=n}^{5n} \frac{n}{k}{k \choose n}(\frac{1}{5})^{n}(\frac{4}{5})^{k-n}}$$
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Context:
$x_{1}, x_{2}, x_{3}, x_{4}$ 为互不相等的正实数， $x_{i 1}, x_{i 2}, x_{i 3}, x_{i 4}$ 为 $x_{1}, x_{2}, x_{3}, x_{4}$ 的任意顺序排列, $X=\max \left\{\min \left\{x_{i 1}, x_{i 2}\right\}, \min \left\{x_{i 3}, x_{i 4}\right\}, Y=\min \left\{\max \left\{x_{i 1}, x_{i 2}\right\}, \max \left\{x_{i 3}, x_{i 4}\right\}\right.\right.$, 求 x 大于 y 的概率.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many positive integers less than 2005 are relatively prime to 1001?
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Context:
There are 3 positions to be filled by selecting from 4 college graduates. If each position must select at least 1 person (not all 4 college graduates necessarily get selected), then the number of different ways to select is \_\_\_\_\_\_\_\_ (answer with a specific number).
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Context:
在平面直角坐标系 xOy 中，圆 \Omega 与抛物线 \Gamma:y^2=4x 恰有一个公共点，且圆 \Omega 与 x 轴相切于 \Gamma 的焦点 F. 求圆 \Omega 的半径.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。
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Context:
Two trains of some length and 200 m are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. They will be clear of each other from the moment they meet in 14.998800095992321 seconds. What is the length of the first train?
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Context:
4. In the stands of the hockey arena, there are several rows with 168 seats in each row. For the final match, 2016 students from several sports schools were invited as spectators, with no more than 40 from each school. Students from any school must be seated in one row. What is the minimum number of rows that must be in the arena to ensure this can always be done?
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Context:
For each integer $1 \le j \le 2017$, let $S_j$ denote the set of integers $0 \le i \le 2^{2017} - 1$ such that $\left\lfloor \frac{i}{2^{j-1}} \right\rfloor$ is an odd integer. Let $P$ be a polynomial such that
\[P\left(x_0, x_1, \ldots, x_{2^{2017} - 1}\right) = \prod_{1 \le j \le 2017} \left(1 - \prod_{i \in S_j} x_i\right).\]
Compute the remainder when
\[ \sum_{\left(x_0, \ldots, x_{2^{2017} - 1}\right) \in \{0, 1\}^{2^{2017}}} P\left(x_0, \ldots, x_{2^{2017} - 1}\right)\]
is divided by $2017$.
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Context:
Let $A > 0$ be a positive real number and $n \geq 4$ be an integer. Evaluate the integral:

$$ \int_0^\infty \log \left (1+\left (\frac A {\sqrt t}\right)^n\right) \, dt $$
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Context:
At 15:00 there were 20 students in the computer lab. At 15:03 and every three minutes after that, 4 students entered the lab. If at 15:10 and every ten minutes after that 8 students left the lab, how many students were in the computer lab at 15:44?
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Context:
Juan makes a list of $2018$ numbers. The first number is $1$. Each subsequent number is obtained by adding one of the numbers $1, 2, 3, 4, 5, 6, 7, 8,$ or $9$ to the previous number. Given that none of the numbers in the list ends in $0$, what is the largest possible value for the last number on the list?
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Context:
Given the function y = f(x) = x + sqrt(13 - 2ax), where a is a positive integer, find the maximum value of y such that it is a positive integer. Express your answer as a single integer.
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Context:
Given the equation \( 4(\sqrt{x} + \sqrt{y-1} + \sqrt{z-2}) = x + y + z + 9 \), calculate the value of \( xyz \).
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Context:
Given real numbers \( x \) and \( y \in (1, +\infty) \), and \( x y - 2 x - y + 1 = 0 \). Find the minimum value of \( \frac{3}{2} x^{2} + y^{2} \).
