Context:
For a positive integer  $n$  that is not a power of two, we define  $t(n)$  as the greatest odd divisor of  $n$  and  $r(n)$  as the smallest positive odd divisor of  $n$  unequal to  $1$ . Determine all positive integers  $n$  that are not a power of two and for which we have  $n = 3t(n) + 5r(n)$ .
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Context:
设k为给定的正整数，求最小正整数n，使得任意n个整数中总存在两个正整数，它们的和或差被2k整除．
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Context:
The numbers on opposite sides of a die total 7. Two dice are shown with some faces visible. Using the properties of dice, find the sum of the numbers on the unseen faces of the two dice. Express your answer as a numerical value.
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Context:
In how many different ways can the name "Bolyai János" be read from the following diagram? The reading can only occur continuously, meaning starting from the letter B, moving one letter at a time diagonally to the right or left, until reaching the letter S.
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Context:
有20个队参加全国足球冠军决赛，为了使任何三个队中都有两个队相互比赛过，问最少要进行多少场比赛？
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Context:
In the diagram, each scale shows the total mass (weight) of the shapes on that scale. Using the information provided by the scales, calculate the mass (weight) of a $\triangle$. Express your answer as a single numerical value.
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Context:
The segment \( PQ \) is divided into several smaller segments. On each of them, a square is constructed (see illustration).

What is the length of the path along the arrows if the length of the segment \( PQ \) is 73? If necessary, round your answer to 0.01 or write the answer as a common fraction.
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Context:
A triangle is composed of circular cells arranged in  $5784$  rows: the first row has one cell, the second has two cells, and so on (see the picture). The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called **diagonal** if the two cells in it *aren't* in the same row. What is the minimum possible amount of diagonal pairs in the division?
An example division into pairs is depicted in the image.
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Context:
Find three numbers in an arithmetic progression with a common difference of 2, such that the sum of their squares is a number consisting of four identical digits.
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Context:
What is the minimum number of squares into which a staircase of 15 steps can be cut (see the figure)? Cutting can only be done along the boundaries of the cells.
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Context:
Let $A,B,C,D$ denote four points in space such that at most one of the distances $AB,AC,AD,BC,BD,CD$ is greater than $1$ . Determine the maximum value of the sum of the six distances.
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Context:
The quotient of the division of a three-digit number by the sum of its digits is 13, and the remainder is 15. Find all such three-digit numbers.
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Context:
Does the following integral converge or diverge?
$$\int_{0}^{\infty}\frac{e^{-at^2}}{\sqrt{t}}\prod_{i=1}^n\left|\frac{1}{\sqrt{t}}-r_i\right|^2 dt$$
Here $r_i\in\mathbb{R}$ and $a>0$.
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Context:
For any finite set $X$, let $| X |$ denote the number of elements in $X$. Define
\[S_n = \sum | A \cap B | ,\]
where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\left\{ 1 , 2 , 3,  \cdots , n \right\}$ with $|A| = |B|$.
For example, $S_2 = 4$ because the sum is taken over the pairs of subsets
\[(A, B) \in \left\{ (\emptyset, \emptyset) , ( \{1\} , \{1\} ), ( \{1\} , \{2\} ) , ( \{2\} , \{1\} ) , ( \{2\} , \{2\} ) , ( \{1 , 2\} , \{1 , 2\} ) \right\} ,\]
giving $S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4$.
Let $\frac{S_{2022}}{S_{2021}} = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the remainder when $p + q$ is divided by
1000.
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Context:
The product
$$\left(\frac{1+1}{1^2+1}+\frac{1}{4}\right)\left(\frac{2+1}{2^2+1}+\frac{1}{4}\right)\left(\frac{3+1}{3^2+1}+\frac{1}{4}\right)\cdots\left(\frac{2022+1}{2022^2+1}+\frac{1}{4}\right)$$
can be written as $\frac{q}{2^r\cdot s}$, where $r$ is a positive integer, and $q$ and $s$ are relatively prime odd positive integers. Find $s$.
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Context:
What is the molecular weight of 9 moles of a certain compound if the molecular weight is 972?
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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:
In the star-shaped figure below, if all side lengths are equal to $3$ and the three largest angles of the figure are $210$ degrees, its area can be expressed as $\frac{a \sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is square-free. Compute $a + b + c$.
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Context:
In three out of six circles in a diagram, the numbers 4, 14, and 6 are written. How many ways can natural numbers be placed in the remaining three circles so that the products of the triplets of numbers along each of the three sides of the triangular diagram are the same?
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Context:
In the game of Guess the Card, two players each have a  $\frac{1}{2}$  chance of winning and there is exactly one winner. Sixteen competitors stand in a circle, numbered  $1,2,\dots,16$  clockwise. They participate in an  $4$ -round single-elimination tournament of Guess the Card. Each round, the referee randomly chooses one of the remaining players, and the players pair off going clockwise, starting from the chosen one; each pair then plays Guess the Card and the losers leave the circle. If the probability that players  $1$  and  $9$  face each other in the last round is  $\frac{m}{n}$  where  $m,n$  are positive integers, find  $100m+n$ .
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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:
The product of the proper positive integer factors of $n$ can be written as $n^{(ax+b)/c}$, where $x$ is the number of positive divisors $n$ has, $c$ is a positive integer, and the greatest common factor of the three integers $a$, $b$, and $c$ is $1$. What is $a+b+c$?
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Context:
In the adjoining figure, \( \triangle CDE \) is an equilateral triangle and \( \square ABCD \) and \( \square DEFG \) are squares. If the measure of \( \angle GDA \) is \( x^\circ \), what is the value of \( x \)?
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Context:
Suppose that  $a$  and  $b$  are natural numbers such that \[p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}\] is a prime number. What is the maximum possible value of  $p$ ?
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Context:
设 $a, b>0$ ，满足：关于 $x$ 的方程 $\sqrt{|x|}+\sqrt{|x+a|}=b$ 恰有三个不同的实数解 $x_{1}, x_{2}, x_{3}$, 且 $x_{1}<x_{2}<x_{3}=b$, 则 $a+b$ 的值为 $\qquad$.
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Context:
Compute the sum of the series:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))))) \]
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Context:
Liquid $X$ does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film $0.1$cm thick. A rectangular box measuring $6$cm by $3$cm by $12$cm is filled with liquid $X$. Its contents are poured onto a large body of water. What will be the radius, in centimeters, of the resulting circular film?
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Context:
Determine the expected value ($\mathbb{E}[X]$) and variance of the number of times it is necessary to roll a dice until the result "1" occurs 4 times in a row.
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Context:
In the diagram, $\triangle ABC$ is isosceles with $AB = AC$ and $BC = 30 \mathrm{~cm}$. Square $EFGH$, which has a side length of $12 \mathrm{~cm}$, is inscribed in $\triangle ABC$, as shown. The area of $\triangle AEF$, in $\mathrm{cm}^{2}$, is
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Context:
Wenlu, Xander, Yasser, and Zoe make the following statements:
Wenlu: "Xander is lying."
Xander: "Yasser is lying."
Yasser: "Zoe is telling the truth."
Zoe: "Wenlu is telling the truth."
Determine the possible numbers of people telling the truth. Express your answer as a list or range of possible numbers.
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Context:
In the diagram, \(\angle AFC = 90^\circ\), \(D\) is on \(AC\), \(\angle EDC = 90^\circ\), \(CF = 21\), \(AF = 20\), and \(ED = 6\). Determine the total area of quadrilateral \(AFCE\).
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Context:
For each even positive integer $x$, let $g(x)$ denote the greatest power of 2 that divides $x.$ For example, $g(20)=4$ and $g(16)=16.$ For each positive integer $n,$ let $S_n=\sum_{k=1}^{2^{n-1}}g(2k).$ Find the greatest integer $n$ less than 1000 such that $S_n$ is a perfect square.
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Context:
A single burger is not enough to satisfy a guy's hunger. The five guys go to Five Guys' Restaurant, which has 20 different meals on the menu. Each meal costs a different integer dollar amount between $1 and $20. The five guys have \$20 to split between them, and they want to use all the money to order five different meals. How many sets of five meals can the guys choose?
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Context:
The product underwent a price reduction from 25 yuan to 16 yuan. Calculate the average percentage reduction for each price reduction.
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Context:
Given a parabola $y=ax^2+bx+c$ that opens downward and has its vertex at $(2, -3)$, determine the type of value (maximum or minimum) this point represents for the parabola and its numerical value. Express your answer in the format: 'The parabola has a [maximum/minimum] value of [numerical value]'.
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\boxed{The parabola has a maximum value of -3.}


