Context:
10. The sum $\sum_{k=1}^{2020} k \cos \left(\frac{4 k \pi}{4041}\right)$ can be written in the form

$$
\frac{a \cos \left(\frac{p \pi}{q}\right)-b}{c \sin ^{2}\left(\frac{p \pi}{q}\right)}
$$

where $a, b, c$ are relatively prime positive integers and $p, q$ are relatively prime positive integers where $p<q$. Determine $a+b+c+p+q$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $S$  be the set of all ordered triples  $(p,q,r)$  of prime numbers for which at least one rational number  $x$  satisfies  $px^2+qx+r=0.$  Which primes appear in seven or more elements of  $S?$
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Context:
$\left[\begin{array}{ll}[\text { Classical combinatorics (other) })]\end{array}\right]$

On a $11 \times 11$ grid, 22 cells are marked such that exactly two cells are marked in each row and each column. Two arrangements of marked cells are equivalent if, by swapping any number of columns among themselves and any number of rows among themselves, one arrangement can be transformed into the other. How many non-equivalent arrangements of marked cells exist?
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Context:
For positive integers $n$, let $h(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point, and $k$ is divisible by 3. How many positive integer divisors does $h(2010)$ have?
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Context:
Given $x>0$, $y>0$, $a=x+y$, $b= \sqrt{x^{2}-xy+y^{2}}$, $c=\lambda \sqrt{xy}$, if $a$, $b$, $c$ can be the lengths of the sides of a triangle, then the range of the positive real number $\lambda$ is \_\_\_\_\_\_.
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Context:
Let $p$ be a prime. We arrange the numbers in ${\{1,2,\ldots ,p^2} \}$ as a $p \times p$ matrix $A = ( a_{ij} )$. Next we can select any row or column and add $1$ to every number in it, or subtract $1$ from every number in it. We call the arrangement [i]good[/i] if we can change every number of the matrix to $0$ in a finite number of such moves. How many good arrangements are there?
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Context:
Given an ellipse x^2/36 + y^2/b^2 = 1 (6 > b > 0) and any point P on the ellipse other than the vertices, it satisfies |OP + OF| = 7, where point F is the left focus of the ellipse, and point O is the origin of the coordinate system. Find the distance |PF|. Express your answer as a whole number.
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Context:
Compute the number of non-empty subsets $S$ of $\{-3, -2, -1, 0, 1, 2, 3\}$ with the following property: for any $k \ge 1$ distinct elements $a_1, \dots, a_k \in S$, we have $a_1 + \dots + a_k \neq 0$.
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Context:
A rectangle is divided into three smaller congruent rectangles. Each smaller rectangle is similar to the large rectangle. Find the ratio of the length of a longer side to that of a shorter side in each of the four rectangles. Express your answer as a simplified ratio (e.g., x:y).
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Context:
Find the probability that the reflected Brownian motion process $Y(t)$ hits $a$ at or after time $t$, given that it hits $a$ before time $t$.
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Context:
For how many integers n, where 0 <= n < 2013, is the polynomial n^4 + 2n^3 - 20n^2 + 2n - 21 congruent to 0 modulo 2013? Provide your answer as a single integer representing the count of integers n.
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Context:
A **Beaver-number** is a positive 5-digit integer whose digit sum is divisible by 17. Call a pair of **Beaver-numbers** differing by exactly $1$ a **Beaver-pair**. The smaller number in a **Beaver-pair** is called an **MIT Beaver**, while the larger number is called a **CIT Beaver**. Find the positive difference between the largest and smallest **CIT Beavers** (over all **Beaver-pairs**).
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Context:
For real numbers $x$, let \[P(x)=1+\cos(x)+i \sin(x)-\cos(2x)-i \sin(2x)+\cos(3x)+i \sin(3x)\] where $i=\sqrt{-1}$. Find the number of values of $x$ with $0 \leq x<2 \pi$ such that $P(x)=0$.
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Context:
If you multiply a number by 2 instead of multiply it by 20 and add 3, you get 22. How much is it when you calculate correctly?
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Context:
In a city with 10 parallel streets and 10 streets crossing them at right angles, what is the minimum number of turns that a closed bus route passing through all intersections can have?
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Context:
设等边三角形 $A B C$ 的内切圆半径为 2, 圆心为 $I$. 若点 $P$ 满足 $P I=1$, 则 $\triangle A P B$ 与 $\triangle A P C$ 的面积之比的最大值为 \frac{m+\sqrt{n}}{k} 的形式，请给出 m + n + k 的值。
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Context:
Let $m$ and $n$ be positive integers such that
$$(m^3 -27)(n^3- 27) = 27(m^2n^2 + 27):$$
Find the maximum possible value of $m^3 + n^3$.
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Context:
Find the smallest value of \( n \) such that given any \( n \) distinct integers, it is always possible to find 4 different integers \( a, b, c, d \) such that \( a + b \equiv c + d \mod 20 \).
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Context:
Evaluate the integral $$\oint_{\verts{z}\ =\ 3} {\dd z \over \root{z^{2} - 1}}$$ using a branch of the function that is analytic outside the interval $[-1.1]$ and positive on $(1,\infty)$.
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Context:
Gunner has a box of mittens with 20 fewer pairs than a box of plugs. He decides to put 30 more pairs of plugs into the box of plugs. If there are now 400 plugs, how many pairs of mittens are in the box of mittens?
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Context:
Solve the equation: $$\begin{aligned}|x+1|-|x|+3|x-1|-2|x-2|=x+2.\end{aligned}$$ Please provide the sum of the minimum and maximum solutions.
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Context:
Consider the recurrence system for an integer $k$ where $1\le k\le \frac{p-1}{2}$, given by:
\[ c_k = \frac{1}{p}\binom{p}{k} - c_{k-1}, \]
with the initial condition $c_0 = 0$. Here, $p$ is an odd prime. The goal is to find a closed-form solution for $c_k$ in terms of elementary functions. It has been observed that $c_k$ can be expressed as a sum:
\[ c_k = \sum_{i=0}^{k-1} \frac{(-1)^i}{p} \binom{p}{k-i}. \]
The task is to either simplify this sum into a closed-form expression or directly solve the recurrence system to obtain a closed-form solution. What is the closed-form solution for $c_k$?
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Context:
In 2021, a first-class postage stamp cost 85 pence and a second-class postage stamp cost 66 pence. What is the smallest total number of stamps that should be purchased in order to spend an exact number of pounds, given that at least one of each type of stamp must be bought? Express your answer as a single integer.
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Context:
Gilbert is an enthusiastic gardener who loves growing herbs in his yard to use in his cooking. At the beginning of the spring, he planted eight different types of herbs: three basil bushes, one parsley plant, two types of mint, one rosemary plant, and one thyme plant. Each herb has a different growth rate: basil doubles every two weeks, parsley grows steadily with one new plant each week, mint triples each month, rosemary and thyme stay the same.