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Context:
19. (3 points) A male and a female track and field athletes are practicing running on a 110-meter slope (the top of the slope is $A$, the bottom is $B$). Both start from point $A$ at the same time, running back and forth between $A$ and $B$ continuously. If the male athlete's uphill speed is 3 meters per second and downhill speed is 5 meters per second; the female athlete's uphill speed is 2 meters per second and downhill speed is 3 meters per second, then the location of their second face-to-face meeting is $\qquad$ meters away from point $A$.
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Context:
John has a donkey, an elephant, a giraffe, and a hippopotamus. The elephant weighs 3 tons (a ton is 2000 pounds). The donkey weighs 90% less than the elephant. The giraffe weighs 1.5 tons and the hippopotamus weighs 4000 kg. Additionally, John bought a bag of food for each of them which weighs 5 lbs, 16 oz, 3 kg, and 5000 g, respectively. What is the combined weight of the animals and their bags of food in pounds?
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Context:
The geometric sequence $\{a_n\}$ satisfies: $a_1=a$ ($a>0$), and $a_{1}+1$, $a_{2}+2$, $a_{3}+3$ form a geometric sequence. If $\{a_n\}$ is unique, then the value of $a$ equals \_\_\_\_\_\_.
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Context:
Thomas started saving for a car almost 2 years ago. His weekly allowance was some amount for the first year. In the second year, he got a job that pays $9 an hour at a coffee shop and worked 30 hours a week, so his parents discontinued his allowance. If the car he wants to buy is $15,000 and he spends $35 a week on himself, Thomas needs $2000 more to buy the car by the end of the 2 years. What was Thomas's weekly allowance in the first year?
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Context:
Given the set M = {1,2,3,4,5,6,7} and the proposition p: ∀n∈M, n > 1, what is the negation of proposition p? Express your answer as a logical statement in the form of ∃n∈M, [condition].
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Context:
For a positive integer $ n$, let $ \theta(n)$ denote the number of integers $ 0 \leq x < 2010$ such that $ x^2 \minus{} n$ is divisible by $ 2010$. Determine the remainder when $ \displaystyle \sum_{n \equal{} 0}^{2009} n \cdot \theta(n)$ is divided by $ 2010$.
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Context:
In each cell of a \(75 \times 75\) table, a natural number is written. Each row contains at least 15 different numbers, and in every three consecutive rows, there are no more than 25 different numbers. What is the maximum number of different numbers that the table can contain?
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Context:
A trapezoidal room has dimensions of 25 feet (length), 15 feet (width), 12 feet (height1) and 8 feet (height2). What is the cost of white washing the four walls of the room at Rs. 10 per square feet, if there are two doors with dimensions of 6 feet * 3 feet and three windows of dimensions 4 feet * 3 feet each, and considering an irregular ceiling decoration that subtracts 10 square feet from the total area to be painted?
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Context:
Given that the Earth's global mean temperature follows a normal distribution $N(\mu, \sigma)$, and we have $N$ samples, denoted as $T_1, \cdots, T_N$. Let $T_{maxN}$ be the maximum of the first $N$ samples. What is the probability of drawing three consecutive samples, $T_{N+1}, T_{N+2}, T_{N+3}$, such that each subsequent sample is greater than the previous and the maximum of the first $N$ samples, i.e., $P(T_{N+3} > T_{N+2} > T_{N+1} > T_{maxN})$?
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Context:
2. Given is the set $A=\{1,2,3, \ldots, 62,63\}$. A subset $X$ of the set $A$ will be called good if for every element $x \in X$ it holds that $x$ does not divide the sum of the remaining elements of the set $X$. Determine the maximum number of elements that a good subset can have.
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Context:
Evaluate $\lim_{n\to \infty}\cos(\pi(3+\sqrt{2})^n)$ where $n\in \mathbb{N}$.