Context:
Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 1, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane that contains this edge.
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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:
Let $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB$, respectively, with $FA=5$ and $CD=2$.  Point $E$ lies on side $CA$ such that $\angle DEF = 60^\circ$.  The area of triangle $DEF$ is $14\sqrt{3}$.  The two possible values of the length of side $AB$ are $p \pm q\sqrt{r}$, where $p$ and $q$ are rational, and $r$ is an integer not divisible by the square of a prime.  Find $r$.
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Context:
Point $P$ is located inside a square $ABCD$ of side length $10$. Let $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of $P AB$, $P BC$, $P CD$, and $P DA$, respectively. Given that $P A+P B +P C +P D = 23\sqrt2$ and the area of $O_1O_2O_3O_4$ is $50$, the second largest of the lengths $O_1O_2$, $O_2O_3$, $O_3O_4$, $O_4O_1$ can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.
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Context:
As shown in Figure 1, a cross-section of cube \(ABCDEFGH\) passes through vertices \(A\), \(C\), and a point \(K\) on edge \(EF\), and divides the cube into two parts with a volume ratio of 3:1. Find the value of \(\frac{EK}{KF}\).
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Context:
有9本书，其中4本是相同的数学书，3本是相同的语文书，2本是相同的外语书，1本是词典，从中取出4本有多少种不同的取法
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Context:
Wendy randomly chooses a positive integer less than or equal to $2020$. The probability that the digits in Wendy's number add up to $10$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
The diagram shows a circle that has been divided into six sectors of different sizes. Two of the sectors are to be painted red, two of them are to be painted blue, and two of them are to be painted yellow. Any two sectors which share an edge are to be painted in different colors. In how many ways can the circle be painted?
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Context:
Circle $B$ has radius $6\sqrt{7}$.  Circle $A$, centered at point $C$, has radius $\sqrt{7}$ and is contained in $B$.  Let $L$ be the locus of centers $C$ such that there exists a point $D$ on the boundary of $B$ with the following property: if the tangents from $D$ to circle $A$ intersect circle $B$ again at $X$ and $Y$, then $XY$ is also tangent to $A$.  Find the area contained by the boundary of $L$.
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Context:
Let $F_{1}$ and $F_{2}$ be the left and right foci of the ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$. A line passing through point $F_{1}$ intersects the ellipse $E$ at points $A$ and $B$ such that $\left|A F_{1}\right|=3\left|B F_{1}\right|$ and $\cos \angle A F_{2} B=\frac{3}{5}$, as shown in the figure. The line $y=\frac{1}{2} x$ intersects the ellipse $E$ at points $P$ and $Q$. Points $C$ and $D$ are other points on the ellipse $E$ distinct from $P$ and $Q$, and lines $P C$ and $Q D$ intersect at point $M$, and lines $P D$ and $Q C$ intersect at point $N$. Find the slope of the line $M N$.
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Context:
We define the function  $f(x,y)=x^3+(y-4)x^2+(y^2-4y+4)x+(y^3-4y^2+4y)$ . Then choose any distinct  $a, b, c \in \mathbb{R}$  such that the following holds:  $f(a,b)=f(b,c)=f(c,a)$ . Over all such choices of  $a, b, c$ , what is the maximum value achieved by
\[\min(a^4 - 4a^3 + 4a^2, b^4 - 4b^3 + 4b^2, c^4 - 4c^3 + 4c^2)?\]
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Context:
For integers $n \geq 3$, let $n! = \prod_{i=1}^{k} p_{i}^{\alpha_{i}}$, where $p_{1}, p_{2}, \cdots, p_{k}$ are prime numbers, and $p_{1}<p_{2}<\cdots<p_{k}$, and $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}$ are positive integers. Find all $n \geq 3$ such that $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}$ form a geometric progression.
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Context:
For all positive integers  $m>10^{2022}$ , determine the maximum number of real solutions  $x>0$  of the equation  $mx=\lfloor x^{11/10}\rfloor$ .
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Context:
Define a sequence by $x_0=3$ and $$ x_{n+1}=\frac{x_{n}^{2}+3 x_{n}+2}{x_{n}+4}. $$ Find the interval in which the least positive integer $m$ lies such that $$ x_{m} \leq 2+\frac{1}{2^{15}}, $$ given the following interval options: $$ [5,14],[15,44],[45,134],[135,404],[405,+\infty]. $$
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Context:
Three brothers returned from fishing. Their mother asked each of them how many fish they caught together. Vasya said, "More than ten," Petya said, "More than eighteen," and Kolya said, "More than fifteen." How many fish could have been caught (list all possibilities), knowing that two brothers told the truth and one lied?
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Context:
Three people want to buy a house for 26,000 livres. They agreed that the first will give half, the second - one third, and the third - one quarter. How much will each give?
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Context:
Three semicircles of radius \(1\) are constructed on diameter \(\overline{AB}\) of a semicircle of radius \(2\). The centers of the small semicircles divide \(\overline{AB}\) into four line segments of equal length, as shown. Find the area of the shaded region that lies within the large semicircle but outside the smaller semicircles. The original answer is in the form \(\frac{k}{m}\pi - \frac{\sqrt{n}}{p}\). Please calculate the value of \(k + m + n + p\).
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Context:
Find the minimum value of
$$
\left|\sin x+\cos x+\frac{\cos x-\sin x}{\cos 2 x}\right|
$$
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Context:
Find the largest positive integer $n$ such that $n\varphi(n)$ is a perfect square.  ($\varphi(n)$ is the number of integers $k$, $1 \leq k \leq n$ that are relatively prime to $n$)
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Context:
Methanol can be gently oxidized with hot copper metal. What is the product of this oxidation reaction? Express your answer as a chemical compound using standard chemical notation.
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Context:
Find all real solutions to $x^4+(2-x)^4=34$.  Enter all the solutions, separated by commas.
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Context:
设有n个X队和n个Y队进行循环比赛，其中每个X队都恰好与k个Y队进行比赛；每个Y队恰好与k个X队进行比赛.问：一共要安排多少场比赛？
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Context:
Find all integer solutions \( x \) and \( y \) to the equation \( x^{2} = 1 + 4y^{3}(y + 2) \).
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Context:
Four numbers from the set of six numbers $\{0, 1, 2, 3, 4, 5\}$ are placed in the circles of Figure 1. The absolute difference between the numbers at the endpoints of each line segment yields 5 differences, which must be exactly $\{1, 2, 3, 4, 5\}$. Figure 2 shows one such arrangement. How many such arrangements are possible? (In the given figures, swapping 3 and 5 or 0 and 1 counts as a new arrangement.)
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Context:
设圆 $O: x^{2}+y^{2}=5$ 与抛物线 $C: y^{2}=2 p x(p>0)$ 交于点 $A\left(x_{0}, 2\right), A B$ 为圆 $O$ 的直径，过 $B$ 的直线与 $C$ 交于两不同点 $D, E$ ，则直线 $A D$ 与 $A E$ 的斜率之积为 ?
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Context:
Define a function  $g :\mathbb{N} \rightarrow \mathbb{R}$ Such that  $g(x)=\sqrt{4^x+\sqrt {4^{x+1}+\sqrt{4^{x+2}+...}}}$ .
Find the last 2 digits in the decimal representation of  $g(2021)$ .
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Context:
At Easter-Egg Academy, each student has two eyes, each of which can be eggshell, cream, or cornsilk. It is known that $30 \%$ of the students have at least one eggshell eye, $40 \%$ of the students have at least one cream eye, and $50 \%$ of the students have at least one cornsilk eye. What percentage of the students at Easter-Egg Academy have two eyes of the same color?
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Context:
In a right triangle \(ABC\), the height \(BH\) is drawn to the hypotenuse \(AC\). Points \(X\) and \(Y\) are the centers of the circles inscribed in triangles \(ABH\) and \(CBH\) respectively. The line \(XY\) intersects the legs \(AB\) and \(BC\) at points \(P\) and \(Q\). Find the area of triangle \(BPQ\), given that \(BH = h\).
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Context:
the diagonal of a square is 4 ? 2 cm . what will be the diameter of another square if their area is double that of the first square ?
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Context:
Determine all possible values of  $m+n$ , where  $m$  and  $n$  are positive integers satisfying \[\operatorname{lcm}(m,n) - \gcd(m,n) = 103.\]
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Context:
Let  $S$  be the set of all positive integers from 1 through 1000 that are not perfect squares.  What is the length of the longest, non-constant, arithmetic sequence that consists of elements of  $S$ ?
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Context:
Let the integer  $n \ge 2$  , and  $x_1,x_2,\cdots,x_n $  be real numbers such that  $\sum_{k=1}^nx_k$  be integer  .  $d_k=\underset{m\in {Z}}{\min}\left|x_k-m\right| $ ,  $1\leq k\leq n$  .Find the maximum  value of  $\sum_{k=1}^nd_k$ .
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Context:
[ [ correct prism $]$

The side of the base $A B C$ of a regular prism $A B C A 1 B 1 C 1$ is 1, and each of the lateral edges has a length of $\frac{1}{\sqrt{3}}$. A right circular cylinder is positioned such that point $A 1$ and the midpoint $M$ of edge $C C 1$ lie on its lateral surface, and the axis of the cylinder is parallel to line $A B 1$ and is at a distance of $\frac{1}{4}$ from it. Find the radius of the cylinder.