Halfway through spring, which was six weeks long, one of the basil plants bloomed flowers and dropped seeds, resulting in an extra basil plant in his garden. Unfortunately, during the same week, weather damage ruined all the mint plants, so they didn't grow any further for the rest of the season. Also, in the fifth week, an infestation of caterpillars devoured two of the parsley plants. By the time spring ended, how many herb plants did Gilbert have in his garden?
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Context:
Masha wrote on the board, in ascending order, all the natural divisors of a certain number \( N \) (the first divisor is 1, and the largest divisor is the number \( N \) itself). It turned out that the third-to-last divisor is 21 times larger than the second divisor. What is the largest possible value of \( N \)?
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Context:
Each unit square of a $4 \times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color?
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Context:
曲线 $C$ 的极坐标方程是 $\rho=1+\cos \theta$, 点 $A$ 的极坐标是 $(2,0)$, 曲线 $C$ 在它所在的平面内绕 $A$ 旋转一周，则它扫过的图形的面积是。原始答案为$\frac{m}{n}\pi$的形式，请给出m + n的值。
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Context:
In the Cartesian coordinate system, the vertices of rhombus $OABC$ are $O(0,0)$ and $A(1,1)$, and $\overrightarrow{OA} \cdot \overrightarrow{OC} = 1$. Then, $\overrightarrow{AB} \cdot \overrightarrow{AC}$ equals \_\_\_\_\_\_.
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Context:
A fruit stand is selling apples for $2 each, bananas for $1 each, and oranges for $3 each. Emmy has $200 while Gerry has $100. They notice that if they buy 10 apples, they get a 20% discount on the apple price. Both Emmy and Gerry want to buy a mix of these three fruits within their budgets, and they want to ensure they have at least 5 of each fruit. Their goal is to maximize the number of apples they buy, considering the discount and the minimum quantity constraint for other fruits. How many apples can Emmy and Gerry buy altogether at most?
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Context:
The expression $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. Find the value of $$\left\lfloor\frac{2002!}{2001!+2000!+1999!+\cdots+1!}\right\rfloor.$$
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Context:
At the bus stop near Absent-minded Scientist's house, buses of two routes stop: No. 152 and No. 251. Both go to the metro station. The interval between buses No. 152 is exactly 5 minutes, and the interval between buses No. 251 is exactly 7 minutes. The intervals are strictly adhered to, but these two routes are not coordinated between each other and their schedules do not depend on each other. At a completely random moment in time, the Absent-minded Scientist comes to the bus stop and gets on the first bus that arrives to get to the metro. What is the probability that the Scientist will get on bus No. 251?
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Context:
Let $m$ and $n$ be two odd integers with $m, n \geqslant 3$. In an $m \times n$ grid, each small square is colored either blue or red. We denote $A$ as the number of rows where blue squares are in the majority, and $B$ as the number of columns where red squares are in the majority.