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Context:
某个国家有 $2023$ 个岛和 $2022$ 座桥，任意一座桥连接两个不同的岛，任意两个岛之间至多有一座桥相连，且可以从任意一个岛通过若干座桥到达其他任何岛. 若三个岛中的某个岛与另两个岛都有桥连接, 则称这三个岛组成"岛群", 已知任两个"岛群"中都有相同的岛，那么恰有一座桥的岛最少有多少个？
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Context:
Calculate

$$
\operatorname{tg} \frac{\pi}{43} \cdot \operatorname{tg} \frac{2 \pi}{43}+\operatorname{tg} \frac{2 \pi}{43} \cdot \operatorname{tg} \frac{3 \pi}{43}+\ldots+\operatorname{tg} \frac{k \pi}{43} \cdot \operatorname{tg} \frac{(k+1) \pi}{43}+\ldots+\operatorname{tg} \frac{2019 \pi}{43} \cdot \operatorname{tg} \frac{2020 \pi}{43}
$$
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Context:
Given that $\overset{→}{a\_n}=\left(\cos \frac{nπ}{6},\sin \frac{nπ}{6}\right)$, $n∈ℕ^∗$, $\overset{→}{b}=\left( \frac{1}{2}, \frac{\sqrt{3}}{2}\right)$, calculate the value of $y={\left| \overset{→}{{a\_1}}+ \overset{→}{b}\right|}^{2}+{\left| \overset{→}{{a\_2}}+ \overset{→}{b}\right|}^{2}+···+{\left| \overset{→}{{a\_2015}}+ \overset{→}{b}\right|}^{2}$.
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Context:
If the value of the algebraic expression $2x^2 + 3x + 7$ is $8$, then the value of the algebraic expression $4x^2 + 6x - 9$ is ______.
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Context:
Let \( n \) be an integer between 1 and 1990, and let \( x^2 + x - 3n \) be factored into the product of two linear factors with integer coefficients. Determine the number of integers \( n \) that satisfy this condition.
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Context:
Example 3 Given an integer $n \geqslant 3$. Find the largest real number $M$ such that for any sequence of positive real numbers $x_{1}, x_{2}, \cdots, x_{n}$, there exists a permutation $y_{1}, y_{2}, \cdots, y_{n}$, satisfying
$$
\sum_{i=1}^{n} \frac{y_{i}^{2}}{y_{i+1}^{2}-y_{i+1} y_{i+2}+y_{i+2}^{2}} \geqslant M,
$$

where, $y_{n+1}=y_{1}, y_{n+2}=y_{2}$.
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Context:
在平面直角坐标系 $x O y$ 中, $A$ 为椭圆 $\frac{x^{2}}{4}+y^{2}=1$ 上一点， $M$ 为线段 $O A$ 上的动点，过 $M$ 作直线与椭圆交于 $P 、 Q$ 两点. 若 $\overrightarrow{P M}=2 \overrightarrow{M Q}$, 求四边形 $O P A Q$ 面积的最大值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Calculate the value of $\dfrac{\sqrt[3]{81}}{\sqrt[4]{81}}$ in terms of 81 raised to what power?
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Context:
Vovochka adds three-digit numbers in a column in the following way: he does not carry over tens, but writes the sum of pairs of digits in the same place value position under each pair of digits, even if the result is a two-digit number. For example, for the sum \(248 + 208\), he would get the result 4416. Find the smallest possible positive difference between Vovochka's answer and the correct answer.
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Context:
9. To what power must the root $x_{0}$ of the equation $x^{11} + x^{7} + x^{3} = 1$ be raised to obtain the number $x_{0}^{4} + x_{0}^{3} - 1 ?$
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Context:
In a math competition consisting of problems $A$, $B$, and $C$, among the 39 participants, each person answered at least one problem correctly. Among the people who answered $A$ correctly, the number of people who only answered $A$ is 5 more than the number of people who answered other problems as well. Among the people who did not answer $A$ correctly, the number of people who answered $B$ is twice the number of people who answered $C$. Additionally, the number of people who only answered $A$ equals the sum of the number of people who only answered $B$ and the number of people who only answered $C$. What is the maximum number of people who answered $A$ correctly?