#
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Context:
What is the smallest possible perimeter of a triangle with integer coordinate vertices, area  $\frac12$ , and no side parallel to an axis?
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Context:
In the diagram, pentagon \( A B C D E \) is symmetrical about altitude \( C F \). Also, \( A E=200 \), \( C F=80 \sqrt{3} \), \( \angle A B C=150^\circ \), and \( \angle B C D=120^\circ \). Determine the vertical distance between \( A E \) and \( B D \).
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Context:
Let the universal set U = {3, 4, 5}, and M = {|a-3|, 3}. If the complement of M in U is {5}, then find all possible values of a. Express your answer as a list of integers.
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Context:
Find the number of elements of orders $p$ and $q$ in a nonabelian group $G$ of order $pq$, where $p$ and $q$ are distinct primes with $p<q$. Assume that $G$ has a unique subgroup of order $q$. How many elements of order $p$ and $q$ are there?
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Context:
Consider a square of side length  $1$  and erect equilateral triangles of side length  $1$  on all four sides of the square such that one triangle lies inside the square and the remaining three lie outside. Going clockwise around the square, let  $A$ ,  $B$ ,  $C$ ,  $D$  be the circumcenters of the four equilateral triangles. Compute the area of  $ABCD$ .
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Context:
Define an ordered triple $(A, B, C)$ of sets to be minimally intersecting if $|A \cap B| = |B \cap C| = |C \cap A| = 1$ and $A \cap B \cap C = \emptyset$. For example, $(\{1,2\},\{2,3\},\{1,3,4\})$ is a minimally intersecting triple. Let $N$ be the number of minimally intersecting ordered triples of sets for which each set is a subset of $\{1,2,3,4,5,6,7\}$. Find the remainder when $N$ is divided by $1000$.
Note: $|S|$ represents the number of elements in the set $S$.
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Context:
Let $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10$. Let $N$ be the smallest positive integer such that $S(N) = 2013$. What is the value of $S(5N + 2013)$?
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Context:
An ellipse has  foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is  tangent to the $x$-axis. What is the length of its major axis?
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Context:
In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$
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Context:
In the diagram, \(PR, PS, QS, QT,\) and \(RT\) are straight line segments. \(QT\) intersects \(PR\) and \(PS\) at \(U\) and \(V\), respectively. If \(PU = PV\), \(\angle UPV = 24^\circ\), \(\angle PSQ = x^\circ\), and \(\angle TQS = y^\circ\), find the value of \(x + y\). Express your answer in degrees as a whole number.
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Context:
Find the mass percentage of Al in Aluminum chloride
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Context:
For a positive real number  $a$ , let  $C$  be the cube with vertices at  $(\pm a, \pm a, \pm a)$  and let  $T$  be the tetrahedron with vertices at  $(2a,2a,2a),(2a, -2a, -2a),(-2a, 2a, -2a),(-2a, -2a, -2a)$ . If the intersection of  $T$  and  $C$  has volume  $ka^3$  for some  $k$ , find  $k$ .
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Context:
Let $a_1 < a_2 < \cdots < a_k$ denote the sequence of all positive integers between $1$ and $91$ which are relatively prime to $91$, and set $\omega = e^{2\pi i/91}$. Define \[S = \prod_{1\leq q < p\leq k}\left(\omega^{a_p} - \omega^{a_q}\right).\] Given that $S$ is a positive integer, compute the number of positive divisors of $S$.
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Context:
A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form $a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},$ where $a^{}_{}$, $b^{}_{}$, $c^{}_{}$, and $d^{}_{}$ are positive integers. Find $a + b + c + d^{}_{}$.
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Context:
A circle of radius \(2\) is cut into four congruent arcs. The four arcs are joined to form the star figure shown. Find the ratio of the area of the star figure to the area of the original circle. The original answer is in the form \(\frac{k-\pi}{m\pi}\). Please provide the value of \(k + m\).
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Context:
As shown in the diagram, the area of parallelogram \(ABCD\) is 60. The ratio of the areas of \(\triangle ADE\) and \(\triangle AEB\) is 2:3. Find the area of \(\triangle BEF\).
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Context:
The diagram shows two congruent equilateral triangles whose overlap is a hexagon. The areas of the smaller triangles, which are also equilateral, are 1, 1, 9, 9, 16, and 16, as shown. Calculate the area of the inner hexagon. Express your answer as a single numerical value.
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Context:
Find the number of ways to partition a set of  $10$  elements,  $S = \{1, 2, 3, . . . , 10\}$  into two parts; that is, the number of unordered pairs  $\{P, Q\}$  such that  $P \cup Q = S$  and  $P \cap Q = \emptyset$ .
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Context:
A black bishop and a white king are placed randomly on a $2000 \times 2000$ chessboard (in distinct squares). Let $p$ be the probability that the bishop attacks the king (that is, the bishop and king lie on some common diagonal of the board). Then $p$ can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m$.
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Context:
A team consisting of Petya, Vasya, and a single-seat scooter is participating in a race. The distance is divided into 42 equal-length segments, with a checkpoint at the beginning of each segment. Petya completes a segment in 9 minutes, Vasya in 11 minutes, and either of them can cover a segment on the scooter in 3 minutes. They all start simultaneously, and the finishing time is determined by whoever finishes last. The boys agree that one will cover the first part of the distance on the scooter and the rest by running, while the other does the opposite (the scooter can be left at any checkpoint). How many segments should Petya cover on the scooter for the team to show the best possible time?
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Context:
Suppose $r^{}_{}$ is a real number for which
$\left\lfloor r + \frac{19}{100} \right\rfloor + \left\lfloor r + \frac{20}{100} \right\rfloor + \left\lfloor r + \frac{21}{100} \right\rfloor + \cdots + \left\lfloor r + \frac{91}{100} \right\rfloor = 546.$
 Find $\lfloor 100r \rfloor$. (For real $x^{}_{}$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x^{}_{}$.)
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Context:
For which \( a \) and \( b \) is the polynomial \( P(x) = (a + b) x^5 + ab x^2 + 1 \) divisible by \( x^2 - 3x + 2 \)?
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Context:
Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How many schools are in the city? Fill in the blank with the number of schools.
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Context:
A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get  $-1$  mark. He will get 0 mark for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.)
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Context:
An ice ballerina rotates at a constant angular velocity at one particular point. That is, she does not translationally move. Her arms are fully extended as she rotates. Her moment of inertia is $I$. Now, she pulls her arms in and her moment of inertia is now $\frac{7}{10}I$. Calculate the ratio of the new kinetic energy (arms in) to the initial kinetic energy (arms out). Express your answer as a simplified fraction.
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Context:
A dance with 2018 couples takes place in Havana. For the dance, 2018 distinct points labeled  $0, 1,\ldots, 2017$  are marked in a circumference and each couple is placed on a different point. For  $i\geq1$ , let  $s_i=i\ (\textrm{mod}\ 2018)$  and  $r_i=2i\ (\textrm{mod}\ 2018)$ . The dance begins at minute  $0$ . On the  $i$ -th minute, the couple at point  $s_i$  (if there's any) moves to point  $r_i$ , the couple on point  $r_i$  (if there's any) drops out, and the dance continues with the remaining couples. The dance ends after  $2018^2$  minutes. Determine how many couples remain at the end.

Note: If  $r_i=s_i$ , the couple on  $s_i$  stays there and does not drop out.
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Context:
Let  $r$  be a real number such that  $\sqrt[3]{r} - \frac{1}{\sqrt[3]{r}}=2$ . Find  $r^3 - \frac{1}{r^3}$ .
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Context:
In a circle,  $15$  equally spaced points are drawn and arbitrary triangles are formed connecting  $3$  of these points. How many non-congruent triangles can be drawn?
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Context:
Suppose four solid iron balls are placed in a cylinder with the radius of 1 cm, such that every two of the four balls are tangent to each other, and the two balls in the lower layer are tangent to the cylinder base. Now put water into the cylinder. Find, in  $\text{cm}^2$ , the volume of water needed to submerge all the balls.
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Context:
What is the mass percentage of Br in Barium bromide?
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Context:
You are racing an Artificially Intelligent Robot, called Al, that you built. You can run at a constant speed of  $10$  m/s throughout the race. Meanwhile, Al starts running at a constant speed of  $ 1$  m/s. Thereafter, when exactly  $ 1$  second has passed from when Al last changed its speed, Al’s speed instantaneously becomes  $1$  m/s faster, so that Al runs at a constant speed of  $k$  m/s in the kth second of the race. (Start counting seconds at  $ 1$ ). Suppose Al beats you by exactly  $1$  second. How many meters was the race?
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Context:
In equality  $$ 1 * 2 * 3 * 4 * 5 * ... * 60 * 61 * 62 = 2023 $$  Instead of each asterisk, you need to put one of the signs “+” (plus), “-” (minus), “•” (multiply) so that the equality becomes true. What is the smallest number of "•" characters that can be used?
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Context:
Given a set $T = \{a, b, c, d, e, f\}$, determine the number of ways to choose two subsets of $T$ such that their union is $T$ and their intersection contains exactly three elements.
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Context:
Positive real numbers are arranged in the form:  $ 1 \ \ \ 3 \ \ \ 6 \ \ \ 10 \ \ \ 15 ...$  $ 2 \ \ \ 5 \ \ \ 9 \ \ \ 14 ...$  $ 4 \ \ \ 8 \ \ \ 13 ...$  $ 7 \ \ \ 12 ...$  $ 11 ...$ Find the number of the line and column where the number 2002 stays.
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Context:
The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let $N$ be the original number of the room, and let $M$ be the room number as shown on the sign. The smallest interval containing all possible values of $\frac{M}{N}$ can be expressed as $\left[\frac{a}{b}, \frac{c}{d}\right)$ where $a, b, c, d$ are positive integers with $\operatorname{gcd}(a, b)=\operatorname{gcd}(c, d)=1$. Compute $1000 a+100 b+10 c+d$.
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Context:
Calamitous Clod deceives the math beasts by changing a clock at Beast Academy. First, he removes both the minute and hour hands, then places each of them back in a random position, chosen uniformly along the circle.

Professor Grok notices that the clock is not displaying a valid time. That is, the hour and minute hands are pointing in an orientation that a real clock would never display. One such example is the hour hand pointed at $6$ and the minute hand pointed at $3$.

The math beasts can fix this, though. They can turn both hands by the same number of degrees clockwise. On average, what is the minimal number of degrees they must turn the hands so that they display a valid time?
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Context:
In the diagram, \( PQR \) is a straight line segment and \( QS = QT \). Also, \( \angle PQS = x^\circ \) and \( \angle TQR = 3x^\circ \). If \( \angle QTS = 76^\circ \), find the value of \( x \). Express your answer in degrees.
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Context:
Call an ordered triple $(a, b, c)$ of integers feral if $b -a, c - a$ and $c - b$ are all prime.
Find the number of feral triples where $1  \le a < b < c \le 20$.
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Context:
A $2$ by $2$ square is divided into four $1$ by $1$ squares. Each of the small squares is to be painted either green or red. In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square? There may be as few as zero or as many as four small green squares. Provide your answer as an integer.
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Context:
Let $\Gamma$ denote the circumcircle of triangle $A B C$. Point $D$ is on $\overline{A B}$ such that $\overline{C D}$ bisects $\angle A C B$. Points $P$ and $Q$ are on $\Gamma$ such that $\overline{P Q}$ passes through $D$ and is perpendicular to $\overline{C D}$. Compute $P Q$, given that $B C=20, C A=80, A B=65$.
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Context:
Petya added together several consecutive even numbers. It turned out that the resulting sum is 30 times the largest addend and 90 times the smallest addend. Find which numbers Petya added.
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Context:
Let  $\alpha < 0 < \beta$  and consider the polynomial  $f(x) = x(x-\alpha)(x-\beta)$ . Let  $S$  be the set of real numbers  $s$  such that  $f(x) - s$  has three different real roots. For  $s\in S$ , let  $p(x)$  the product of the smallest and largest root of  $f(x)-s$ . Determine the smallest possible value that  $p(s)$  for   $s\in S$ .
Please reason step by step, and put your final answer within \boxed{}.