What is the largest possible value of $A + B$?
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Context:
Given vectors $\vec{a}$ and $\vec{b}$ satisfy $|\vec{a}|=2$, $|\vec{b}|=1$, and $|\vec{a}-2\vec{b}|=2\sqrt{3}$, then the angle between $\vec{a}$ and $\vec{b}$ is __________.
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Context:
Convex quadrilateral $ABCD$ satisfies the following conditions: \( \angle{CAB} = \angle{ADB} = 30^{\circ} \), \( \angle{ABD} = 77^{\circ} \), \( BC = CD \), and \( \angle{BCD} = n^{\circ} \) for some positive integer \( n \). Compute \( n \).
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Context:
Given that point \( P \) lies on the hyperbola \(\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1\), and the distance from \( P \) to the right directrix of this hyperbola is the arithmetic mean of the distances from \( P \) to the two foci of this hyperbola, find the x-coordinate of \( P \).
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Context:
Let $ABC$ be a triangle with $AB = 6$, $AC = 8$, $BC = 7$. Let $H$ be the orthocenter of $\triangle ABC$. Let $D \neq H$ be a point on $\overline{AH}$ such that $\angle HBD = \frac{3}{2} \angle CAB + \frac{1}{2} \angle ABC - \frac{1}{2} \angle BCA$. Find $DH$.
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Context:
设数列$\left\{ {{a}_{n}} \right\}$满足${{a1}}=\frac{2}{3}$，且对任意的$n\in {{N}^{*}}$，满足${{a}_{n+2}}-{{an}}\le {{2n}}$，${{a}_{n+4}}-{{an}}\ge 5\times {{2}^{n}}$,则${{a}_{2017}}=$_________
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Context:
Given a sequence $\{a_{n}\}$ that satisfies $a_{n+1}+(-1)^{n}a_{n}=2n-1$, then the sum of the first $40$ terms of $\{a_{n}\}$ is ______.
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Context:
Six lines are drawn in the plane. Determine the maximum number of points through which at least three lines pass.
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Context:
In the arithmetic sequence {a_n}, a_5 > 0 and a_4 + a_7 < 0. Find the term S_n in the sum of the first n terms of {a_n} that corresponds to the maximum sum. Express your answer in the form S_n, specifying the value of n.
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Context:
On an $11 × 11$ checker-board 22 squares are marked so that exactly two of the marked squares lie in each column and each row. Two arrangements of marked squares are considered equivalent if in any number of permutations of the columns and/or (independent) permutations of rows one arrangement can be obtained from the other. How many nonequivalent arrangements of marked squares are there?
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Context:
The average of 20, 40, and 60 is 5 more than the average of some numbers and 35. If the other numbers are 60 and a third number, what is the first number?
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Context:
A $8 \times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maximum value of $k$.
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Context:
Let $a$ and $b$ be real constants such that
\[x^4 + ax^3 + 3x^2 + bx + 1 \ge 0\]for all real numbers $x.$  Find the largest possible value of $a^2 + b^2.$
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Context:
An equilateral triangle \( \triangle ABC \) has each of its sides divided into four equal parts. Through each division point, lines parallel to the other two sides of the triangle are drawn. The 15 intersection points of these lines, including the vertices of \( \triangle ABC \), are called lattice points. Out of these 15 lattice points, find the minimum number \( n \) such that among any \( n \) chosen points, there are always three points that form an isosceles triangle (including an equilateral triangle).
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Context:
How many solutions are there to the equation $x^2(x+1)^2\equiv 0\pmod{1000}$?
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Context:
Mike, Ted, and Sarah planted tomatoes and peppers. In the morning, Mike planted 50 tomato seeds and 30 pepper seeds, while Ted planted twice as many tomato seeds as Mike and half as many pepper seeds. Sarah planted 30 seeds more than Mike for both tomatoes and peppers. In the afternoon, Mike planted 60 tomato seeds and 40 pepper seeds, while Ted planted 20 fewer tomato seeds than Mike and the same number of pepper seeds. Sarah planted 20 more tomato seeds and 10 more pepper seeds in the afternoon than she did in the morning. How many tomato and pepper seeds did they plant altogether?
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Context:
From the vertex of the obtuse angle $A$ of triangle $ABC$, a perpendicular $AD$ is drawn. A circle with center $D$ and radius $DA$ is drawn, which intersects sides $AB$ and $AC$ at points $M$ and $N$ respectively. Find $AC$ if $AB = c$, $AM = m$, and $AN = n$.
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Context:
On a table lie two spheres that touch each other externally. A cone touches the table with its lateral surface as well as both spheres (externally). The vertex of the cone is on the segment connecting the points where the spheres touch the table. It is known that the rays connecting the vertex of the cone with the centers of the spheres form equal angles with the table. Find the maximum possible angle at the vertex of the cone. (The angle at the vertex of the cone is defined as the angle between its generatrices in the axial cross-section.)
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Context:
How many 9-digit numbers are there in the decimal system that are divisible by 11 and in which every digit except zero occurs?
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Context:
Find \( g(2021) \) if for any real numbers \( x \) and \( y \) the following equality holds:
\[ g(x-y) = 2021(g(x) + g(y)) - 2022xy \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $x$ cows give $x+1$ cans of milk in $x+2$ days, how many days will it take $x+3$ cows to give $x+5$ cans of milk?
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Context:
6. Through the midpoints of sides $A B$ and $A D$ of the base of a regular quadrilateral pyramid $S A B C D$, a plane is drawn parallel to the median of the lateral face $S D C$, drawn from vertex $D$. Find the area of the section of the pyramid by this plane, if the side of the base of the pyramid is 2, and