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Context:
Find the last two digits of  $\tbinom{200}{100}$ . Express the answer as an integer between  $0$  and  $99$ . (e.g. if the last two digits are  $05$ , just write  $5$ .)
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Context:
The surface area of a spherical segment is given as $S$ (consider the spherical part of the segment). Determine the maximum value of the volume of this segment.
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Context:
Evan has $66000$ omons, particles that can cluster into groups of a perfect square number of omons. An omon in a cluster of $n^2$ omons has a potential energy of $\frac{1}{n}$. Evan accurately computes the sum of the potential energies of all the omons. Compute the smallest possible value of his result.
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Context:
Determine the largest integer $n \geq 3$ for which the edges of the complete graph on $n$ vertices can be assigned pairwise distinct non-negative integers such that the edges of every triangle have numbers which form an arithmetic progression.
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Context:
If @ is a binary operation defined as the difference between an integer n and the product of n and a certain number, then the largest positive integer n such that the outcome of the binary operation of n is less than 14 is 3. What is the number that n is multiplied with in the binary operation?
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Context:
Marcy keeps a 2-liter bottle of water by her desk. She takes a sip every five minutes. During the first hour, each sip is 40 ml. However, in the second hour, her sips increase to 50 ml, and in the third hour, they increase again to 60 ml. At the end of the third hour, Marcy takes a 30-minute break, during which she doesn't drink. How many minutes does it take her to drink the whole bottle of water?
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Context:
Evaluate the sum of the following series: $$\sum_{n=0}^\infty \left[ \frac{1}{(2n+1)^2} - \frac{(-1)^n}{(2n+1)^3} \right]$$
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Context:
In a simple graph with 300 vertices, no two vertices of the same degree are adjacent. What is the maximal possible number of edges in such a graph?
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Context:
Find the polar coordinates of the center of the circle given by the polar equation $\rho=\cos \left(\theta+ \frac{\pi}{3}\right)$. Express your answer as an ordered pair of polar coordinates.
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Context:
Given a quadrilateral pyramid $P-ABCD$ where the lengths of edges $AB$ and $BC$ are $\sqrt{2}$, and the lengths of all other edges are 1. Find the volume of the quadrilateral pyramid.
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Context:
Let  $a_1$ ,  $a_2$ ,  $\ldots$  be an infinite sequence of (positive) integers such that  $k$  divides  $\gcd(a_{k-1},a_k)$  for all  $k\geq 2$ .  Compute the smallest possible value of  $a_1+a_2+\cdots+a_{10}$ .
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Context:
8.1. Points $K, L, M, N$ are the centers of the circles inscribed in the faces $S A B, S A C$, $S B C$ and $A B C$ of the tetrahedron $S A B C$. It is known that $A B=S C=5, A C=S B=7, B C=S A=8$. Find the volume of the tetrahedron $K L M N$. Round your answer to the nearest hundredth if necessary.
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Context:
A set  $\mathcal{S}$  of distinct positive integers has the following property: for every integer  $x$  in  $\mathcal{S},$  the arithmetic mean of the set of values obtained by deleting  $x$  from  $\mathcal{S}$  is an integer.  Given that 1 belongs to  $\mathcal{S}$  and that 2002 is the largest element of  $\mathcal{S},$  what is the greatet number of elements that  $\mathcal{S}$  can have?
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Context:
Let \(ABC\) be a triangle with circumradius \(R = 17\) and inradius \(r = 7\). Find the maximum possible value of \(\sin \frac{A}{2}\).
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Context:
Determine the number of subsets \( S \) of \(\{1, 2, \ldots, 1000\}\) that satisfy the following conditions:
- \( S \) has 19 elements, and
- the sum of the elements in any non-empty subset of \( S \) is not divisible by 20.