\boxed{ -\frac{1}{4} (\beta - \alpha)^2 }


Context:
There are $10$ birds on the ground. For any $5$ of them, there are at least $4$ birds on a circle. Determine the least possible number of birds on the circle with the most birds.
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Context:
Gabriela found an encyclopedia with  $2023$  pages, numbered from  $1$  to  $2023$ . She noticed that the pages formed only by even digits have a blue mark, and that every three pages since page two have a red mark. How many pages of the encyclopedia have both colors?
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Context:
The sequence  $\{a_{n}\}$  is defined as follows. \[a_{1}=\frac{\pi}{4},\ a_{n}=\int_{0}^{\frac{1}{2}}(\cos \pi x+a_{n-1})\cos \pi x\ dx\ \ (n=2,3,\cdots)\] Find  $\lim_{n\to\infty}a_{n}$ .
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Context:
The lengths of the three sides of $\triangle ABC$ are 4 cm, 7 cm, and $a$ cm. Using the triangle inequality theorem, determine the possible integer values of $a$. Express your answer as a list or range of integers.
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Context:
2015 people sit down at a restaurant. Each person orders a soup with probability $\frac{1}{2}$. Independently, each person orders a salad with probability $\frac{1}{2}$. What is the probability that the number of people who ordered a soup is exactly one more than the number of people who ordered a salad?
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Context:
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let $S$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $S$?
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Context:
In a clinical trial, only 0.1% of new treatments are actually effective. Given a significance level of 0.05 and a power of 0.8, calculate the proportion of tests that declare a new treatment effective which are actually false positives.
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Context:
从正15边形的顶点中选出3个连成钝角三角形，不同的选法有多少种？
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Context:
Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability $\frac{1}{2}$. What is the expected value of the number of games they will play?
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Context:
In the four-digit number 4753, three two-digit numbers are formed by successive pairs of digits (47, 75, 53). Exactly two of these two-digit numbers are prime. Find all four-digit numbers in which all four digits are prime, and all three two-digit numbers formed by successive digits are prime.
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Context:
Andrea is three times as old as Jim was when Jim was twice as old as he was when the sum of their ages was  $47$ . If Andrea is  $29$  years older than Jim, what is the sum of their ages now?
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Context:
In a group of 2009 people, each pair of people shares exactly one common friend. Determine the smallest possible difference between the maximum and minimum number of friends any person in the group can have.
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Context:
Determine the values of \( \alpha \in \mathbb{R} \) for which the integral \( \int_0^1 \frac{\sin(\frac{1}{x})}{(\sqrt{x} - x)^\alpha} \, dx \) converges.
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Context:
Compute the smallest positive integer $M$ such that there exists a positive integer $n$ such that

[list] [*] $M$ is the sum of the squares of some $n$ consecutive positive integers, and
[*] $2M$ is the sum of the squares of some $2n$ consecutive positive integers.
[/list]

[i]Proposed by Jaedon Whyte[/i]
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Context:
In triangle $\triangle ABC$, $\angle C=90^{\circ}$, $\angle B=30^{\circ}$, and $AC=2$. Let $M$ be the midpoint of $AB$. The triangle $\triangle ACM$ is folded along $CM$ such that the distance between points $A$ and $B$ becomes $2\sqrt{2}$. Determine the distance from point $M$ to the plane $ABC$. \(\text{Express your answer in simplest form.}\)
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Context:
Calculate the flux of the vector field \( \vec{F}(x,y) = (x^2+y^2)^{-1}\begin{bmatrix} x \\ y \end{bmatrix} \) across the circle \( C \) of radius \( a \) centered at the origin, with positive orientation.
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Context:
What is the smallest three-digit number in Pascal's triangle that is also a multiple of 5?
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Context:
Find the remainder when the infinite power tower \(3^{3^{3^{\:
\phantom{}^{.^{.^.}}}}}\) with 2020 levels is divided by 46. Use Euler's totient theorem and modular order reduction to solve this problem.
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Context:
## Task B-3.7.

The base of the pyramid is a right-angled triangle with sides of length 1, $a, a^{2}, a>1$. The apex of the pyramid is orthogonally projected onto the vertex of the right angle of the base. The acute angle opposite the side of length 1 is equal to the angle at which one lateral face is inclined to the base. Calculate the volume of the pyramid.
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Context:
In $\triangle ABC$, $\angle ABC = 50^{\circ}$ and $\angle ACB = 20^{\circ}$. Point $N$ is inside $\triangle ABC$ such that $\angle NAB = 40^{\circ}$ and $\angle NBC = 30^{\circ}$. Find the measure of $\angle NCB$.
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Context:
Let \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a function that satisfies

\[ f(1) = 2, \]
\[ f(2) = 1, \]
\[ f(3n) = 3f(n), \]
\[ f(3n + 1) = 3f(n) + 2, \]
\[ f(3n + 2) = 3f(n) + 1. \]

Find how many integers \( n \leq 2014 \) satisfy \( f(n) = 2n \).
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Context:
A rectangle and an isosceles triangle share a common side, which lies along the $x$-axis, with the right vertex of the rectangle and the left vertex of the triangle located at $(15, 0)$. The dimensions of the rectangle are $15$ units wide (along the $x$-axis) and $10$ units high, while the base and height of the triangle are both $10$ units. A segment is drawn from the top left vertex of the rectangle to the top vertex of the triangle. Calculate the area of the triangle formed by this segment and part of the $x$-axis.
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Context:
若实数 $x 、 y$ 满足
$2^{x}+4 x+12=\log _{2}(y-1)^{3}+3 y+12=0$,则 $x+y=$          .
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Context:
Benjamin walks to work and home five days a week, walks his dog twice a day every day, walks to his best friend’s house once a week, and walks to the convenience store twice a week. Work is six miles away, dog walks are two miles, and his best friend’s house is one mile. He walks 95 miles in a week. How far is the convenience store?
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Context:
Consider all the positive integers  $N$  with the property that all of the divisors of  $N$  can be written as  $p-2$  for some prime number  $p$ . Then, there exists an integer  $m$  such that  $m$  is the maximum possible number of divisors of all
numbers  $N$  with such property. Find the sum of all possible values of  $N$  such that  $N$  has  $m$  divisors.

*Proposed by **FedeX333X***
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Context:
A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k \ge 1$, the circles in $\bigcup_{j=0}^{k-1}L_j$ are ordered according to their points of tangency with the $x$-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer $L_k$ consists of the $2^{k-1}$ circles constructed in this way. Let $S=\bigcup_{j=0}^{6}L_j$, and for every circle $C$ denote by $r(C)$ its radius. What is
\[\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?\]
[asy] import olympiad; size(350); defaultpen(linewidth(0.7)); // define a bunch of arrays and starting points pair[] coord = new pair[65]; int[] trav = {32,16,8,4,2,1}; coord[0] = (0,73^2); coord[64] = (2*73*70,70^2); // draw the big circles and the bottom line path arc1 = arc(coord[0],coord[0].y,260,360); path arc2 = arc(coord[64],coord[64].y,175,280); fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75)); fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75)); draw(arc1^^arc2); draw((-930,0)--(70^2+73^2+850,0)); // We now apply the findCenter function 63 times to get // the location of the centers of all 63 constructed circles. // The complicated array setup ensures that all the circles // will be taken in the right order for(int i = 0;i<=5;i=i+1) { int skip = trav[i]; for(int k=skip;k<=64 - skip; k = k + 2*skip) { pair cent1 = coord[k-skip], cent2 = coord[k+skip]; real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2); real shiftx = cent1.x + sqrt(4*r1*rn); coord[k] = (shiftx,rn); } // Draw the remaining 63 circles } for(int i=1;i<=63;i=i+1) { filldraw(circle(coord[i],coord[i].y),gray(0.75)); }[/asy]
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Context:
Simplify the expression:

