Context:
Omar has four fair standard six-sided dice. Omar invented a game where he rolls his four dice over and over again until the number 1 does not appear on the top face of any of the dice. Omar wins the game if on that roll the top faces of his dice show at least one 2 and at least one 5. The probability that Omar wins the game is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
Given positive integer  $n$ , find the biggest real number  $C$  which satisfy the condition that if the sum of the reciprocals of a set of integers (They can be the same.) that are greater than  $1$  is less than  $C$ , then we can divide the set of numbers into no more than  $n$  groups so that the sum of reciprocals of every group is less than  $1$ .
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Context:
For a natural number $n \ge 3$, we draw $n - 3$ internal diagonals in a non self-intersecting, but not necessarily convex, $n$-gon, cutting the $n$-gon into $n - 2$ triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural number and no two of these angle values are equal. What is the largest possible value of $n$?
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Context:
John bought a shirt on sale for 25% off the original price and another 25% off the discounted price. The final price was $17. What was the price before the first discount?
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Context:
Let  $n$  be a positive integer. Each of the numbers  $1,2,3,\ldots,100$  is painted with one of  $n$  colors in such a way that two distinct numbers with a sum divisible by  $4$  are painted with different colors. Determine the smallest value of  $n$  for which such a situation is possible.
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Context:
1. The numbers from 1 to 2000 are written on a board. In one move, it is allowed to erase any number $n$, as well as no more than $n$ numbers not less than $n$. In what minimum number of moves can all the numbers be erased?
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Context:
For how many values of q, does | | | q - 5 | - 10 | - a certain number | equals 2?
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Context:
Let $ABCD$ be a rectangle with $AB = 6$ and $BC = 6 \sqrt{3}$. We construct four semicircles $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ whose diameters are the segments $AB$, $BC$, $CD$, $DA$. It is given that $\omega_i$ and $\omega_{i+1}$ intersect at some point $X_i$ in the interior of $ABCD$ for every $i=1,2,3,4$ (indices taken modulo $4$). Compute the square of the area of $X_1X_2X_3X_4$.
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Context:
Given that \( P \) is a point on the hyperbola \( C: \frac{x^{2}}{4} - \frac{y^{2}}{12} = 1 \), and \( F_{1} \) and \( F_{2} \) are the left and right foci of the hyperbola \( C \), and \( M \) and \( I \) are the centroid and incenter of the triangle \( \triangle P F_{1} F_{2} \) respectively. If \( M I \perp x \)-axis, then find the radius of the incircle of \( \triangle P F_{1} F_{2} \).
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Context:
Find the dimension of the null space of the linear transformation $T: P_{n} \rightarrow P_{n-1}$ defined by $T(p(x)) = p'(x) - \int_{0}^{x} p(t) dt$, where $P_{n}$ is the vector space of all polynomials with real coefficients of degree at most $n$.
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Context:
Consider the four points \(A=(2,4)\), \(B=(0,0)\), \(C=(6,0)\), and \(D=(6,3)\) in the plane. For which point \(P\) in the plane is the sum \(PA+PB+PC+PD\) minimized?
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Context:
Let $ABC$ be a triangle with $AB=11$, $BC=12$, and $CA=13$. Let $M$ and $N$ be the midpoints of sides $CA$ and $AB$, respectively. The incircle touches sides $CA$ and $AB$ at points $X$ and $Y$, respectively. Suppose that $R$, $S$, and $T$ are the midpoints of line segments $MN$, $BX$, and $CY$, respectively. Then $\sin \angle{SRT} = \frac{k\sqrt{m}}{n}$, where $k$, $m$, and $n$ are positive integers such that $\gcd(k,n) = 1$ and $m$ is not divisible by the square of any prime. Compute $100k + 10m + n$.
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Context:
Given that the cubic function $f(x)= \frac {1}{3}x^{3}-(4m-1)x^{2}+(15m^{2}-2m-7)x+2$ is increasing on $x∈(-∞,+∞)$, the range of values for $m$ is \_\_\_\_\_\_.
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Context:
8,9

In trapezoid $ABCD$, side $AD$ is the larger base. It is known that $AD=CD=4 \frac{2}{3}$, $\angle BAD=90^{\circ}$, and $\angle BCD=150^{\circ}$. On base $AD$, triangle $AED$ is constructed, and points $B$ and $E$ lie on the same side of line $AD$ with $AE=DE$. The height of this triangle, drawn from vertex $E$, is $1 \frac{2}{5}$. Find the area of the common part of trapezoid $ABCD$ and triangle $AED$.
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Context:
If 1 is an element of the set $\{a+2, (a+1)^2, a^2+3a+3\}$, then the real number $a=$ .
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Context:
The digits 2, 3, 5, 7, and 8 can be used, each exactly once, to form many five-digit integers. Of these integers, find the one that is as close as possible to 30000. What is the tens digit of this integer? Express your answer as a single digit.
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Context:
2020 magicians are divided into groups of 2 for the Lexington Magic Tournament. After every 5 days, which is the duration of one match, teams are rearranged so no 2 people are ever on the same team. If the longest tournament is $n$ days long, what is the value of $n$?
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Context:
Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. When at the beginning of the turn there are $n\geq 1$ marbles on the table, then the player whose turn it is removes $k$ marbles, where $k\geq 1$ either is an even number with $k\leq \frac{n}{2}$ or an odd number with $\frac{n}{2}\leq k\leq n$. A player win the game if she removes the last marble from the table.
Determine the smallest number $N\geq 100000$ such that Berta can enforce a victory if there are exactly $N$ marbles on the tale in the beginning.
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Context:
A 16-step path is to go from $(-4,-4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $-2 \le x \le 2$, $-2 \le y \le 2$ at each step?
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Context:
What is the maximum number of rooks that can be placed on a $3n \times 3n$ chessboard so that each of them is attacked by no more than one of the others?
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Context:
Find $\frac{a^{8}-6561}{81 a^{4}} \cdot \frac{3 a}{a^{2}+9}$, given that $\frac{a}{3}-\frac{3}{a}=4$.
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Context:
What is the maximum number of disjoint groups into which all the integers from 1 to 25 can be divided so that the sum of the numbers in each group is a perfect square?
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Context:
In a tournament, each chess player scored half of all their points in matches against participants who took the last three places. How many people participated in the tournament in total?
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Context:
7.1. Two cars are driving on a highway at a speed of 80 km/h and with an interval of 10 m. At the speed limit sign, the cars instantly reduce their speed to 60 km/h. What will be the interval between them after the speed limit sign?
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Context:
Given that the odd function $f(x)$ is a differentiable function defined on $\mathbb{R}$, with its derivative denoted as $f'(x)$. When $x > 0$, it satisfies $2f(x) + xf'(x) > x^2$. Find the solution set for the inequality $(x+2014)^2f(x+2014) + 4f(-2) < 0$. Express your answer in interval notation.
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Context:
The library has received $n$ types of new books, with $m$ copies of each type. There are $r$ people ($n \leq r \leq m$) who want to borrow books, with each person limited to borrowing 1 book. How many ways are there for each type of new book to be borrowed by at least one person?
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Context:
For the equation involving $x$, $\frac{2x+1}{x-3}=\frac{m}{3-x}+1$, if it has a root that increases, the value of $m$ is ______.
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Context:
Let $T$ denote the value of the sum\[\sum_{n=0}^{432} (-1)^{n} {1500 \choose 3n}\]Determine the remainder obtained when $T$ is divided by $100$.
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Context:
There are digits $a$ and $b$ such that the 15-digit number $7a7ba7ab7ba7b77$ is divisible by 99. Find $10a + b$.
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Context:
Tas and his friends put up a t-shirt sale with multiple colors and prices. They ended up selling 600 t-shirts in 40 minutes. The number and cost for each color were as follows:

- 250 black shirts at $30 each
- 200 white shirts at $25 each
- 100 red shirts at $28 each
- 50 blue shirts at $25 each

Additionally, they offered a bulk purchase discount for the customers who bought a specific number of shirts. The discount details are:

- 5% discount for purchasing 5 or more black shirts
- 8% discount for purchasing 10 or more white shirts
- 10% discount for purchasing 8 or more red shirts