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Context:
Bus stop \(B\) is located on a straight highway between stops \(A\) and \(C\). After some time driving from \(A\), the bus finds itself at a point on the highway where the distance to one of the three stops is equal to the sum of the distances to the other two stops. After the same amount of time, the bus again finds itself at a point with this property, and 25 minutes later it arrives at \(B\). How much time does the bus need for the entire journey from \(A\) to \(C\) if its speed is constant and it stops at \(B\) for 5 minutes?
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Context:
At a club, James buys 2 rounds for his 5 friends, 6 drinks for himself, and some fried chicken. The club charges $20 to enter, drinks cost $6 each, and he leaves a 30% tip on everything he orders. He spent $163 for the night. How much did the fried chicken cost?
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Context:
A right circular cone and a right cylinder with the same height $20$ do not have the same circular base, but the circles are coplanar, and their centers are the same. If the cone and the cylinder are on the same side of the plane and their base radii are $20$ and $10$, respectively, find the ratio of the volume of the part of the cone inside the cylinder over the volume of the part of the cone outside the cylinder.
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Context:
In a movie theater row with 10 seats, three people, A, B, and C, go to watch a movie. They want to sit in the same row, and each person wants to have empty seats on both sides, with A sitting in the middle. How many ways can they sit like this?
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Context:
Sasha loves to drink coffee at work, so every morning before work he prepares a thermos of his favorite drink according to a strict ritual. He brews 300 ml of very strong aromatic coffee in a cezve. Then, Sasha repeats the following procedure 6 times: he pours 200 ml from the cezve into the thermos, adds 200 ml of water into the cezve, and mixes thoroughly. After the 6th time, Sasha pours all 300 ml from the cezve into the thermos. What will be the concentration of the original strong aromatic coffee in the thermos? Initially, the thermos is empty.

Concentration is a number between 0 and 1. Provide the answer in the form of a fraction (either common or decimal), not as a percentage.
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Context:
For a special event, the five Vietnamese famous dishes including Phở, (Vietnamese noodle), Nem (spring roll), Bún Chả (grilled pork noodle), Bánh cuốn (stuffed pancake), and Xôi gà  (chicken sticky rice) are the options for the main courses for the dinner of Monday, Tuesday, and Wednesday. Every dish must be used exactly one time. How many choices do we have?
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Context:
Let $a,b,c,d,e$ be positive real numbers. Find the largest possible value for the expression
$$\frac{ab+bc+cd+de}{2a^2+b^2+2c^2+d^2+2e^2}.$$
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Context:
Evaluate $\int_{1}^{4} \frac{x-2}{(x^2+4)\sqrt{x}}\,dx$
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Context:
Given that $d(\phi(n)) = 2017$, where $d()$ represents the divisor function and $\phi()$ represents the totient function, find the smallest natural number $n$.
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Context:
Given the vertices of a triangle are $A(3,4)$, $B(0,0)$, and $C(c,2c-6)$, if $\angle BAC$ is an obtuse angle, then the range of values for $c$ is \_\_\_\_\_\_.
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Context:
In the tetrahedron \(KLMN\), the edge lengths are given as \(KL = MN = 9\), \(KM = LN = 15\), and \(KN = LM = 16\). The points \(P, Q, R,\) and \(S\) are the centers of the circles inscribed in the triangles \(KLM\), \(KLN\), \(KMN\), and \(LMN\), respectively. Find the volume of the tetrahedron \(PQRS\). Round the answer to two decimal places if necessary.
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Context:
We color some unit squares in a $99 \times 99$ square grid with one of $5$ given distinct colors, such that each color appears the same number of times. On each row and on each column, there are no differently colored unit squares. Find the maximum possible number of colored unit squares.
Please reason step by step, and put your final answer within \boxed{}.