\[
\frac{\sin 2 \alpha + \cos 2 \alpha - \cos 6 \alpha - \sin 6 \alpha}{\sin 4 \alpha + 2 \sin^2 2 \alpha - 1}
\]
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Context:
Suppose that the polynomial $p(x) = x^{2010} \pm x^{2009} \pm \ldots \pm x \pm 1$ does not have a real root. What is the maximum number of coefficients that can be $-1$?
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Context:
Our class got 125 books from the library. Then we gave 42 books away to a different class, and later added 68 books from another source. After that, we decided to donate 31 more books to the school's book drive. How many books from the library does our class have left?
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Context:
Determine the number of integer pairs \((x, y)\) such that \((|x| - 2)^2 + (|y| - 2)^2 < 5\).
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Context:
ΔABC是边长为1的正三角形，PAL平面ABC，且$P A = \frac { \sqrt { 6 } } { 4 } , A$点关于平面PBC的对称点为$A ^ { \prime }$，则直线$A ^ { \prime } C$与AB所成角的余弦值是 .
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Context:
A triangle has sides \(a, b, c\). Semicircles are constructed outwardly on each of its sides as diameters, forming a figure \(F\) composed of the triangle and the three semicircles. Find the diameter of \(F\) (the diameter of a set in the plane is the greatest distance between its points).
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Context:
Given that when $-1 < x < 0$, the value of the quadratic function $y = x^2 - 4mx + 3$ is always greater than 1, find the range of values for $m$.
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Context:
Given \(0 < b < 1\) and \(0 < a < \frac{\pi}{4}\), determine the correct ordering of the three numbers defined as follows:
\[ x = \sin a^{\log_{b} (b^{\sin a})}, \]
\[ y = (\cos a)^{\log_{6} (b^{\cos a})}, \]
\[ z = (\sin a)^{\log_{b} (\cos a)}. \]
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Context:
Two trains of different lengths run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. One train is 300 m long, and they take 17.998560115190788 seconds to cross each other. How long is the other train?
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Context:
Rafael works 10 hours on Monday and 8 hours on Tuesday on his delivery job. He must work a total of 40 hours per week. However, there is a 5-day work week constraint, meaning that he can only work a maximum of 8 hours per day from Wednesday to Friday. Given that he is paid $20 per hour for the first 8 hours of work each day and $30 per hour for any overtime (hours worked after the first 8 hours), calculate his total pay for the week, taking into account a 10% tax deduction on his earnings.
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Context:
Evaluate the integral: $$\int_{0}^{1}\dfrac{x^4+1}{x^6+1}\,dx.$$
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Context:
Find the maximum value of \( \sum_{i=1}^n \frac{1}{|x-p_i|} \) for \( 0 \le x \le 1 \), where \( 0 \le p_i \le 1 \) for \( i = 1, 2, \dots, n. \)
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Context:
Determine the condition for repeated roots in the fourth-order polynomial equation: \( x^4 + ax^3 + bx^2 + cx + d = 0 \).
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Context:
A man can buy a new house in 4 years with all of his earnings. He spends the same amount on household expenses in some years and on savings in 12 years. He needs 24 years to buy the house. How many years does it take for him to spend the same amount on household expenses?
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Context:
在平面直角坐标系 $x O y$ 中, 设一条动直线 $l$ 与抛物线 $\Gamma: y^{2}=4 x$ 相切，且与双曲线 $\Omega: x^{2}-y^{2}=1$ 交于左、右两支各一点 $A 、 B$ 。求 $\triangle A O B$ 的面积的最小值。原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。
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Context:
In the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \ (a>b>0)\), let the left focus be \(F\), the right vertex be \(A\), and the endpoint on the upper side of the minor axis be \(B\). If the eccentricity of this ellipse is \(\frac{\sqrt{5}-1}{2}\), then find \(\angle ABF\).
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Context:
Given $m\in R$, let point $A$ be on the moving line $x+my=0$ and point $B$ be on the moving line $mx-y-m+3=0$. These two lines intersect at point $P(x,y)$. Find the maximum value of $|PA|\cdot|PB|$.
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Context:
Petya is thinking of a four-digit number of the form \( \overline{20 * *} \).

Vasya consecutively checks whether the number chosen by Petya is divisible by 1, 3, 5, 7, 9, 11. If the number is divisible, Vasya pays Petya 1, 3, 5, 7, 9, or 11 rubles respectively. For example, for the number 2000, Vasya would pay Petya \(1+5=6\) rubles.

What is the maximum number of rubles Petya can receive?
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Context:
Given a four-digit number where the digits are distinct non-negative integers and their sum is 12, find the number of such four-digit numbers.
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Context:
In the rectangular coordinate plane, the point \((x, y)\) corresponds one-to-one with the complex number \(x + yi\). If points \(A\) and \(B\) correspond to the complex numbers \(z\) and \(z^{-1} \, (z \notin \mathbb{R})\) respectively, then the complex number corresponding to the intersection of the line \(AB\) with the \(x\)-axis (expressed in terms of \(z\)) is:
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\boxed{\frac{z + \bar{z}}{1 + z \bar{z}}}


Context:
Given linearly independent vectors $\left\{a,b,c\right\}\in \mathbb{R}^3$, find the value of $\lambda $ so that the dimension of the subspace generated by the vectors: $2a-3b,\:\:\left(\lambda -1\right)b-2c,\:\:3c-a,\:\:\lambda c-b$ is 2.
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Context:
已知抛物线 $\Omega: y=x^{2}$, 动线段 $A B$ 在直线 $y=\sqrt{3} x-3$ 上 ( $B$ 在 $A$ 右侧), 且 $|A B|=2 \sqrt{3}$. 过 $A$ 作 $\Omega$的切线, 取左边的切点为 $M$. 过 $B$ 作 $\Omega$ 的切线, 取右边的切点为 $N$. 当 $M N / / A B$ 时, 求点 $A$ 的横坐标.
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Context:
Given a sequence ${a_n}$ with its first $n$ terms sum $S_n$, it satisfies $S_n=(-1)^{n}a_{n}+ \frac {1}{2^{n}}$. Let $T_n$ be the sum of the first $n$ terms of ${S_n}$. Find $T_{2017}$.
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Context:
已知数列 \{x_n\} 满足：x_1=a, x_2=b, x_{n+2}=3x_{n+1}+2x_n. 若存在整数 k\geq3，使得 x_k=2019，则这样的有序正整数对 (a,b) 的个数为__________.
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Context:
In a convex pentagon \(ABCDE\), \(AB = BC\), \(CD = DE\), \(\angle ABC = 100^\circ\), \(\angle CDE = 80^\circ\), and \(BD^2 = \frac{100}{\sin 100^\circ}\). Find the area of the pentagon.
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Context:
Given $a=\sin 11^\circ$, $b=\cos 10^\circ$, and $c=\sin 168^\circ$, find the relationship between the magnitudes of $a$, $b$, and $c$.
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Context:
Find the limit of the following integral as n approaches infinity:

$$I_n = \int_0^3 \underbrace{\sin\left(\frac{\pi}{3} \sin\left(\frac{\pi}{3} \sin \left(\frac{\pi}{3} \dots \sin\left(\frac{\pi}{3} x\right) \dots \right)\right)\right)}_\text{n times of sines}dx$$
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Context:
The average age of 15 men is increased by years when one of them whose age is a certain number of years and another whose age is 23 years are replaced by two new men. The average age of the two new men is 37. What is the age of the first man who was replaced?
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Context:
Assume that each airplane engine has a failure rate of $1-P$ during flight, and whether each engine fails is independent. If at least $50\%$ of the engines are functioning normally, the airplane can successfully fly. To make a 4-engine airplane safer than a 2-engine airplane, what is the range of the value of $P$?
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Context:
The taxi fare in Metropolis City is $3.00 for the first $\frac{3}{4}$ mile and additional mileage charged at the rate $0.30 for each additional 0.1 mile. You plan to give the driver a $3 tip. Calculate the number of miles you can ride for $15.
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Context:
What is the minimum number of factors that need to be removed from the number 99! (99! is the product of all numbers from 1 to 99) so that the product of the remaining factors ends in 2?
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Context:
Let $ABCD$ be a convex quadrilateral with $AB=5$, $BC=6$, $CD=7$, and $DA=8$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, and $DA$ respectively. Compute $MN^2 - PQ^2$.
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Context:
Given six people, including A, B, and C, are arranged in a row for a photo, and A and B are not on the same side as C, calculate the total number of different arrangements.
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Context:
Leticia has a $9 \times 9$ board. She says that two squares are "friends" if they share a side, if they are at opposite ends of the same row, or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of three colors: green, blue, or red. In each square, a number will be written based on the following rules:

- If the square is green, write the number of red friends plus twice the number of blue friends.
- If the square is red, write the number of blue friends plus twice the number of green friends.
- If the square is blue, write the number of green friends plus twice the number of red friends.

Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares.
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Context:
化简$C _ { n } ^ { 0 } C _ { n } ^ { 1 } + C _ { n } ^ { 1 } C _ { n } ^ { 2 } + \cdots + C _ { n } ^ { n - 1 } C _ { n } ^ { n } =$ .
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Context:
Consider a sequence defined as $500, x, 500 - x, \ldots$ where each term of the sequence after the second one is obtained by subtracting the previous term from the term before it. The sequence terminates as soon as a negative term appears. Determine the positive integer $x$ that leads to the longest sequence.
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Context:
Among the 99 positive integers from 1 to 99, find the minimum value of k such that any selection of k numbers must include at least two numbers a and b (a!= b) satisfying 1/2 <= b/a <= 2.
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Context:
ерешин D.A.

Once, Rabbit was in a hurry to meet with Donkey Eeyore, but Winnie-the-Pooh and Piglet unexpectedly came to him. Being well-mannered, Rabbit offered his guests some refreshments. Pooh tied a napkin around Piglet's mouth and single-handedly ate 10 pots of honey and 22 cans of condensed milk, with each pot of honey taking him 2 minutes and each can of milk taking 1 minute. Upon learning that there was no more sweet food in the house, Pooh said goodbye and took Piglet away. Rabbit sadly thought that he wouldn't have been late for the meeting with the donkey if Pooh had shared with Piglet. Knowing that Piglet eats a pot of honey in 5 minutes and a can of milk in 3 minutes, Rabbit calculated the minimum time it would take for the guests to finish his supplies. What is this time? (A can of milk and a pot of honey can be divided into any parts.)