Assuming the customers took full advantage of the discounts for all available shirts, calculate the total income per minute during the 40-minute sale, considering the unique prices, quantities of each color sold, and the discounts applied.
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Context:
The ratio 2 : 3 expressed as a percent equals what value?
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Context:
A circle can be inscribed in quadrilateral \(ABCD\). Let \(K\) be the intersection point of its diagonals. It is known that \(BC > AB > BK\), \(KC = \sqrt{7} - 1\), the cosine of angle \(KBC\) is \(\frac{\sqrt{7} + 1}{4}\), and the perimeter of triangle \(BKC\) is \(2\sqrt{7} + 4\). Find \(DC\).
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Context:
9. (10 points) A four-digit number "HaoShiChengShuang" divided by a two-digit number "ChengShuang" has a remainder exactly equal to "HaoShi". If different Chinese characters represent different digits and "HaoShi" and "ChengShuang" are not coprime, then the largest four-digit number "HaoShiChengShuang" is
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Context:
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. Find how many distinct integers can be written as the sum of two, not necessarily different, special fractions.
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Context:
What is the expected value of the ratio between the second largest and the largest number in a set of $n$ independent, identically distributed (i.i.d.) random variables following a uniform distribution from $0$ to $1$? In other words, given $X_1, X_2, \ldots, X_n \sim U[0,1]$, where $x_p = \arg\max_i X_i$ represents the largest number and $x_q = \arg\max_{i \neq p} X_i$ represents the second largest number, find $\mathbb{E}\left[\frac{x_q}{x_p}\right]$.
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Context:
已知动点 $P(x, y)$ 满足二次方程 $10 x-2 x y-2 y+1=0$, 则此二次曲线的商心率为 $\qquad$ -
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Context:
Let \( A B C \) be a triangle such that \( A B = 7 \), and let the angle bisector of \(\angle B A C \) intersect line \( B C \) at \( D \). If there exist points \( E \) and \( F \) on sides \( A C \) and \( B C \), respectively, such that lines \( A D \) and \( E F \) are parallel and divide triangle \( A B C \) into three parts of equal area, determine the number of possible integral values for \( B C \).
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Context:
What is the smallest $n$ for which there exists an $n$-gon that can be divided into a triangle, quadrilateral, ..., up to a 2006-gon?
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Context:
Let $n$ be the answer to this problem. Box $B$ initially contains $n$ balls, and Box $A$ contains half as many balls as Box $B$. After 80 balls are moved from Box $A$ to Box $B$, the ratio of balls in Box $A$ to Box $B$ is now $\frac{p}{q}$, where $p, q$ are positive integers with $\operatorname{gcd}(p, q)=1$. Find $100p+q$.
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Context:
How many different-looking arrangements are possible when four balls are selected at random from six identical red balls and three identical green balls and then arranged in a line?
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Context:
In a right trapezoid $ABCD$ where $AB \parallel CD$, the angle at vertex $B$ measures $75^\circ$. Point $H$ is the foot of the perpendicular from point $A$ to the line $BC$. If $BH = DC$ and $AD + AH = 8$, find the area of $ABCD$.
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Context:
Given a positive arithmetic-geometric sequence $\{a_n\}$ that satisfies $a_7 = a_6 + 2a_5$, if there exist two terms $(a_m, a_n)$ such that $\sqrt{a_m \cdot a_n} = 4a_1$, find the minimum value of $\frac{1}{m} + \frac{4}{n}$.
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Context:
Let \( ABCDEF \) be a hexagon such that the diagonals \( AD, BE, \) and \( CF \) intersect at the point \( O \), and the area of the triangle formed by any three adjacent points is 2 (for example, the area of \(\triangle BCD\) is 2). Find the area of the hexagon.
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Context:
A mouse has a wheel of cheese which is cut into  $2018$  slices. The mouse also has a  $2019$ -sided die, with faces labeled  $0,1,2,\ldots, 2018$ , and with each face equally likely to come up. Every second, the mouse rolls the dice. If the dice lands on  $k$ , and the mouse has at least  $k$  slices of cheese remaining, then the mouse eats  $k$  slices of cheese; otherwise, the mouse does nothing. What is the expected number of seconds until all the cheese is gone?

*Proposed by Brandon Wang*
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Context:
In  $\triangle ABC$  with  $AB = 10$ ,  $BC = 12$ , and  $AC = 14$ , let  $E$  and  $F$  be the midpoints of  $AB$  and  $AC$ . If a circle passing through  $B$  and  $C$  is tangent to the circumcircle of  $AEF$  at point  $X \ne A$ , find  $AX$ .
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Context:
Given an ellipse \(\varepsilon: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) where \(a > b > 0\), and a moving circle \(\Gamma: x^{2}+y^{2}=R^{2}\) such that \(b < R < a\), let \(A\) be a point on the ellipse \(\varepsilon\) and \(B\) a point on the moving circle \(\Gamma\). The line segment \(AB\) is tangent to both the ellipse \(\varepsilon\) and the moving circle \(\Gamma\). Find the maximum distance \(|AB|\) between the points \(A\) and \(B\).
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Context:
On the test, the teacher gave five problems and assigned a test grade equal to the number of problems solved. All students, except for Petya, solved the same number of problems, while Petya solved one more. The first problem was solved by 9 people, the second by 7 people, the third by 5 people, the fourth by 3 people, and the fifth by one person.

How many fours and fives were received on the test?
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Context:
Given the function $f(x) = x^3 + 3ax^2 - 6ax + 2$ ($a \in \mathbb{R}$), if $f(x)$ takes its minimum value at $x = x_0 \in (1, 3)$, find the value of $a$.
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Context:
A container holds one liter of wine, and another holds one liter of water. From the first container, we pour one deciliter into the second container and mix thoroughly. Then, we pour one deciliter of the mixture back into the first container. Calculate the limit of the amount of wine in the first container if this process is repeated infinitely many times, assuming perfect mixing with each pour and no loss of liquid.
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Context:
Triangles \(ABC\) and \(ABD\) are inscribed in a semicircle with diameter \(AB = 5\). A perpendicular from \(D\) to \(AB\) intersects segment \(AC\) at point \(Q\), ray \(BC\) at point \(R\), and segment \(AB\) at point \(P\). It is known that \(PR = \frac{27}{10}\), and \(PQ = \frac{5}{6}\). Find the length of segment \(DP\). If necessary, round the answer to hundredths.
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Context:
若 3 个整数 a,b,c 满足 a^2+b^2+c^2+3<ab+3b+3c，则这样的有序整数组 (a,b,c) 共有__________组.
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Context:
Calculate the limit of the sequence defined by $f(n) = |B_{2n}|^{1/2n}$, where $B_{2n}$ denotes the $(2n)$-th Bernoulli number, as $n$ approaches infinity. Specifically, find the value of $$\lim\limits_{n\to\infty}(f(n+1) - f(n)).$$
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Context:
Tim has a working analog 12-hour clock with two hands that run continuously (instead of, say, jumping on the minute). He also has a clock that runs really slow—at half the correct rate, to be exact. At noon one day, both clocks happen to show the exact time. At any given instant, the hands on each clock form an angle between $0^\circ$ and $180^\circ$ inclusive. At how many times during that day are the angles on the two clocks equal?
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Context:
Points \( M, N, \) and \( K \) are located on the lateral edges \( A A_{1}, B B_{1}, \) and \( C C_{1} \) of the triangular prism \( A B C A_{1} B_{1} C_{1} \) such that \( \frac{A M}{A A_{1}} = \frac{5}{6}, \frac{B N}{B B_{1}} = \frac{6}{7}, \) and \( \frac{C K}{C C_{1}} = \frac{2}{3} \). Point \( P \) belongs to the prism. Find the maximum possible volume of the pyramid \( M N K P \), given that the volume of the prism is 35.
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Context:
How many ordered quadruples $(a, b, c, d)$ of four distinct numbers chosen from the set $\{1,2,3, \ldots, 9\}$ satisfy $b<a, b<c$, and $d<c$?
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Context:
(MON 1) Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.
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Context:
What is the largest possible area of a quadrilateral with sides 1, 4, 7, and 8? Express your answer as a single number.
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Context:
The average length of some strings is 80 cm. If the average length of one third of the strings is 70 cm, the average of the other strings is 85 cm. How many strings are there in total?
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Context:
Given that $\sqrt{23.6}=4.858$ and $\sqrt{2.36}=1.536$, calculate the value of $\sqrt{0.00236}$. Express your answer as a decimal value.
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Context:
双曲线$C:\frac{{{x}^{2}}}{{{a2}}}-\frac{{{y}^{2}}}{{{b2}}}=1\left( a>0,b>0 \right)$的右焦点为$\textit{F}$，$\textit{O}$为坐标原点，以$\textit{F}$为圆心，$\left| OF \right|$为半径的圆与$\textit{C}$和$\textit{C}$的渐近线在第一象限分别交于$\textit{M}$，$\textit{N}$两点，线段$\textit{MF}$的中点为$\textit{P}$.若$\left| ON \right|=\left| OP \right|$，则$\textit{C}$的离心率为________.
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Context:
The acute angle formed by the intersection of line $l_1$: $\sqrt{3}x-y+1=0$ and line $l_2$: $x+5=0$ is ______.
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Context:
设实数$x,y$满足$3\le x{{y}^{2}}\le 8,4\le \frac{{{x2}}}{y}\le 9$,则$\frac{{{x}^{3}}}{{{y4}}}$的最大值是_______.
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Context:
For some positive integers $c$ and $d$, the product \[\log_c(c+1) \cdot \log_{c+1} (c+2) \dotsm \log_{d-2} (d-1)  \cdot\log_{d-1} d\]contains exactly $930$ terms, and its value is $3.$ Compute $c+d.$
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Context:
Find the interval(s) where the function $f(x)=2\cos (-2x+ \frac {π}{4})$ is monotonically increasing.
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Context:
The center of a semicircle, inscribed in a right triangle such that its diameter lies on the hypotenuse, divides the hypotenuse into segments of 30 and 40. Find the length of the arc of the semicircle that is enclosed between the points where it touches the legs.
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Context:
Given a function y = f(x) such that f(x) = f(4-x) for all x in R, and f(x) is an increasing function on the interval [2, +inf), determine the correct ordering of the function values f(4), f(1), and f(0.5). Express your answer as an inequality or ordered relationship using function notation (e.g., f(4) > f(0.5) > f(1)).
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Context:
Given that the proposition "$\exists x \in (0, +\infty), x^2 - ax + 1 < 0$" is true, determine the range of values for the real number $a$.
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Context:
$\frac{4}{x+1}+\frac{9}{x+2}+\frac{16}{x+3}=(4 x+5)(2-x)$ 有几个正实数解?
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Context:
Reema took a loan of Rs 45,000 with simple interest for as many years as the rate of interest. If she paid Rs 12,500 as interest at the end of the loan period, what was the rate of interest?
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Context:
Given the function $f(x)$ that satisfies $f\left(\frac{\pi}{2} - x\right) + f(x) = 0$ and $f(\pi + x) = f(-x)$, calculate the value of $f\left(\frac{79\pi}{24}\right)$.
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Context:
3. Call a lattice point visible if the line segment connecting the point and the origin does not pass through another lattice point. Given a positive integer $k$, denote by $S_{k}$ the set of all visible lattice points $(x, y)$ such that $x^{2}+y^{2}=k^{2}$. Let $D$ denote the set of all positive divisors of $2021 \cdot 2025$. Compute the sum
$$
\sum_{d \in D}\left|S_{d}\right|
$$