#
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Context:
## Problem Statement

Calculate the indefinite integral:

$$
\int \frac{1-\sqrt{x}}{\sqrt{x} \cdot(x+1)} d x
$$
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Context:
The diagram shows a smaller rectangle made from three squares, each of area \(25 \ \mathrm{cm}^{2}\), inside a larger rectangle. Two of the vertices of the smaller rectangle lie on the mid-points of the shorter sides of the larger rectangle. The other two vertices of the smaller rectangle lie on the other two sides of the larger rectangle. What is the area, in \(\mathrm{cm}^{2}\), of the larger rectangle?
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Context:
In $\triangle ABC$, if $\sin^{2} B + \sqrt{2} \sin B \sin C = \sin^{2} A - \sin^{2} C$, find the value of $A$.
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Context:
14. First group (from 8th grade. 3 points). The sequence consists of 19 ones and 49 zeros, arranged in a random order. We will call a group the maximal subsequence of identical symbols. For example, in the sequence 110001001111 there are five groups: two ones, then three zeros, then one one, then two zeros, and finally four ones. Find the expected value of the length of the first group.
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Context:
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length 5, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are 12 possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point $(2021,2021)$ ?
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Context:
A triangular pyramid is inscribed in a cone, with its lateral edges pairwise mutually perpendicular. Find the angle between the slant height of the cone and its height.
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Context:
The sum of the ages of five people \( A, B, C, D, \) and \( E \) is 256 years. The age difference between any two people is not less than 2 years and not more than 10 years. What is the minimum age of the youngest person among them?
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Context:
Elliot has 4 books to read. The first book has 512 pages and he has already read 194 of them. The second book has 298 pages and he hasn't started it yet. The third book is 365 pages long and he has already finished 50 pages. The fourth book has 421 pages and he's yet to start this one as well. If Elliot aims to read 30 pages a day from the first book, 20 pages a day from the second, 25 pages a day from the third book, and 15 pages a day from the fourth book, how many pages are still left to be read after three weeks?
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Context:
Calculate the definite integral:

$$
\int_{2 \operatorname{arctg} \frac{1}{3}}^{2 \operatorname{arctg} \frac{1}{2}} \frac{d x}{\sin x(1-\sin x)}
$$
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Context:
How many natural numbers $n$ satisfy the equation

$$\varphi(n)+\tau(n^2)=n$$

where $\varphi$ is the Euler's totient function and $\tau$ is the divisor function i.e. number of divisors of an integer?
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Context:
A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square?
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Context:
Let $k:\Bbb R\to\Bbb R$ be a twice differentiable function such that $ k(a)=k(e)=0 ,k(b)=k(d)=2 , k(c)=1$, where $a<b<c<d<e$. Find the minimum number of solutions of $(k'(x))^2+k(x)k''(x)=0$ in $[a,e]$.
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Context:
How many solutions in natural numbers does the equation $\left\lfloor \frac{x}{10} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor + 1$ have?
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Context:
Calculate the integral $I= \int_{\partial B_1 \cap \{x_1 \geq 0 \}} x_1 \, d \sigma$ over the intersection of the upper hemisphere of the unit ball $B_1$ in $\mathbb{R}^n$ with the boundary $\partial B_1$, where $\sigma$ is the surface measure.
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Context:
A motorcyclist departed from point $A$ with an initial speed of 90 km/h, uniformly increasing the speed (i.e., the speed increases by the same amount over equal time intervals). After three hours, the motorcyclist arrived at point $B$, passing through $C$ along the way. Then, the motorcyclist turned around and, still uniformly increasing speed, set off back. Two hours later, he passed by point $C$ again at a speed of 110 km/h and continued on to $A$. Find the distance between points $A$ and $C$.
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Context:
Three numbers are chosen independently at random, one from each of the three intervals $[0, L_i](i=1, 2,3)$. If the distribution of each random number is uniform with respect to length in the interval it is chosen from, determine the expected value of the smallest of the three numbers chosen. The original answer is in the format $\frac{a}{b} L_{1}^{2} L_{2} L_{3}-\frac{c}{d} L_{1}^{3}\left(L_{2}+L_{3}\right)+\frac{e}{f} L_{1}^{4}$, please give the value of a+b+c+d+e+f.
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Context:
Boy Vasya wrote the non-zero coefficients of an 8th-degree polynomial \( P(x) \) in a notebook. He then calculated the derivative of the resulting polynomial and noted its non-zero coefficients, continuing in this manner until a constant was obtained, which he also noted.

What is the minimum number of different numbers he could have come up with?

Coefficients are written with their signs, constant terms are also written if there is a term of the form \( \pm x^{n} \), \(\pm 1\) is written.
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Context:
[color=darkred] Let $m$ and $n$ be two nonzero natural numbers. Determine the minimum number of distinct complex roots of the polynomial $\prod_{k=1}^m\, (f+k)$ , when $f$ covers the set of $n^{\text{th}}$ - degree polynomials with complex coefficients.
[/color]
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Context:
Simplify first, then evaluate: $\frac{4y^2-x^2}{x^2+2xy+y^2}÷\frac{x-2y}{2x^2+2xy}$, where $x=\sqrt{2}$ and $y=2\sqrt{2}$.
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Context:
Given that \( O \) is the circumcenter of \( \triangle ABC \), and the equation
\[
\overrightarrow{A O} \cdot \overrightarrow{B C} + 2 \overrightarrow{B O} \cdot \overrightarrow{C A} + 3 \overrightarrow{C O} \cdot \overrightarrow{A B} = 0,
\]
find the minimum value of \( \frac{1}{\tan A} + \frac{1}{\tan C} \).
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Context:
77 Ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$ on: different points $A\left(x_{1}, y_{1}\right), B\left(4, \frac{9}{5}\right), C\left(x_{2}, y_{2}\right)$ and the focus $F(4,0)$ have distances that form an arithmetic sequence. If the perpendicular bisector of line segment $A C$ intersects the $x$-axis at point $T$, then the slope $k$ of line $B T$ is $\qquad$.
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Context:
How many distinct, positive factors does $1320$ have?
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Context:
因为实数的乘法满足交换律与结合律，所以若干个实数相乘的时候，可以有不同的次序，例如，三个实数 $a, b, c$ 相乘的时候，可以有 $(a b) c,(b a) c$ ， $b(c a)$ ，等等不同的次序。记 $n$ 个实数相乘时不同的次序有 $I_{n}(n \geqslant 2)$ 种，则求 $I_{3}$ 的值。
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Context:
Given a hyperbola with asymptotes $2x \pm y=0$, that passes through the intersection of the lines $x+y-3=0$ and $2x-y+3t=0$, where $-2 \leq t \leq 5$. Find the maximum possible length of the real axis of the hyperbola.
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Context:
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\frac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive integers. Find $r+s+t$.
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Context:
Amir enters Fine Hall and sees the number $2$ written on the blackboard. Amir can perform the following operation: he flips a coin, and if it is heads, he replaces the number $x$ on the blackboard with $3x + 1$; otherwise, he replaces $x$ with $\lfloor x/3 \rfloor$. If Amir performs the operation four times, let $\frac{m}{n}$ denote the expected number of times that he writes the digit $1$ on the blackboard, where $m, n$ are relatively prime positive integers. Find $m + n$.
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Context:
The integer sequence \( \left\{a_{n}\right\} \) is defined as follows:
$$
a_{0}=3, a_{1}=4, a_{n+2}=a_{n+1} a_{n}+\left\lceil\sqrt{\left(a_{n+1}^{2}-1\right)\left(a_{n}^{2}-1\right)} \right\rceil \text{ for } n \geq 0.
$$

Find the value of \( \sum_{n=0}^{+\infty}\left(\frac{a_{n+3}}{a_{n+2}}-\frac{a_{n+2}}{a_{n}}+\frac{a_{n+1}}{a_{n+3}}-\frac{a_{n}}{a_{n+1}}\right) \).
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Context:
How many multiples of 7 between $10^{6}$ and $10^{9}$ are perfect squares?
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Context:
Let $ABC$ be a triangle with $BC = 7$, $AB = 5$, and $AC = 8$. Let $M$ and $N$ be the midpoints of sides $AC$ and $AB$, respectively, and let $O$ be the circumcenter of $\triangle ABC$. Let $BO$ and $CO$ meet $AC$ and $AB$ at $P$ and $Q$, respectively. If $MN$ meets $PQ$ at $R$ and $OR$ meets $BC$ at $S$, then the value of $OS^2$ can be written in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $100m + n$.
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Context:
Let $m = 2007^{2008}$. How many natural numbers $n$ are there such that $n < m$ and $n(2n + 1)(5n + 2)$ is divisible by $m$ (i.e., $m \mid n(2n + 1)(5n + 2)$)?
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Context:
Consider the function $g(x) = \frac{x^2}{2} + 2x - 1$. Determine the sum of all distinct numbers $x$ such that $g(g(g(x))) = 1$.
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Context:
Let $n$ be a positive integer. A pair of $n$-tuples \left(a_{1}, \ldots, a_{n}\right)$ and \left(b_{1}, \ldots, b_{n}\right)$ with integer entries is called an exquisite pair if $$\left|a_{1} b_{1}+\cdots+a_{n} b_{n}\right| \leq 1$$ Determine the maximum number of distinct $n$-tuples with integer entries such that any two of them form an exquisite pair.
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Context:
Given that the domain of the function $f(x)$ is $\mathbb{R}$, if $f(x+1)$ and $f(x-1)$ are both odd functions, then the function $y=f(x)$ has at least \_\_\_\_\_ zeros in the interval $[0,100]$.
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Context:
5. Point $M$ divides the diagonal $A C$ of square $A B C D$ in the ratio $A M: M C=3: 2$. A line passing through point $M$ divides the square into two parts, the areas of which are in the ratio $9: 11$. In what ratio does this line divide the perimeter of the square?
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Context:
Define a "spacy" set of integers such that it contains no more than one out of any four consecutive integers. How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are spacy?
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Context:
The percentage of seventh-grade students participating in the gymnastics section is between 2.9% and 3.1%. Determine the smallest possible number of students in this class.
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Context:
Find the smallest positive integer $n$ such that a cube with sides of length $n$ can be divided into exactly 2007 smaller cubes, each with integer side lengths.
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Context:
现有 6 人会英语， 4 人会日语， 2 人都会（共 12 人），从其中选出 3 人做翻译，若要求两种语言都有人翻译，则符合条件的选法共           种。
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Context:
A *strip* is the region between two parallel lines. Let $A$ and $B$ be two strips in a plane. The intersection of strips $A$ and $B$ is a parallelogram $P$. Let $A'$ be a rotation of $A$ in the plane by $60^\circ$. The intersection of strips $A'$ and $B$ is a parallelogram with the same area as $P$. Let $x^\circ$ be the measure (in degrees) of one interior angle of $P$. What is the greatest possible value of the number $x$?
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Context:
Compute the value of the integral $$\int ^{\infty}_0 \frac{1}{1+2ax+x^2} dx $$
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Context:
The graph of the rational function $\frac{q(x)}{2x^5+x^4-7x^2+1}$ has a horizontal asymptote. What is the largest possible degree of $q(x)$?
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Context:
The numbers $1, 2, \ldots, 10000$ are placed in the cells of a $100 \times 100$ square grid, each number appearing exactly once. In this arrangement, numbers that differ by $1$ are located in adjacent cells. We then calculate the distances between the centers of every pair of cells where the numbers differ by exactly $5000$. Let $S$ be the minimum of these distances. What is the largest value $S$ can take?
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Context:
Evaluate the limit \( \lim_{n\to\infty} nt_n \), where \( t_n=\sum_{i=1}^{2n}\frac{(-1)^{i+1}}{2n+i} \). Relate the given limit to suitable Riemann sums for the function \( f(x)=(1+x)^{-2} \).
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Context:
What additional property must an isometry have to be considered a symmetry of a set of points $X$ in a plane with a metric structure?
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Context:
Determine the number of downward-closed subsets of the set \( \mathbb{N}^{<\infty} \), where \( \mathbb{N}^{<\infty} \) is the set of all finite sequences of natural numbers, including the empty sequence, ordered by extension. A subset \( C \subset \mathbb{N}^{<\infty} \) is downward-closed if for any sequence \( t \in C \), all initial segments \( s < t \) are also in \( C \).
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Context:
Find the derivative of the quadratic form \( z^t M z \) with respect to the vector \( z \), where \( M \) is an \( n \times n \) matrix and \( z \) is an \( n \times 1 \) vector. Express your answer in terms of \( M \), \( M^t \), and \( z \).
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Context:
Suppose there are \(N\) realizations of a Gaussian process denoted as the vectors \(z_{j} \in \mathbb{R}^{n}\) for \(j = 1, \ldots, N\). Let \(y\) be a random variable such that \(y = \sum_{j=1}^{N}(Bz_{j})[i]\), where \(B\) is a unitary matrix satisfying \(B^* B = BB^* = I\). Determine the variance of \(y^{2}\).
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Context:
Find the smallest integer \( n \) such that the symmetric group \( S_n \) contains an even permutation of order 312, given that there exists an 8-cycle, a 3-cycle, and a 13-cycle in the group.
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Context:
Calculate the average number of messages required for a stat to increase from 100 to its maximum value of 300, given that the probability of the stat increasing by 1 with each message is \( \frac{300 - \text{current stat}}{300} \).
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Context:
Determine the minimum time after which it can be concluded that the remaining bricks in a $2N \times 2N$ grid will never be hit by a ball. The ball travels 1 unit in both the x and y directions, bounces off perpendicularly upon hitting a boundary or a brick, and starts its trajectory from a given point.
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Context:
Evaluate the limit: \[ \lim_{x\to0}\ln(e + 2x)^\frac{1}{\sin x} \]
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Context:
Simplify the expression \( \frac{1}{1+2x} - \frac{1-x}{1+x} \) for \( x \) much smaller than 1, using appropriate approximations or series expansions.
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Context:
A player randomly selects digits, either 1 or 0, to form a sequence. The player wins if the last four digits of the sequence are '0010' and loses if they are '1001'. Determine the probability that the player wins.
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Context:
Evaluate the limit of the following triple integral as \( m \to \infty \):
\[
\lim_{m\to\infty} \int_{\frac{1}{m}}^1 \int_0^\pi \int_{-\pi}^\pi \frac{r^6 \sin^4(a) \cos^2(a) \cos^2(b) \sin^2(b)}{r^{17/2}} \, db \, da \, dr
\]
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Context:
Sherry is waiting for a train. Every minute, there is a $75\%$ chance that a train will arrive.
However, she is engrossed in her game of sudoku, so even if a train arrives she has a $75\%$ chance of not noticing it (and hence missing the train).
What is the probability that Sherry catches the train in the next five minutes?
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Context:
Simplify the expression of the quotient ring:

$$(\mathbb{Z}/25\mathbb{Z})[X] / \langle 5(X+2), 2X -22, X^2 +26 \rangle $$
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Context:
Evaluate the limit: \[ \lim_{x\rightarrow +\infty}\sqrt{x}e^{-x}\left(\sum_{k=1}^{\infty}\frac{x^{k}}{k!\sqrt{k}}\right) \]
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Context:
Evaluate the integral \( \int_0^1 \frac{x^{10}-1}{\log (x)}\,dx \).
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Context:
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{x \sin(x)}{(x^2+1)^2} \, dx \).
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Context:
Consider a circle with an area of $2n$, divided into $n$ regions by joining each of $n$ evenly spaced points on the circle with every other point. Determine the limit of the product of the areas of these regions as $n$ approaches infinity.
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Context:
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has a length of $5$, and after each hop, the grasshopper is at a point whose coordinates are both integers. Thus, there are $12$ possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point $(2021,2021)$?
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Context:
Find all real  $x$  that satisfy the equation   $$ \frac{1}{x+1}+\frac{1}{x+2}=\frac{1}{x} $$  *2015 CCA Math Bonanza Lightning Round #2.2*
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Context:
设 \mathcal{A} 是所有关于 x,y,z 的整系数多项式构成的集合. 设 \mathcal{B} 是由可以表示为如下形式的多项式构成的 \mathcal{A} 的子集：(x+y+z)P(x,y,z)+(xy+yz+zx)Q(x,y,z)+xyzR(x,y,z),其中 P,Q,R\in\mathcal{A}. 求最小的非负整数 n，使得对任意满足 i+j+k\geq n 的非负整数 i,j,k，均有 x^iy^jz^k\in\mathcal{B}.
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Context:
满足正整数 $n \leqslant 2022$ 且 $2n+1$ 与 $3n+1$ 均为完全平方数的 $n$ 的个数是 _______.
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Context:
Find the leading coefficient of the polynomial $R(x)$ of degree $2015$ that satisfies $R(k+1)-2R(k)=2^{k+1}$ for $k=0,1,2,\dots,2014$ and $R(0)=0$.
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Context:
##

Side $A B$ of parallelogram $A B C D$ is equal to $2, \angle A=45^{\circ}$. Points $E$ and $F$ are located on diagonal $B D$, such that

$\angle A E B=\angle C F D=90^{\circ}, B F=\frac{3}{2} B E$.

Find the area of the parallelogram.
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Context:
Let the sets \( A = \{ x \mid x^2 + 2x - 3 = 0 \} \) and \( B = \{ x \mid x^2 - (k+1)x + k = 0, \, x \in \mathbb{R} \} \). Given that \( A \cap B = B \), find the set of real numbers \( k \).
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Context:
In quadrilateral \( \square ABCD \), \(\angle B < 90^\circ\) and \(AB < BC\). Tangents are drawn from point \( D \) to the circumcircle \( \Gamma \) of triangle \( \triangle ABC \), touching the circle at points \( E \) and \( F \). Given that \(\angle EDA = \angle FDC\), find \(\angle ABC\).
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Context:
Let $\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k$. Compute the following expression in terms of $k$:
\[ E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}.  \]
[i]Ciprus[/i]
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Context:
已知点$P(4,2)$，过点$P$的直线$l$与$x$轴、$y$轴的正半轴分别交于$A、B$两点，$O$为坐标原点。则$\triangle AOB$周长的最小值为_______。
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Context:
有一个 2024\times2024 的方格表，初始时所有格都是白色的. 一次操作可以选择每个格都是白色的一行或一列，将其中 1000 个格染为红色. 求有限次操作后，红格个数的最大可能值.
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Context:
Xiao Li spent $108 to buy some bottled milk at the supermarket. A few days later, when she went to the same supermarket, she encountered a "75% off storewide" promotion and only spent $90 to buy 1 more bottle of milk than last time. What was the original price per bottle of this milk?
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Context:
The glass gauge on a cylindrical coffee maker shows that there are $45$ cups left when the coffee maker is $36\%$ full.  How many cups of coffee does it hold when it is full?
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Context:
Dave had 15 apps on his phone. He added some new apps. After deleting some, he had 14 left and had deleted 1 more app than he added. How many new apps did Dave add to his phone?
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Context:
Consider the bilinear form \( f(A,B) = n\operatorname{Tr}(AB) - \operatorname{Tr}(A)\operatorname{Tr}(B) \) defined on \( M_n(\mathbb{C}) \). Determine the dimension of the set \( U^\perp \) of all matrices \( A \) such that \( f(A,B) = 0 \) for every \( B \in M_n(\mathbb{C}) \).
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Context:
Determine the minimum degree of the Taylor polynomial for \( \sin(x) \) needed to approximate \( \sin(100) \) such that the error is less than \( 10^{-4} \).
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Context:
## Task B-4.4.