Here, a lattice point is a point $(x, y)$ on the plane where both $x$ and $y$ are integers, and $|A|$ denotes the number of elements of the set $A$.
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Context:
In an isosceles triangle \(ABC \) (\(AB = BC\)), the angle bisectors \(BD\) and \(AF\) intersect at point \(O\). The ratio of the area of triangle \(DOA\) to the area of triangle \(BOF\) is \(\frac{3}{8}\). Find the ratio \(\frac{AC}{AB}\).
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Context:
Given three composite numbers \( A, B, \) and \( C \) which are pairwise coprime, and \( A \times B \times C = 11011 \times 28 \). What is the maximum value of \( A + B + C \)?
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Context:
Let $N$ denote the number of subsets of $\{1,2,3, \ldots, 100\}$ that contain more prime numbers than multiples of 4. Compute the largest integer $k$ such that $2^{k}$ divides $N$.
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Context:
A rectangular piece of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called basic if
 (i) all four sides of the rectangle are segments of drawn line segments, and
 (ii) no segments of drawn lines lie inside the rectangle.
 Given that the total length of all lines drawn is exactly 2007 units, let $N$ be the maximum possible number of basic rectangles determined. Find the remainder when $N$ is divided by 1000.
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Context:
Let \( G, A_{1}, A_{2}, A_{3}, A_{4}, B_{1}, B_{2}, B_{3}, B_{4}, B_{5} \) be ten points on a circle such that \( G A_{1} A_{2} A_{3} A_{4} \) is a regular pentagon and \( G B_{1} B_{2} B_{3} B_{4} B_{5} \) is a regular hexagon, and \( B_{1} \) lies on the minor arc \( G A_{1} \). Let \( B_{5} B_{3} \) intersect \( B_{1} A_{2} \) at \( G_{1} \), and let \( B_{5} A_{3} \) intersect \( G B_{3} \) at \( G_{2} \). Determine the degree measure of \( \angle G G_{2} G_{1} \).
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Context:
For which values of the parameter \( a \) does the equation \( x^{3}+6x^{2}+ax+8=0 \) have exactly three solutions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two different digits from 1 to 9 are chosen. One digit is placed in each box to complete the two 2-digit numbers shown. The result of subtracting the bottom number from the top number is calculated. How many of the possible results are positive? Express your answer as a whole number.
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Context:
You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out, you must open all the doors and disarm all the mines. In the room, there is a panel with $3$ buttons, which conveniently contains an instruction manual:

- The red button arms a mine.
- The yellow button disarms two mines and closes a door.
- The green button opens two doors.

Initially, $3$ doors are closed, and $3$ mines are armed. The manual warns that attempting to disarm two mines or open two doors when only one is armed/closed will reset the system to its initial state. What is the minimum number of buttons you must push to get out?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest positive integer  $n$  for which there exist pairwise different sets  $\mathbb{S}_1 , ..., \mathbb{S}_n$  with the following properties:   $1$ )  $|\mathbb{S}_i \cup \mathbb{S}_j | \leq 2004$  for any two indices  $1 \leq i, j\leq n$ , and  $2$ )  $\mathbb{S}_i \cup \mathbb{S}_j \cup \mathbb{S}_k = \{ 1,2,...,2008 \}$  for any  $1 \leq i < j < k \leq n$ *Proposed by Ivan Matic*
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Context:
Determine the maximal possible length of the sequence of consecutive integers which can be expressed in the form $x^3 + 2y^2$, where $x$ and $y$ are integers.
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Context:
Joel's garden is circular with a diameter of 16 feet. He wants to use 40% of the garden for fruits, 35% for vegetables, and 25% for flowers. In the fruit section, he wants to use 30% for strawberries, 50% for apples, and 20% for oranges. How many square feet of the circular garden will be used for strawberries?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a positive integer $N$, let $f_N^{1}$ be the function defined by
$$f_N(x)=\sum_{n=0}^N\frac{N+1/2-n}{(N+1)(2n+1)}\sin((2n+1)x).$$
Determine the smallest constant $M$ such that $f_N(x)\leq M$
 for all $N$ and all real $x.$ The original answer is in $\frac{m\pi}{n}$ format, please give the value of m + n.
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Context:
Given a grid strip (one cell wide), infinite in both directions. Two cells of the strip are traps, with $-N$ cells between them, one of which is occupied by a grasshopper. On each move, we call out a natural number, after which the grasshopper jumps that number of cells to the left or right (at its choice). For which $N$ can we call out numbers such that we can guarantee to drive the grasshopper into one of the traps, no matter where it initially is between the traps and how it chooses the directions of its jumps? (We always see where the grasshopper is sitting.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of all divisors \( d \) of \( N = 19^{88} - 1 \) that are of the form \( d = 2^a \cdot 3^b \) where \( a \) and \( b \) are natural numbers.
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Context:
已知自然数序列$\{ x _ { n } \}$满足$x _ { 1 } = a$ $x _ { 2 } = b ,$ $, x _ { n + 2 } = x _ { n } + x _ { n + 1 } , n = 1 , 2 , 3 ,$，···如果序列中某项为1000，试问a+b的最小可能值是多少？
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Context:
Find the value of $a$ such that:

\[101a = 6539\int_{-1}^1 \frac{x^{12}+31}{1+2011^{x}}\ dx.\]
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Context:
In an economy following supply and demand rules, two different types of articles (Article A and Article B) experience price fluctuations. Article A has a selling price of "x" dollars. If this article were sold at half of that price, there would be a loss of 20%. Article B has a selling price of "y" dollars. Due to an increase in demand, the selling price of Article B has increased by 5% and now it equals the selling price of Article A after a 10% discount. In addition, a sales tax of 8% is applied to both articles. What profit percent is made by selling Article A after considering the tax, discount, and price fluctuations of both articles?
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Context:
Tom's cat needs an expensive surgery. He has had pet insurance for 24 months that cost $20 per month. The procedure cost $5000 but the insurance covers a certain percentage of this. He saved $3520 by having insurance. What percentage of the procedure cost does the insurance not cover?
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Context:
Mary works in a restaurant a maximum of 70 hours a week. For the first 40 hours, she is paid $10 per hour. For the next 20 hours (overtime), she is paid at a rate which is 30% higher than her regular rate. For any additional hours worked beyond 60, her pay rate is increased to 60% higher than her regular rate. Furthermore, there are tiered deductions for taxes and insurance based on her earnings:

- For hourly earnings up to $400, she pays a flat 15% in taxes and $50 for insurance.
- For hourly earnings between $401 and $600, she pays an additional 10% tax on that portion.
- For hourly earnings above $600, she pays an additional 25% tax on that portion.

On weekends, Mary gets a flat bonus of $75 for every 8-hour shift she works. If Mary works a combination of regular shifts and overtime during the week, and a single 8-hour shift on weekends, how much will she earn in a week after accounting for deductions?
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Context:
In a pirate's trunk, there are 5 chests. In each chest, there are 4 boxes, and in each box, there are 10 gold coins. The trunk, the chests, and the boxes are all locked. Blind Pew unlocks 9 locks and takes all the coins in all the boxes he unlocks. What is the smallest number of gold coins he could take?
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Context:
In a zoo, there were 200 parrots. One day, they each made a statement in turn. Starting from the second parrot, all statements were: "Among the previous statements, more than 70% are false." How many false statements did the parrots make in total?
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Context:
Given that a hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ has only one common point with the parabola $y=x^{2}+1$, calculate the eccentricity of the hyperbola.
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Context:
Find all values that the expression

$$
5 \arcsin x - 2 \arccos y
$$

can take under the condition \(x^{2} + y^{2} = 1\).
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Context:
In the cube $ABCD-A_1B_1C_1D_1$, among all the edges, face diagonals, and space diagonals, how many line segments are perpendicular to $AB$? Express your answer as a whole number.
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Context:
In the tetrahedron \( ABCD \), it is known that \( \angle ADB = \angle BDC = \angle CDA = 60^\circ \), \( AD = BD = 3 \), and \( CD = 2 \). Find the radius of the circumscribed sphere of the tetrahedron \( ABCD \).
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Context:
Call a positive integer $N \ge 2$ "special" if for every $k$ such that $2 \leq k \leq N$, $N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). How many special integers are there less than $100$?
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Context:
Let $ABCD$ be a quadrilateral such that $\angle ABC = \angle  CDA = 90^o$, and $BC = 7$. Let $E$ and $F$ be on $BD$ such that $AE$ and $CF$ are perpendicular to BD. Suppose that $BE = 3$. Determine the product of the smallest and largest possible lengths of $DF$.
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Context:
In the Cartesian coordinate system \(xOy\), point \(A\) is on the circle \(x^{2} + y^{2} = 1\), and the coordinates of point \(B\) are \((3,0)\). If point \(C\) makes \(\triangle ABC\) an equilateral triangle, then the maximum value of \(|OC|\) is \(\quad\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A regular tetrahedron has a square shadow of area 16 when projected onto a flat surface (light is shone perpendicular onto the plane). Compute the sidelength of the regular tetrahedron.
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Context:
a train is 300 meter long is running at a speed of 35 km / hour . in what time will it pass a bridge of 115 meter length ?
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Context:
On March $6$, $2002$, during the celebrations of the $500$th anniversary of the birth of mathematician Pedro Nunes, ten people entered the Viva bookstore for science. Each person bought exactly $3$ different books. Furthermore, any two people bought at least one copy of the same book. "The Adventures of Mathematics" by Pedro Nunes was one of the books that achieved the highest number of sales that morning. What is the smallest value this number could have taken?
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Context:
In the land of Draconia, there are red, green, and blue dragons. Each dragon has three heads, and each head always either tells the truth or lies consistently. Moreover, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each dragon said the following:

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

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


Context:
Given a tetrahedron \(K L M N\) with \(K L = 5\), \(N M = 6\), \(\angle L M N = \angle K N M = 35^\circ\), and \(\angle L N M = \angle K M N = 55^\circ\). A sphere is circumscribed around the tetrahedron. Consider the set of all points on this sphere such that the sum of spherical distances from these points to \(K\), \(L\), \(M\), and \(N\) is no more than \(6 \pi\). What is the area of this set? Note that the spherical distance between two points on a sphere is the length of the shortest arc of a great circle connecting these points.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Line $l_1$ has equation $4x - 3y = 2$ and passes through point $D = (-2, -3)$. Line $l_2$ has equation $y = 2$ and intersects line $l_1$ at point $E$. Line $l_3$ has a positive slope, passes through point $D$, and meets $l_2$ at point $F$. The area of $\triangle DEF$ is $6$. What is the slope of $l_3$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of positive integers $n \le 2014$ such that there exists an integer $x$ satisfying the condition that $\frac{x+n}{x-n}$ is an odd perfect square.
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Context:
Evin's calculator is broken and can only perform three operations:

1. Given a number $x$, output $2x$.
2. Given a number $x$, output $4x + 1$.
3. Given a number $x$, output $8x + 3$.

Initially starting with the number $0$, determine how many distinct numbers at most $128$ can be generated using these operations.
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Context:
Let  $n$  be a positive integer with  $k\ge22$  divisors  $1=d_{1}< d_{2}< \cdots < d_{k}=n$ , all different. Determine all  $n$  such that \[{d_{7}}^{2}+{d_{10}}^{2}= \left( \frac{n}{d_{22}}\right)^{2}.\]
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Context:
In trapezoid  $ABCD$ ,  $AD \parallel BC$  and  $\angle ABC + \angle CDA = 270^{\circ}$ .  Compute  $AB^2$  given that  $AB \cdot \tan(\angle BCD) = 20$  and  $CD = 13$ .
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Context:
Find the function $g(x)$ if it satisfies the equation $$(e^x-1)g(4x)=\left(e^{4x}-1\right)g(x)$$ and $g'(0) = 2$.
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Context:
Find the smallest number written using only the digits 0 and 1 that is divisible by the product of the six smallest natural numbers.
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Context:
In the Cartesian coordinate plane, define d(P, Q) = |x₁ - x₂| + |y₁ - y₂| as the "manhattan distance" between two points P(x₁, y₁) and Q(x₂, y₂). Find the minimum "manhattan distance" between a point P on the ellipse $\frac{x^2}{2} + y^2 = 1$ and a point Q on the line 3x + 4y - 12 = 0.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find \( k \) such that, for all \( n \), the following expression is a perfect square:

$$
4 n^{2} + k n + 9
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways are there to label the faces of a regular octahedron with the integers 18, using each exactly once, so that any two faces that share an edge have numbers that are relatively prime? Physically realizable rotations are considered indistinguishable, but physically unrealizable reflections are considered different.
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Context:
In the faculty of reverse-engineering, 240 second-year students study numeric methods, 423 second-year students study automatic control of airborne vehicles, and 134 second-year students study them both. There are 663 students in the faculty. What percentage of the total students are second-year students?
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Context:
Among the nine digits 1, 2, 3, …, 9, select four digits to form a four-digit number where the first and last digits are odd. How many such four-digit numbers are there?
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Context:
One mole of an ideal monoatomic gas is first heated isobarically. In this process, it does 20 J of work. Then, it is heated isothermally, receiving the same amount of heat as in the first case. How much work (in joules) does the gas do in the second case?
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Context:
Let \(ABCDEF\) be a regular hexagon with side length \(1\). Denote by \(X\), \(Y\), and \(Z\) the midpoints of sides \(\overline{AB}\), \(\overline{CD}\), and \(\overline{EF}\), respectively. Find the area of the convex hexagon whose interior is the intersection of the interiors of \(\triangle ACE\) and \(\triangle XYZ\). The original answer is in the form \(\frac{k}{m}\sqrt{n}\), where \(\frac{k}{m}\) is in its simplest form. Please give the value of \(k + m + n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define the \textit{bigness} of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer $N$ for which there exists a rectangular prism with integer side lengths and \textit{bigness} $N$ and another one with integer side lengths and \textit{bigness} $N + 1$.
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Context:
(8) Let $S(n)$ denote the sum of the digits of the positive integer $n$, then $\sum_{n=1}^{2011} S(n)=$
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Context:
Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from $3$ pears and $8$ ounces of orange juice from $2$ oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What is the percentage of the blend that is pear juice?
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Context:
Find the smallest natural number \( n \) such that every convex 100-gon can be represented as the intersection of \( n \) triangles.
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Context:
Given an odd prime number \( p \), there exists a positive integer \( k \) such that \( \sqrt{k - p k} \) is also a positive integer. Find the value of \( k \).
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Context:
Find the minimum value of k such that every k-element subset of S = {1, 2,..., 32} contains three different elements a, b, c such that a divides b and b divides c. Express your answer as a single integer.
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Context:
在 $\triangle A B C$ 中, 若 $O$ 为形外一点, 满足 $\angle B O C=2 \angle B A C$, 线段 $O C$ 与线段 $A B$ 交于 $D$, 且 $O B=O C=3$, $O D=2$, 求 $B D \cdot A D$.
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Context:
Evaluate the limit $\lim_{x\to0^+}\frac{x^{x^x}- (\sin x)^{x^x}}{x^4}$.
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Context:
A product was introduced into a shopping mall for sale. In the first year, the mall did not charge a management fee, so the product was priced at 70 yuan per piece, with an annual sales volume of 11.8 thousand pieces. Starting from the second year, the mall charged a management fee of x% of the sales amount (i.e., for every 100 yuan of sales, x yuan was charged as a fee). As a result, the price per piece of the product increased by 70 * x% / (1 - x%) yuan compared to the first year, and the expected annual sales volume decreased by x thousand pieces. Find the maximum value of x such that the management fee collected by the mall from this product in the second year is not less than 140,000 yuan.
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Context:
有限项等差数列的公差为 $4$ ，第二项起各项之和+首项的平方 $<100$ ，则该数列可能有几项?
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Context:
Let  $x,y,z$  is positive. Solve: $\begin{cases}{x\left( {6 - y} \right) = 9}
{y\left( {6 - z} \right) = 9}
{z\left( {6 - x} \right) = 9}\end{cases}$
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Context:
Let $a, b, c$, and $d$ be real numbers. The six sums of two numbers $x$ and $y$, different from the previous four, are $117$, $510$, $411$, and $252$, in no particular order. Determine the maximum possible value of $x + y$.
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Context:
Let the positive integer \( n \) satisfy \( n \leq 2016 \), and \( \left\{\frac{n}{2}\right\}+\left\{\frac{n}{4}\right\}+\left\{\frac{n}{6}\right\}+\left\{\frac{n}{12}\right\}=3 \). The number of such \( n \) is \(\quad\). Here, \(\{x\} = x - \lfloor x \rfloor\), where \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \( x \).
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Context:
Petya and Vasya are playing a game. Initially, each of the 2022 boxes contains one matchstick. In one move, all matchsticks from any non-empty box can be transferred to any other non-empty box. They take turns, with Petya starting. The winner is the one who, after their move, has at least half of all the matchsticks in one box for the first time. Who will win with the correct play?
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Context:
After a typhoon, 13 trees in Haley's backyard died. If she had grown 3 trees initially, how many more trees died in the typhoon than those that survived?
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Context:
Let the sum of the first $n$ terms of a geometric sequence be $S_n$. If the ratio $S_{12} : S_6 = 1 : 2$, then the ratio $S_{18} : S_6 =$ \_\_\_\_\_\_.
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Context:
The symbol \( R_{k} \) represents an integer whose decimal representation consists of \( k \) consecutive 1s. For example, \( R_{3} = 111 \), \( R_{5} = 11111 \), and so on. If \( R_{4} \) divides \( R_{24} \), the quotient \( Q = \frac{R_{24}}{R_{4}} \) is an integer, and its decimal representation contains only the digits 1 and 0. How many zeros are in \( Q \)?
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Context:
Let $D$ be a point on the side $[BC]$ of $\triangle ABC$ such that $|AB| + |BD| = |AC|$ and $m(\widehat{BAD}) = m(\widehat{DAC}) = 30^\circ$. If the answer is $x^\circ$, what is the value of $x$ for $m(\widehat{ACB})$? Please provide the value of $x$ as an integer.
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Context:
Given \( f_{1}(x)=|1-2 x| \) for \( x \in [0,1] \) and \( f_{n}(x)=f_{1}(f_{n-1}(x)) \), determine the number of solutions to the equation \( f_{2005}(x)=\frac{1}{2} x \).
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Context:
A regular triangular pyramid \(SABC\) is given, with the edge of its base equal to 1. Medians of the lateral faces are drawn from the vertices \(A\) and \(B\) of the base \(ABC\), and these medians do not intersect. It is known that the edges of a certain cube lie on the lines containing these medians. Find the length of the lateral edge of the pyramid.
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Context:
Four points \( B, A, E, L \) are on a straight line. \( G \) is a point off the line such that \(\angle B A G = 120^\circ\) and \(\angle G E L = 80^\circ\). Find the reflex angle at \( G \). Express your answer in degrees as a whole number.
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Context:
Let $\Omega_1$ be a circle with center $O$, and let $AB$ be the diameter of $\Omega_1$. Let $P$ be a point on the segment $OB$, different from $O$. Suppose another circle $\Omega_2$ with center $P$ lies in the interior of $\Omega_1$. Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$, intersecting $\Omega_1$ again at $A_1$ and $B_1$, respectively, such that $A_1$ and $B_1$ are on opposite sides of $AB$. Given that $A_1B = 5$, $AB_1 = 15$, and $OP = 10$, find the radius of $\Omega_1$.
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Context:
Calculate $\lim\limits_{n\to\infty} V_{n}$ ($\forall n\in \mathbb{N}$), where $V_{n}=\left(\sum\limits_{k=0}^{2n}\sqrt{k}\right)-(2n+1)\sqrt{n}$.
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Context:
In a tournament with 12 chess players, each player plays against every other player exactly once. After the tournament, each player creates a series of lists: the first list contains only their own name, the second list includes their name and the names of all opponents they defeated, and each subsequent list includes all names from the previous list plus the names of players defeated by anyone on the previous list. It is known that for every player, the eleventh and twelfth lists have different numbers of names. Determine the number of games that ended in a draw during the tournament.
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Context:
Suppose the scores of an exam follow a normal distribution, and the correlation coefficient \( \rho = 0.8 \) for exam1 and exam2. If a student's score percentile rank (PR) is 90% in exam1, estimate his/her PR in exam2.
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Context:
在三棱锥 $P-A B C$ 中, 三条棱 $P A$ 、 $P B 、 P C$ 两两垂直，且 $P A=1, P B=P C=2$. 若 $Q$ 为三棱锥 $P-A B C$ 外接球的球面上任一点, 则 $Q$ 到面 $A B C$ 距离的最大值减去$\frac{\sqrt{6}}6$的结果的2倍为 $\qquad$.
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Context:
Let \(ABC\) be an acute-angled triangle with \(AB \neq BC\) and circumcircle \(k\). Let \(P\) and \(Q\) be the intersections of \(k\) with the angle bisector and the external angle bisector of \(\angle CBA\) respectively. Let \(D\) be the intersection of \(AC\) and \(PQ\). Determine the ratio \(AD:DC\).
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Context:
How many ways can you fill a table of size $n \times n$ with integers such that each cell contains the total number of even numbers in its row and column, excluding itself? Two tables are considered different if they differ in at least one cell.
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Context:
Let $N = 12!$ and denote by $X$ the set of positive divisors of $N$ other than $1$. A pseudo-ultrafilter $U$ is a nonempty subset of $X$ such that for any $a,b \in X$:
\begin{itemize}
\item If $a$ divides $b$ and $a \in U$ then $b \in U$.
\item If $a,b \in U$ then $\gcd(a,b) \in U$.
\item If $a,b \notin U$ then $\operatorname{lcm}(a,b) \notin U$.
\end{itemize}
How many such pseudo-ultrafilters are there?
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Context:
A sum fetched a total simple interest of Rs. 6016.75 at the rate of 8% per annum in 5 years. What is the sum?
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Context:
Molly has a set of cards numbered 1 to 12. She places eight of them at the vertices of an octagon so that the sum of every pair of numbers at opposite ends of an edge of the octagon is a multiple of 3. Identify the numbers that Molly did not place on the vertices of the octagon. Express your answer as a list of numbers.
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Context:
Distinct points \(A\) and \(B\) are on a semicircle with diameter \(MN\) and center \(C\). The point \(P\) is on \(CN\) and \(\angle CAP = \angle CBP = 10^{\circ}\). If \(\stackrel{\frown}{MA} = 40^{\circ}\), then \(\stackrel{\frown}{BN}\) equals how many degrees? Please provide your answer as an integer representing the number of degrees.
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Context:
已知集合 $A, B, C \subseteq\{1,2,3, \cdots, 2020\}$, 且 $A \subseteq B \subseteq C$, 求有序集合组 $(A, B, C)$ 的个数。答案的形式为x^y，请给出x+y的值。
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Context:
## Task A-1.1.