How many passwords consisting of 4 characters from the set $\{a, b, c, 1,2,3,4,5,+,!\}$ are there such that all characters in the password are different or the password contains exactly two, not necessarily different letters and two, not necessarily different digits?
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Context:
Two teams are in a best-two-out-of-three playoff: the teams will play at most $3$ games, and the winner of the playoff is the first team to win $2$ games. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a $\frac{2}{3}$ chance of winning at home, and its probability of winning when playing away from home is $p$. Outcomes of the games are independent. The probability that Team A wins the playoff is $\frac{1}{2}$. Then $p$ can be written in the form $\frac{1}{2}(m - \sqrt{n})$, where $m$ and $n$ are positive integers. What is $m + n$?
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Context:
Find the norm of the integral operator $A$ in $L_2(0,1)$, where $Ax(t) = \int_0^t x(s) \, ds$.
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Context:
Calculate the limit \( \lim\limits_{r\to 0^+}\dfrac{1}{\pi r^2}\iint\limits_{|(x,y)|\le r} f(x,y) \, dS \) given that \( f(x,y) \) is continuous at the origin and integrable in that region. Assume \( f(x,y) \) is bounded in the neighborhood of the origin.
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Context:
Let $p$ be the largest prime less than $2013$ for which\[ N = 20 + p^{p^{p+1}-13} \]is also prime. Find the remainder when $N$ is divided by $10^4$.
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Context:
Given the AR(1) process \( \{X_t\} \) defined by:
\[ X_t = 2.62 - 0.84X_{t-1} + \epsilon_t, \quad \epsilon_t \sim WN(0, 2.27) \]
compute the variance of the average:
\[ \overline{X} = \frac{1}{3}\sum_{t=1}^{3} X_t \]
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Context:
Given the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$, let the left and right foci of the ellipse be $F_1$ and $F_2$, respectively. The line passing through $F_1$ and perpendicular to the x-axis intersects the ellipse at points $A$ and $B$. If the line $AF_2$ intersects the ellipse at another point $C$, and the area of triangle $\triangle ABC$ is three times the area of triangle $\triangle BCF_2$, determine the eccentricity of the ellipse.
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Context:
Given that point $M$ is a moving point on the parabola $y^2 = 8x$, with $F$ being the focus of the parabola, and point $A$ lies on the circle $C: (x-3)^2 + (y+1)^2 = 1$, find the minimum value of $|AM| + |MF|$.
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Context:
A palindrome between $10000$ and $100000$ is chosen at random. What is the probability that it is divisible by $11$?
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Context:
Calculate the limit as \( n \to \infty \) for the expression: \[ \lim\limits_{n\to \infty} \sqrt [n]{\frac{(3n)!}{n!(2n+1)!}} \]
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Context:
Find the equation of the orthogonal projection of the space curve defined by \( z = x^y \) onto the x-y plane, given that the curve is symmetrical about the plane \( y = x \) and lies to the right of the line \( y = e \) and above the line \( y = x \), where \( x^y > y^x \).
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Context:
A box contains 14 balls: 7 black balls numbered from 1 to 7 and 7 white balls numbered from 1 to 7. A person picks 2 balls at a time and repeats the process until the box is empty. Determine the number of ways to draw the balls such that each time two balls are picked, one black and one white, the difference between their numbers is at most one.
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Context:
Find the residue of the function \( f(z) = \frac{\sin z}{(z^2+1)^2} \) at \( z = \infty \).
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Context:
Given that $A = (\sqrt{2008}+\sqrt{2009}),$ $B = (-\sqrt{2008}-\sqrt{2009}),$ $C = (\sqrt{2008}-\sqrt{2009}),$ and $D = (\sqrt{2009}-\sqrt{2008}),$ find $ABCD.$
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Context:
In the monotonically increasing sequence $\{a_n\}$, $a_1=2$, the inequality $(n+1)a_n \geq na_{2n}$ holds for any $n \in \mathbb{N}^*$.
Find the range of values for $a_2$.
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Context:
Evaluate the limit: \[ \lim_{n \to +\infty}\left(\frac{1^p + 3^p + \ldots + (2n-1)^p}{(2n)^p} - \frac{n}{p + 1}\right) \] where \( p \in \mathbb{N} \).
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Context:
Estimate the number of floating-point operations (flops) required to solve the linear system \(Ax = b\) to ten-digit accuracy using the Conjugate Gradient method, where \(A\) is a dense symmetric positive definite \(1000 \times 1000\) matrix with a condition number \(\kappa(A) = 100\).
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Context:
Determine the value of \(a\) such that the matrix \(A\) is not diagonalizable, where \(A = \begin{pmatrix} 3 & 0 & 0 \\ 0 & a & a-2 \\ 0 & -2 & 0 \end{pmatrix}\).
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Context:
Determine the values of \(\theta\) for which the set \(S = \{x \in \mathbb{R}^n : |x-a| \leq \theta|x-b|\}\) forms a ball.
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Context:
Determine the Big-O notation for the function \((n^n + n2^n + 5^n)(n! + 2^n)\).
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Context:
Calculate the surface area formed by the rotation around the polar axis of the lemniscate $\rho = a \sqrt{\cos 2 \varphi}$ over the interval $0 \leq \varphi \leq \pi / 4$.
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Context:
Let $f(x)$ be a differentiable function on $(-\infty, 0)$, with the derivative $f'(x)$, and suppose $2f(x) + xf'(x) > x^2$. Determine the solution set for the inequality $(x+2016)^2f(x+2016) - 9f(-3) > 0$.
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Context:
Find the value of the infinite sum:
$$1 + \frac{0.9}{11} +\frac{0.99}{11^2} +\frac{0.999}{11^3}+\cdots$$
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Context:
Equilateral triangles $A B F$ and $B C G$ are constructed outside regular pentagon $A B C D E$. Compute $\angle F E G$.
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Context:
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. If $\overrightarrow{m}=(a+c,-b)$, $\overrightarrow{n}=(a-c,b)$, and $\overrightarrow{m}\cdot \overrightarrow{n}=bc$, then $A=$ ______.
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Context:
A cone with a base radius of 2 and a slant height of \( c \) contains three spheres, each with radius \( r \). The spheres touch each other (externally), the lateral surface of the cone, and the first two spheres touch the base of the cone. Find the maximum value of \( r \).
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Context:
Suppose $f$ is a function defined on the real numbers $\mathbb{R}$ that is differentiable and not identically zero. If $f$ satisfies the functional equation $xf(x) - yf(y) = (x - y)f(x + y)$ for all $x, y \in \mathbb{R}$, what is the value of $f'(2x)$?
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Context:
Given $0 \leq x \leq \pi$ and $-\frac{1}{2} < a < 0$, find the minimum value of the function $f(x) = \cos^2x - 2a\sin x - 1$.
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Context:
Given the complex number $z$ that satisfies the equation $|z-1|=|z+2i|$ (where $i$ is the imaginary unit), find the minimum value of $|x-1-i|$.
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Context:
Find all positive integers $x$ and $y$ that satisfy the Diophantine equation: $$12x^4 - 6x^2 + 1 = y^2$$
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Context:
Calculate $\displaystyle \iiint_A(xy+z+1)dxdydz$, where $A$ is the region bounded by the cones $z=2+\sqrt{x^2+y^2}$ and $z=6-\sqrt{x^2+y^2}$.
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Context:
A tractor is dragging a very long pipe on sleds. Gavrila walked along the entire pipe in the direction of the tractor's movement and counted 210 steps. When he walked in the opposite direction, the number of steps was 100. What is the length of the pipe if Gavrila's step is 80 cm? Round the answer to the nearest whole number of meters.
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Context:
Calculate the following limit:
$$\lim_{n\to\infty} \frac{\sum_{m=1}^{\lfloor n/2 \rfloor} \frac {n!}{(n-2m)! 2^m m!}}{\sum_{m=1}^{\lfloor n/4 \rfloor} \frac {n!}{(n-4m)! 2^{2m} m!}}.$$
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Context:
$1991^{2000}$ 除以 $10^{6}$ ，余数是 $\qquad$.
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Context:
已知 $I_{n}=
\left(1+\frac{i}{2}\right)\left(1+\frac{i}{4}\right)\left(1+\frac{i}{16}\right) \cdots\left(1+\frac{i}{2^{2^{n}}}\right)$, 求 $\left|I_{n}\right|$.\n\n原始答案的格式为\sqrt{\frac{m}{n}\left(1-\frac{1}{2^{2^{n+2}}}\right)}, 请给出m + n的值。
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Context:
A snowman is built on a level plane by placing a ball of radius $6$ on top of a ball of radius $8$, which is on top of a ball of radius $10$, as shown. If the average height above the plane of a point in the snowman is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m + n$.
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Context:
In a 20 over cricket match, taking into account no wide balls, no wides, no extras, no overthrows, as well as a maximum of 3 sixes allowed per over, and only two fielders allowed outside the 30-yard circle for the first 10 overs, and five fielders allowed outside for the last 10 overs. No bowler is allowed to bowl more than 4 overs. What is the maximum number of runs that a batsman can score in an ideal case scenario with these additional constraints?
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Context:
Four square pieces of paper with side lengths of $2, 3, 4,$ and $5$ are placed on a table. If one piece of paper of each side length is removed, the areas covered by the remaining three pieces decrease by $2, 3, 4,$ and $5$ respectively. What is the total area covered by the four pieces of paper?
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Context:
Find the greatest exponent $k$ for which $2001^k$ divides $2000^{2001^{2002}} + 2002^{2001^{2000}}$.
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Context:
In $\triangle ABC$, $\angle C = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle?
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Context:
Sasha has  $10$  cards with numbers  $1, 2, 4, 8,\ldots, 512$ . He writes the number  $0$  on the board and invites Dima to play a game. Dima tells the integer  $0 < p < 10, p$  can vary from round to round. Sasha chooses  $p$  cards before which he puts a “ $+$ ” sign, and before the other cards he puts a “ $-$ " sign. The obtained number is calculated and added to the number on the board. Find the greatest absolute value of the number on the board Dima can get on the board after several rounds regardless Sasha’s moves.
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Context:
There are 15 boys and 10 girls in a class. If a certain number of students are selected at random, there are 1050 ways that 1 girl and 2 boys are selected. How many students are selected at random?
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Context:
A square $ABCD$ has a side length of $1$. A circle passes through the vertices of the square. Let $P, Q, R, S$ be the midpoints of the arcs which are symmetrical to the arcs $AB$, $BC$, $CD$, $DA$ when reflected on sides $AB$, $BC$, $CD$, $DA$, respectively. The area of square $PQRS$ is $a + b\sqrt{2}$, where $a$ and $b$ are integers. Find the value of $a + b$.
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Context:
A small group consists of $3$ members. Each member participates in labor for $1$ day within a week (calculated as $7$ days). If the labor dates can be arranged randomly, what is the probability that the $3$ members will participate in labor on different days?
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Context:
Suppose $X_1, \ldots, X_n$ are independent and uniformly distributed on the interval $[0, 1]$. What is the expected value of the ratio of the second largest to the largest of these random variables?
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