Determine the smallest natural number $a$ such that the value of the expression

$$
\frac{n^{8}+n^{6}+n^{4}+n^{2}+a}{n^{2}-1}
$$

for $n=2014$ is an integer divisible by 3.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How can we determine the number of possible arrangements of the pieces in a $2 \times 2 \times 2$ Rubik's cube, considering both their orientations and relative positions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
From a group of 5 male teachers and 4 female teachers, select 3 teachers to be class leaders for 3 different classes (1 class leader per class). The selection must include both male and female teachers. How many different appointment plans are there?
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Context:
There were some roses and 84 orchids in the vase. Jessica cut some more roses and orchids from her flower garden. There are now 91 orchids and 14 roses in the vase. How many roses were initially in the vase before Jessica cut more?
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Context:
Given an ellipse \( C_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0) \) and a hyperbola \( C_{2}: \frac{x^{2}}{m^{2}}-\frac{y^{2}}{n^{2}}=1(m>0, n>0) \) with the same foci \( F_{1}, F_{2} \), where \( F_{1} \) is the left focus, point \( P \) is the point of intersection of the two curves in the first quadrant, and \( e_{1}, e_{2} \) are the eccentricities of curves \( C_{1} \) and \( C_{2} \) respectively. If triangle \( \triangle P F_{1} F_{2} \) is an isosceles triangle with \( P F_{1} \) as the base, then the range of values for \( e_{2} - e_{1} \) is \(\qquad\).
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Context:
Nicky and Jill are trading baseball cards internationally. Nicky, who lives in the United States, trades two cards worth $8 each and three cards worth $5 each. Jill, who lives in Canada, trades 1 card worth CAD $21 and 2 cards worth CAD $6 each. If the exchange rate is 1 CAD = 0.8 USD and there is a 5% tax rate on the total value of the traded cards in USD, how many dollars of profit or loss does Nicky make on this trade?
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Context:
11. (20 points) For any real numbers $a_{1}, a_{2}, \cdots, a_{5}$ $\in[0,1]$, find the maximum value of $\prod_{1 \leq i<j \leq 5}\left|a_{i}-a_{j}\right|$.
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Context:
A point is randomly thrown on the segment [3, 8] and let \( k \) be the resulting value. Find the probability that the roots of the equation \((k^{2}-2k-3)x^{2}+(3k-5)x+2=0\) satisfy the condition \( x_{1} \leq 2x_{2} \).
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Context:
A triangle has an area of 114 square units and sides of integer length. What is the perimeter of the triangle?
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Context:
Given the function $f(x)=\sin (ωx+φ)(ω > 0,|φ|\leqslant \dfrac {π}{2})$, $y=f(x- \dfrac {π}{4})$ is an odd function, $x= \dfrac {π}{4}$ is the symmetric axis of the graph of $y=f(x)$, and $f(x)$ is monotonic in $(\dfrac {π}{14}, \dfrac {13π}{84})$, determine the maximum value of $ω$.
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Context:
Given a finite group $G$ with $100$ elements, what is the maximum probability that the equation $ab = cba$ holds for a randomly chosen pair of elements $a$ and $b$, where $c$ is a fixed element of $G$ that is not the identity?
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Context:
A mahogany bookshelf has four identical-looking books which are $200$, $400$, $600$, and $800$ pages long. Velma picks a random book off the shelf, flips to a random page to read, and puts the book back on the shelf. Later, Daphne also picks a random book off the shelf and flips to a random page to read. Given that Velma read page $122$ of her book and Daphne read page $304$ of her book, the probability that they chose the same book is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.
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Context:
Square $ABCD$ has side length $s$, and a circle centered at $E$ has radius $r$. Both $r$ and $s$ are rational numbers. The circle passes through $D$, and $D$ lies on $\overline{BE}$. Point $F$ lies on the circle, on the same side of $\overline{BE}$ as $A$. Segment $AF$ is tangent to the circle, and $AF = \sqrt{9 + 5\sqrt{2}}$. Find the ratio \(\frac{r}{s}\) in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of \(k + m\).
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Context:
In the tetrahedron \(A B C D\), it is known that \(DA \perp\) the base \(ABC\), the face \(ABD \perp\) the face \(BCD\), and \(BD = BC = 2\). The sum of the squares of the areas of the three faces \(DAB\), \(DBC\), and \(DCA\) is 8. Find \(\angle ADB\).
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Context:
The sequence $\left\{a_{n}\right\}_{n \geq 1}$ is defined by $a_{n+2}=7 a_{n+1}-a_{n}$ for positive integers $n$ with initial values $a_{1}=1$ and $a_{2}=8$. Another sequence, $\left\{b_{n}\right\}$, is defined by the rule $b_{n+2}=3 b_{n+1}-b_{n}$ for positive integers $n$ together with the values $b_{1}=1$ and $b_{2}=2$. Find \operatorname{gcd}\left(a_{5000}, b_{501}\right).
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Context:
有 $m$ 个男孩, $n$ 个女孩, ( $m, n$ 为正整数) 围坐在一个圆周上。我们在两端为男孩的弧段上插上一个数字 2 , 在相邻两端为女孩的弧段上插上一个数字 -2 , 相邻两端为一男一女的弧段上插上数字 0 , 把所插的数字总和记作 $a$, 则 $a$ 的可取值为 $\qquad$
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Context:
Let $w, x, y,$ and $z$ be nonzero complex numbers, and let $n$ be a positive integer. Suppose that the following conditions hold:

- $\frac{1}{w} + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 3,$
- $wx + wy + wz + xy + xz + yz = 14,$
- $(w + x)^3 + (w + y)^3 + (w + z)^3 + (x + y)^3 + (x + z)^3 + (y + z)^3 = 2160,$
- $w + x + y + z + i\sqrt{n} \in \mathbb{R}.$

Compute $n$.
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Context:
For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\dots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties: \begin{enumerate} \item[(a)] $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1,\dots,t_n$; \item[(b)] $f(t_0) = 1/2$; \item[(c)] $\lim_{t \to t_k^+} f'(t) = 0$ for $0 \leq k \leq n$; \item[(d)] For $0 \leq k \leq n-1$, we have $f''(t) = k+1$ when $t_k < t< t_{k+1}$, and $f''(t) = n+1$ when $t>t_n$. \end{enumerate} Considering all choices of $n$ and $t_0,t_1,\dots,t_n$ such that $t_k \geq t_{k-1}+1$ for $1 \leq k \leq n$, what is the least possible value of $T$ for which $f(t_0+T) = 2023$?
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Context:
如图，在正方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 中， $E$ 是 $B C$ 的中点， $F$ 在 $A A_{1}$ 上，且 $A_{1} F: F A=1: 2$, 求平面 $B_{1} E F$ 与底面 $A_{1} B_{1} C_{1} D_{1}$ 所成的二面角。已知原始答案为\( \arctan \frac{\sqrt{m}}{n} \)的格式，请给出m + n的值。
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Context:
If the function $y = x^2 - 2ax + a$ has an inverse function on the interval $x \in [1, 3]$ and $|a-1| + |a-3| \leq 4$, find the range of possible values for $a$.
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Context:
Problem 2. (4 points) In a box, there are red, blue, yellow, and white balls, a total of 111. If 100 balls are drawn from the box with closed eyes, among them there will definitely be four balls, each two of which are of different colors. How many balls at least need to be drawn from the box (with closed eyes) to be able to assert with certainty that among them there are three balls, each two of which are of different colors?
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Context:
Calculate $1,000,000,000,000 - 888,777,888,777$.
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Context:
How many distinct strings of length 8 can be formed using the letters $\{a, b, c, d\}$ if each string contains exactly one pair of adjacent letters that are identical?
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Context:
Find the weight function for the Sturm-Liouville problem: $$\frac{d}{dx}\left[(1+x^2)\frac{dy(x)}{dx}\right] + \lambda \sigma (x)y(x)=0,\; y'(0)=y'(1)=0,\; 0 \leq x \leq 1.$$
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Context:
Given the integral $I_n=\int_0^{\frac{\pi}{4}}e^{nx}(\tan^{n-1}x+\tan^nx+\tan^{n+1}x)\,dx$, determine the limit $\lim\limits_{n\to \infty} n\left(\sqrt[n^2]{nI_n}-1\right)$.
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Context:
Mad scientist Kyouma writes $N$ positive integers on a board. Each second, he chooses two numbers $x, y$ written on the board with $x > y$, and writes the number $x^2 - y^2$ on the board. After some time, he sends the list of all the numbers on the board to Christina. She notices that all the numbers from 1 to 1000 are present on the list. Aid Christina in finding the minimum possible value of $N$.
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Context:
Let \( x \) be a positive integer, and write \( a = \left\lfloor \log_{10} x \right\rfloor \) and \( b = \left\lfloor \log_{10} \frac{100}{x} \right\rfloor \). Here \( \lfloor c \rfloor \) denotes the greatest integer less than or equal to \( c \). Find the largest possible value of \( 2a^2 - 3b^2 \).
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Context:
Find the area of the region of the \( xy \)-plane defined by the inequality \( |x|+|y|+|x+y| \leq 1 \).
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Context:
Write $2017$ numbers on the blackboard: $-\frac{1008}{1008}, -\frac{1007}{1008}, \ldots, -\frac{1}{1008}, 0, \frac{1}{1008}, \frac{2}{1008}, \ldots, \frac{1007}{1008}, \frac{1008}{1008}$.

One processes some steps as follows: erase two arbitrary numbers $x, y$ on the blackboard and then write the number $x + 7xy + y$. After $2016$ steps, there is only one number left. Find the last number on the blackboard in the form of a simplified fraction \(-\frac{k}{m}\). Please provide the value of \(k + m\).
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Context:
Given a population of $100$ individuals randomly numbered from $0$ to $99$, and a sample of size $10$ is drawn, with the units digit of the number drawn from the $k$-th group being the same as the units digit of $m + k$, where $m = 6$, find the number drawn from the 7-th group.
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Context:
There are a batch of wooden strips with lengths of \(1, 2, 3, 4, 5, 6, 7, 8, 9, 10,\) and 11 centimeters, with an adequate quantity of each length. If you select 3 strips appropriately to form a triangle with the requirement that the base is 11 centimeters long, how many different triangles can be formed?
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Context:
Trodgor the dragon is burning down a village consisting of $ 90$ cottages.  At time $ t \equal{} 0$ an angry peasant arises from each cottage, and every $ 8$ minutes ($ 480$ seconds) thereafter another angry peasant spontaneously generates from each non-burned cottage.  It takes Trodgor $ 5$ seconds to either burn a peasant or to burn a cottage, but Trodgor cannot begin burning cottages until all the peasants around him have been burned. How many [b]seconds[/b] does it take Trodgor to burn down the entire village?
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Context:
A conical glass is in the form of a right circular cone. The slant height is $21$ and the radius of the top rim of the glass is $14$. An ant at the midpoint of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If $d$ is the shortest distance it should crawl to reach the honey drop, what is the integer part of $d$?
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Context:
A girl multiplies 987 by a certain number. If in the answer, both 9's are wrong but the other digits are correct, and the correct answer is 555681. What is the number she multiplied 987 by?
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Context:
If $\frac{x-y}{x+y}=5$, what is the value of $\frac{2x+3y}{3x-2y}$?
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Context:
What is the maximum area of the projection of a regular tetrahedron with edge length \(a\) onto a plane?
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Context:
On a quiz, every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions on the quiz.
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Context:
8. Find the largest real number $a$ such that for all positive integers $n$ and all real numbers $x_{0}, x_{1}, \cdots, x_{n}\left(0=x_{0}<x_{1}<\cdots\right.$ $\left.<x_{n}\right)$, we have
$$
\sum_{i=1}^{n} \frac{1}{x_{i}-x_{i-1}} \geqslant a \sum_{i=1}^{n} \frac{i+1}{x_{i}} .
$$
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Context:
Seven students in a class receive one failing grade every two days of school, while nine other students receive one failing grade every three days. The remaining students in the class never receive failing grades. From Monday to Friday, 30 new failing grades appeared in the class register. How many new failing grades will appear in the class register on Saturday?
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Context:
设方程 xyz=900 的全部正整数解为 (x_i,y_i,z_i) (1\leq i\leq n). 则 \ds{k=1}{n}(x_k+y_k+z_k)=\underline{\qquad\qquad}.
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Context:
A robot vacuum cleaner "Orthogonal" decides to clean the coordinate plane. It starts from point $O$ located at the origin and travels a straight distance $X_{1}$. At the point it reaches, the vacuum cleaner turns and describes a circle centered at the origin. After completing the circle, it continues travelling in the same direction for a straight distance $X_{2}$. It then turns again to describe a circle centered at the origin, and after completing it, travels straight for a distance $X_{3}$. This process continues. The quantities $X_{1}, X_{2}, X_{3}, \ldots$ follow some distribution with expected value $a$ and variance $d$. Find the expected value of the area of the circle bounded by the $n$-th circle that the vacuum cleaner travels.
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Context:
Given the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \ldots$, find $n$ such that the sum of the first $n$ terms is $2008$ or $2009$.
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Context:
Given an ellipse \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) where \( a > b > 0 \), with the left focus at \( F \). A tangent to the ellipse is drawn at a point \( A \) on the ellipse and intersects the \( y \)-axis at point \( Q \). Let \( O \) be the origin of the coordinate system. If \( \angle QFO = 45^\circ \) and \( \angle QFA = 30^\circ \), find the eccentricity of the ellipse.
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Context:
A $10$-digit number is called interesting if its digits are distinct and the number is divisible by $11111$. Determine the total number of interesting numbers.
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Context:
Given a 4-inch cube constructed from 64 smaller 1-inch cubes, with 50 red and 14 white cubes, arrange these cubes such that the white surface area exposed on the larger cube is minimized, and calculate the fraction of the total surface area of the 4-inch cube that is white.
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Context:
In a map with a scale of $1:50000$, if the area of a polygonal land is $100cm^{2}$, what is the actual area of the land in $m^{2}$ (represented in scientific notation)?
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Context:
Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$, and let $q(x) \in \mathbb{F}_p[x]$ be given by \[ q(x) = \sum_{k=1}^{p-1} a_k x^k, \] where \[ a_k = k^{(p-1)/2} \mod{p}. \] Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\mathbb{F}_p[x]$.
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Context:
Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.
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Context:
Let \( AL, BM, CN \) be the medians of triangle \( ABC \), intersecting at point \( K \). It is known that a circumcircle can be drawn around quadrilateral \( CLK M \), and \( AB = 2 \). Find the length of the median \( CN \).
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Context:
A car A is chasing a car B, both moving at the same speed. Car B is moving directly north, while car A is initially located 1 mile west of car B. If car A always heads directly towards car B, find the distance between the cars as time approaches infinity.
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Context:
The ratio of irises to roses in Nikki's garden is 3:7. Initially, she has 35 roses but gives away 15 roses. After this, she plans to add 25 more roses and wants to plant enough new irises to maintain the same ratio. How many irises will she need to have in total after these adjustments?
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Context:
A train of a certain length passes an oak tree in 11.999040076793857 seconds. The speed of the train is 54 km/hr. How long is the train?
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Context:
Find the number of unitary ring homomorphisms from the ring \( \mathbb{Z}[X]/(X^3-1) \) to \( \mathbb{Z}_{12} \).
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Context:
Given n points on a plane, it is known that the distances between some pairs of points are 1, 2, 4, 8, 16, 32. What is the smallest possible value of n?
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Context:
Find the value of the expression \( 2a - \left( \frac{2a - 3}{a + 1} - \frac{a + 1}{2 - 2a} - \frac{a^2 + 3}{2a^{2-2}} \right) \cdot \frac{a^3 + 1}{a^2 - a} + \frac{2}{a} \) when \( a = 1580 \).
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Context:
Find the value of the sum: $$\sum_{k=1}^{\infty}\frac{(k-1)(k-2)(k-3)\cdot\ldots\cdot(k-n)}{n!z^k}$$ where $n,z>1$ are fixed natural numbers.
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Context:
Casey wants to decorate her toenails and fingernails. She wants to do a base coat, a coat of paint, and a coat of glitter on every nail. It takes some time to apply each coat and the same amount of time for each coat to dry before applying the next one. It takes her 120 minutes to finish decorating her fingernails and toenails. How many minutes does it take to apply and dry each coat?
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Context:
On April 22, 2015, the Asia-Africa Leaders' Meeting was held in Jakarta, Indonesia. Among the leaders of five countries—A, B, C, D, and E—, all pairs except for B & E and D & E had one-on-one meetings. These meetings were arranged to take place over two days, either in the morning or the afternoon sessions, with each leader participating in at most one meeting per session. How many different ways can these one-on-one meetings be arranged?
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Context:
Monroe Vitamin Shop sold equal quantities of Vitamin A, Vitamin B, and Vitamin C supplements yesterday morning. The Vitamin A supplements come in packs of 11, the Vitamin B supplements come in packs of 23, and the Vitamin C supplements come in packs of 37. What is the smallest number of each type of vitamin that the store could have possibly sold?
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Context:
In a regular triangular pyramid, the side of the base is equal to \(a\), and the angle between the slant height (apothem) and a lateral face is \(\frac{\pi}{4}\). Find the height of the pyramid.
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Context:
[ Transfer of side, diagonal, etc.] The Law of Sines

In trapezoid $ABCD (AB \| CD)$, $AB + CD = b$, diagonals $AC$ and $BD$ are related by $5AC = 3BD$. Find the area of the trapezoid if $\angle BAC = 2 \angle DBA$.
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Context:
Consider the function $f(x)$ defined as follows:
- $f(x)$ has domain $\mathbb{R}$.
- $f(x+2)=4f(x)$ for all $x \in \mathbb{R}$.
- $f(x)=x^2-1$ for $x \in [0,2]$.

Find the maximum value of $k$ such that $f(x) \leq 15$ for all $x \in [k, \infty)$.
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Context:
Given two sets with common elements that are not subsets of each other, we say these sets "intersect". For the sets $M=\{x|ax^{2}-1=0,a > 0\}$ and $N=\{- \frac {1}{2}, \frac {1}{2},1\}$, if sets $M$ and $N$ "intersect", then $a=$ \_\_\_\_\_\_.
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Context:
How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n$? Provide your answer as an integer.
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Context:
An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations, the urn contains six balls. Calculate the probability that the urn contains three balls of each color. Express your answer as a fraction in simplest form.
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Context:
The side lengths $a, b, c$ of a triangle $ABC$ are positive integers. Let:

$$T_{n} = (a+b+c)^{2n} - (a-b+c)^{2n} - (a+b-c)^{2n} + (a-b-c)^{2n}$$

for any positive integer $n$. If $\frac{T_{2}}{2T_{1}} = 2023$ and $a > b > c$, determine all possible perimeters of the triangle $ABC$.
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Context:
A $8 \times 8 \times 8$ cube has three of its faces painted red and the other three faces painted blue (ensuring that any three faces sharing a common vertex are not painted the same color), and then it is cut into 512 $1 \times 1 \times 1$ smaller cubes. Among these 512 smaller cubes, how many have both a red face and a blue face?
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Context:
Alexander goes to school at 12 pm. On Tuesday, Alexander's classes are in the following order : Maths, History, Geography, Science and Music. Each of his classes varies in duration: Maths is 45 minutes, History lasts 1 hour and 15 minutes, Geography is 30 minutes long, Science class takes 50 minutes, and there is a 10-minute break between each class. Just now, Alexander's Science class got over. Can you estimate the time now in pm?
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Context:
A square is inscribed in a unit circle, then a circle is inscribed in this square, followed by inscribing another square in this circle. This process is repeated infinitely. Let \( C \) represent the total area of all circles and \( S \) represent the total area of all squares. Calculate the ratio \( \frac{C}{S} \). Express your answer as a simplified mathematical expression.
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Context:
\( S \) is the set of all ordered tuples \((a, b, c, d, e, f)\) where \(a, b, c, d, e, f\) are integers and \(a^2 + b^2 + c^2 + d^2 + e^2 = f^2\). Find the largest \( k \) such that \( k \) divides \( a b c d e f \) for all elements in \( S \).
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Context:
A student was determining the interval $D$ where the root $x_0$ of the equation $x^3 = -x + 1$ lies. They set the function $f(x) = x^3 + x - 1$, calculated $f(-1) < 0$, and $f(1) > 0$. In the process, they used the "bisection method" and chose 3 more values, which are $x_1$, $x_2$, $x_3$, to finally determine the interval $D$. What is the interval $D$?
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Context:
Let $\mathcal{S}$ be a regular 18-gon. For two vertices in $\mathcal{S}$, define the \textit{distance} between them as the length of the shortest path along the edges of $\mathcal{S}$ between them (e.g., adjacent vertices have distance 1). Find the number of ways to choose three distinct vertices from $\mathcal{S}$ such that no two of them have distances 1, 8, or 9.
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Context:
Given an ellipse \( C: \frac{x^{2}}{ a^{2}}+\frac{y^{2}}{ b^{2}}=1 (a > b > 0) \) with foci \( F_{1} \) and \( F_{2} \) on the left and right, respectively, and \( A \) as the right vertex. A line \( l \) passing through \( F_{1} \) intersects the ellipse at points \( P \) and \( Q \), such that \(\overrightarrow{A P} \cdot \overrightarrow{A Q}=\frac{1}{2}(a+c)^{2} \). Find the minimum eccentricity of the ellipse \( C \).
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Context:
In the plane, there are eight distinct points. We will call a numbering of these points from 1 to 8 "good" if the following condition is met:

There exists a line such that all points lie on one side of this line and at different distances from it, with the distances from the points to this line increasing as the numbering increases. That is, the closest point is numbered 1, the next closest point is numbered 2, and so on.

What is the maximum number of different "good" numberings that can be obtained for the given set of eight points?
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Context:
6. Given that a regular tetrahedron can accommodate 10 small spheres with a radius of 1, the minimum edge length of the regular tetrahedron is $\qquad$
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Context:
Let  $a$  and  $b$  be two real numbers and let $M(a,b)=\max\left \{ 3a^{2}+2b; 3b^{2}+2a\right \}$ . Find the values of  $a$  and  $b$  for which  $M(a,b)$  is minimal.
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Context:
Consider \(13\) marbles that are labeled with positive integers such that the product of all \(13\) integers is \(360\). Moor randomly picks up \(5\) marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different products that Moor can obtain?
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Context:
John adopts a dog from a shelter. The dog ends up having health problems and this requires 3 vet appointments. After the first appointment, John paid $100 for pet insurance that covers 80% of the subsequent visits. He paid $660 in total. How much did each vet appointment cost?
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Context:
A moth starts at vertex \(A\) of a certain cube and is trying to get to vertex \(B\), which is opposite \(A\), in five or fewer "steps," where a step consists of traveling along an edge from one vertex to another. The moth will stop as soon as it reaches \(B\). How many ways can the moth achieve its objective?
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Context:
已知凸五边形 $A B C D E$ 的对角线 $C E$ 分别与对角线 $B D, A D$ 交于点 $F, G$, 若 $B F: F D=5: 4, A G: G D=1: 1, C F: F G: G E=2: 2: 3$, 求 $\triangle C D F$ 与 $\triangle A B E$ 面积的比。答案应以 \(\frac{m}{n}\) 的形式给出，请计算 $m + n$ 的值。
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Context:
How many ways are there to choose 2010 functions $f_{1}, \ldots, f_{2010}$ from \{0,1\} to \{0,1\} such that $f_{2010} \circ f_{2009} \circ \cdots \circ f_{1}$ is constant? Note: a function $g$ is constant if $g(a)=g(b)$ for all $a, b$ in the domain of $g$.
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Context:
Determine whether $0$ belongs to or is not in the empty set $\varnothing$. Fill in the blank with $\in$ or $\notin$.
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Context:
What is the real parameter $m$ for which the substitution $y = z^m$ transforms the given differential equation $2x^4yy' + y^4 = 4x^6$ into a first-order homogeneous differential equation?
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Context:
After 100 new people move into a town, 400 of the original population move out. Every year afterward, the town’s population is halved. After 4 years, the population is 60 people. Before the new people moved in, how many people lived in the town?
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Context:
Working alone, Ryan can dig a 10 ft by 10 ft hole in a certain amount of time. Castel can dig the same hole in six hours. It would take them 2.72727272727 hours if they worked together. How many hours does it take for Ryan to dig the hole alone?
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Context:
The quadrilateral formed by the line $l_1$: $x+3y-7=0$, the line $l_2$: $kx+y-2=0$, and the positive semi-axes of the x and y axes has a circumscribed circle. Find the value of $k$.
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Context:
In how many ways can $10$ distinct books be placed onto a $3$-shelf bookcase such that no shelf is empty? Express your answer in the form \( n \cdot 10! \), where \( n \) is an integer.
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Context:
A regular pentagon \(A B C D E\) is given. On side \(A E\), there is a point \(K\), and on side \(C D\), there is a point \(L\). It is known that \(\angle L A E + \angle K C D = 108^\circ\) and \(A K : K E = 3:7\). Find the ratio \(C L : A B\).

A regular pentagon is a pentagon in which all sides are equal and all angles are equal.
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Context:
In a joint-stock company, there are 2017 shareholders, and any group of 1500 of them has a controlling stake (at least 50% of the shares). What is the maximum share of the company that one shareholder can own?
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Context:
In order to complete a large job, $1000$ workers were hired, just enough to complete the job on schedule. All the workers stayed on the job while the first quarter of the work was done, so the first quarter of the work was completed on schedule. Then $100$ workers were laid off, so the second quarter of the work was completed behind schedule. Then an additional $100$ workers were laid off, so the third quarter of the work was completed still further behind schedule. Given that all workers work at the same rate, what is the minimum number of additional workers, beyond the $800$ workers still on the job at the end of the third quarter, that must be hired after three-quarters of the work has been completed so that the entire project can be completed on schedule or before?
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Context:
A gardener wants to plant trees in his garden in such a way that the number of trees in each row should be the same. The garden has a rectangular shape and 4 distinct tree types (A, B, C, and D) which must be planted in alternating patterns (ABCDABCD... or DCBADCBA...). The gardener also wants the total number of trees planted to be at least 100. If there are 7 rows or 6 rows or 4 rows, then no tree will be left. Find the least number of trees required that satisfy all these conditions.
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Context:
We mark point $O_1$ on a unit circle and, using $O_1$ as the center, we mark (by means of a compass) we mark point $O_2$ on the circle (clockwise starting with $O_1$). Using point $O_2$ as a new center, we repeat the procedure in the same direction with the same radius; and so on. After we had marked point $O_{1968}$ a circle is cut through each mark so we get 1968 arcs. How many different arc lengths can we thus procure?
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Context:
Determine the values of \( \alpha \) for which the integral \( \int_0^1 \frac{1-x^\alpha}{1-x} \, \mathrm{d}x \) converges.
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Context:
Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with left and right foci $F_{1}$ and $F_{2}$ respectively, draw a line $l$ through the right focus that intersects the ellipse at points $P$ and $Q$. Find the maximum area of the inscribed circle of triangle $F_{1} PQ$.
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Context:
Find the least positive integer $n$ such that when $3^n$ is written in base $143^2$ , its two right-most digits in base $143$ are $01$ .
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Context:
A bag of potatoes weighs some amount divided by half of its weight. The bag of potatoes weighs 36 lbs. What is the weight of the bag of potatoes before dividing by half of its weight?
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Context:
Define $f(x) = x^2 - 45x + 21$. Find the sum of all positive integers $n$ with the following property: there is exactly one integer $i$ in the set $\{1, 2, \ldots, n\}$ such that $n$ divides $f(i)$.
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Context:
At a certain university, 2/3 of the students are females and 3/5 of the total student population is composed of full-time students. The remaining students are part-time. One-tenth of the male students are foreign students. If the university has 3600 students, how many students are non-foreign male part-time students?
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Context:
Point $D$ lies on side $AC$ of isosceles triangle $ABC$ (where $AB = AC$) such that angle $BAC$ is $100^\circ$ and angle $DBC$ is $30^\circ$. Determine the ratio of the area of triangle $ADB$ to the area of triangle $CDB$.
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Context:
Problem 7.7. The numbers from 1 to 200 were arranged in a random order on a circle such that the distances between adjacent numbers on the circle are the same.

For any number, the following is true: if you consider 99 numbers standing clockwise from it and 99 numbers standing counterclockwise from it, there will be an equal number of numbers that are smaller than it in both groups. Which number stands opposite the number 113?
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Context:
Given a positive integer \( n \geq 2 \) and non-negative real numbers \( a_{i} \) (\(1 \leq i \leq n\)) satisfying \( \sum_{i=1}^{n} a_{i}=1 \), find the maximum value of \( \sum_{1 \leq i<j \leq n}(j - i) a_{i} a_{j} \).
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Context:
Given an integrated-circuit production line with a long-term yield of 80% (defective rate of 20%) and a standard deviation of 0.04 in the proportion defective from day to day, assume a Beta prior distribution for the proportion defective ($\theta$). On a specific day, a sample of 100 units is inspected, and 38 defective units are found. Calculate the mean of the posterior distribution of $\theta$ for this day.
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Context:
A 3 × 3 grid of blocks is labeled from 1 through 9. Cindy paints each block orange or
lime with equal probability and gives the grid to her friend Sophia.
Sophia then plays with the grid of blocks. She can take the top row of blocks and move
it to the bottom, as shown.
1 2 3
4 5 6
7 8 9
4 5 6
7 8 9
1 2 3
Grid A Grid A0
She can also take the leftmost column of blocks and move it to the right end, as shown.
1 2 3
4 5 6
7 8 9
2 3 1
5 6 4
8 9 7
Grid B Grid B0
Sophia calls the grid of blocks citrus if it is impossible for her to use a sequence of the
moves described above to obtain another grid with the same coloring but a different numbering
scheme. For example, Grid B is citrus, but Grid A is not citrus because moving the
top row of blocks to the bottom results in a grid with a different numbering but the same
coloring as Grid A.
What is the probability that Sophia receives a citrus grid of blocks?
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Context:
Determine the values of \( p \) for which the integral \( \int_1^2 \frac{\ln(x-1)}{(4-x^2)^p} \, \mathrm{d}x \) converges.
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Context:
In the cube \( A C_{1} \), determine the magnitude of the dihedral angle \( C_{1}-D_{1} B-C \).
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Context:
Let $a, b \in \mathbb{R}_+$ with $a+b-ab=0$. If the value of $\ln \left( \frac{m^2}{a+b} \right)$ is always non-positive, then the range of $m$ is _______.
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Context:
Find the quadratic subfield of $\mathbb{Q}(\zeta_7)$ that can be expressed in the form $\mathbb{Q}(\sqrt{D})$, where $D$ is an integer.
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Context:
Each face of a cube is painted either  red or blue, each with probability 1/2. The color of each face is determined independently.  What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?
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Context:
Find the integer $n$, $-180 \le n \le 180,$ such that $\sin n^\circ = \cos 682^\circ.$
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Context:
If the order of the letters in the English word "good" is written incorrectly, there are a total of $\_\_\_\_\_$ possible errors.
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Context:
The sum of certain positive integers (not necessarily distinct) less than or equal to 10 is $S$. Find all values of $S$ such that, regardless of these integers, they can always be partitioned into two groups, each with a sum less than or equal to 70.
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Context:
Given the sequence \(\left(a_{n}\right)_{n=1}^{\infty}\) with \(a_{1}=a_{2}=2\), and the recurrence relation
\[ \frac{2 a_{n-1} a_{n}}{a_{n-1} a_{n+1}-a_{n}^{2}}=n^{3}-n \]
for all \(n=2,3,4, \ldots\), find the integer that is closest to the value of \(\sum_{k=2}^{2011} \frac{a_{k+1}}{a_{k}}\).
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Context:
On the sides \( AB \) and \( AC \) of an equilateral triangle \( ABC \) with side length 10, points \( P \) and \( Q \) are taken such that the segment \( PQ \) touches the incircle of the triangle and has a length of 4. Find the area of triangle \( APQ \).
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Context:
Divide the set \( M = \{1, 2, \ldots, 12\} \) of the first 12 positive integers into four subsets each containing three elements, such that in each subset, one number is the sum of the other two. Find the number of different ways to do this.
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Context:
Tony read 23 books, Dean read 12 books and Breanna read 17 books. Tony and Dean read 3 of the same books and all three had read the same book as well. In total how many different books have Tony, Dean, and Breanna read?
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Context:
The Fahrenheit temperature ( $F$ ) is related to the Celsius temperature ( $C$ ) by  $F = \tfrac{9}{5} \cdot C + 32$ . What is the temperature in Fahrenheit degrees that is one-ﬁfth as large if measured in Celsius degrees?
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Context:
Given that $m$ and $n$ are both positive numbers, the curve $\frac {x^{2}}{m} + \frac {y^{2}}{n}=1$ passes through the fixed point $A(1, \sqrt {2})$. Find the minimum value of $m+n$.
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Context:
The product of two two-digit numbers is 1656. Calculate the sum of these two numbers. Express your answer as a whole number.
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Context:
A list of $11$ positive integers has a mean of $10$, a median of $9$, and a unique mode of $8$. What is the largest possible value of an integer in the list?
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Context:
How many factors are there in the product $1 \cdot 2 \cdot 3 \cdot \ldots \cdot n$ if we know that it ends with 1981 zeros?
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Context:
设正三棱锥 $P-A B C$ 的底面边长为 1 、高为 $\sqrt{2}$ ，过底边 $B C$ 作此三棱锥的截面.则截面面积的最小值为 $\frac{m \sqrt{n}}{k}$ 的形式，请给出$m + n + k$ 的值。
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Context:
A bag of potatoes weighs 12 lbs divided by half of its weight. How much does the bag of potatoes weigh?
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Context:
The hyperbola $C$ has its center at the origin and its foci on the $x$-axis. $C$ intersects the directrix of the parabola $y^{2}=4x$ at points $A$ and $B$, with $AB= \sqrt {3}$. Determine the length of the real axis of $C$.
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Context:
### 1.26. Find the center of gravity of a hemisphere

$$
0 \leqslant z \leqslant \sqrt{R^{2}-x^{2}-y^{2}},
$$

where the volume density \( p \) of the material at each point is proportional to the distance from the origin.
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Context:
Evaluate the integral $$\int _{ c } \frac { \sin(iz) }{ { z }^{ 3 }({ z }^{ 2 }+2i) } dz$$ using residue calculus.
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Context:
In the convex quadrilateral $ABCD$, the sum of $AB+BD+DC$ is at most 2, and the area of the quadrilateral is $1/2$. What can be the length of diagonal $AC$?
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Context:
Given the parabola $y^{2}=2px (p\neq 0)$ and the fixed points $A(a,b)$, $B(-a,0)$, where $ab\neq 0$ and $b^{2}\neq 2pa$. Let $M$ be a point on the parabola, and let $M_{1}$ and $M_{2}$ be the other intersection points of lines $AM$ and $BM$ with the parabola, respectively. As $M$ varies, the line $M_{1}M_{2}$ passes through a fixed point. Find the coordinates of this fixed point.
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Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $b\cos B$ is the arithmetic mean of $a\cos C$ and $c\cos A$, then the angle $B=$ ______.
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Context:
Given the function:
$$
\begin{aligned}
y= & |x-1|+|2x-1|+|3x-1|+|4x-1|+|5x-1|
\end{aligned}
$$
Find the value of x that minimizes the function y. Express your answer as a single real number.
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Context:
If 20 men can build a water fountain 56 metres long in 7 days, and they also need to construct a surrounding stone wall that takes an additional 3 days to build, what total length of a similar water fountain with its surrounding wall can be built by 35 men in 9 days, considering that they will be constructing the wall using a more time-efficient material that reduces wall construction time by 50%?
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Context:
Timmy plans to ride a skateboard ramp that is 50 feet high, with a 30-degree incline. He knows he needs to go 40 mph at the start to make it all the way to the top. He measures his speed on three trial runs and goes 36, 34, and 38 mph. The wind resistance varies between 3 mph and 5 mph during each trial run. Considering the impact of varying wind resistance on the ramp with different inclines, how much faster does he have to go than his average speed, taking into account the additional factors, to make it up the ramp?
Please reason step by step, and put your final answer within \boxed{}.