the lateral edge is 6.
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Context:
A person moved along a slope with a ratio of 1:2 and traveled 100√5 meters. Calculate the increase in the person's position. Express your answer in meters.
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Context:
An outlet pipe A empties a tank which is full in 10 hours. There is also another outlet pipe B with a different emptying rate. If the inlet pipe is kept open, which lets water in at the rate of 16 litres/min, and both outlet pipes A and B are working together, the tank would take 8 hours longer to be emptied. Find the capacity of the tank.
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Context:
Somewhere in the universe, $n$ students are taking a 10-question math competition. Their collective performance is called laughable if, for some pair of questions, there exist 57 students such that either all of them answered both questions correctly or none of them answered both questions correctly. Compute the smallest $n$ such that the performance is necessarily laughable.
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Context:
We multiply the number $1234567 . . . 1000$ (juxtaposed are all natural numbers 1 to 1000) by a number from 1 to 9, and strike out all 1’s in the product. We multiply the number obtained by a nonzero one-digit number once again, and strike out the 1’s, and so on, many times over. What is the least number one can obtain in this manner?
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Context:
A TV station is broadcasting 6 advertisements in a row, including 3 different commercial ads, 2 different Olympic promotional ads, and 1 public service ad. The last ad cannot be a commercial ad, and the Olympic promotional ads cannot be played consecutively with each other or with the public service ad. How many different broadcasting methods are there?
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Context:
Find all naturals $k$ such that $3^k+5^k$ is the power of a natural number with the exponent $\ge 2$.
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Context:
The center of one sphere is on the surface of another sphere with an equal radius. How does the volume of the intersection of the two spheres compare to the volume of one of the spheres?
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Context:
Given \(a, b, c \in (0, 1]\) and \(\lambda \) is a real number such that
\[ \frac{\sqrt{3}}{\sqrt{a+b+c}} \geqslant 1+\lambda(1-a)(1-b)(1-c), \]
find the maximum value of \(\lambda\).
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Context:
Let \( A, B, C, D \) be four points in space that are not coplanar. Each pair of points is connected by an edge with a probability of \( \frac{1}{2} \), and whether or not there is an edge between any two pairs of points is independent of the others. Determine the probability that \( A \) and \( B \) can be connected by a path (composed of one or multiple edges).
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Context:
Find all pairs  $(m, n)$  of positive integers, for which number  $2^n - 13^m$  is a cube of a positive integer.

*Proposed by Oleksiy Masalitin*
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Context:
The phrase "COLORFUL TARTAN'' is spelled out with wooden blocks, where blocks of the same letter are indistinguishable. How many ways are there to distribute the blocks among two bags of different color such that neither bag contains more than one of the same letter?
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Context:
During the 2011 Universiade in Shenzhen, a 12-person tour group initially stood in two rows with 4 people in the front row and 8 people in the back row. The photographer plans to keep the order of the front row unchanged, and move 2 people from the back row to the front row, ensuring that these two people are not adjacent in the front row. Calculate the number of different ways to adjust their positions.
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Context:
Petya cut an 8x8 square along the borders of the cells into parts of equal perimeter. It turned out that not all parts are equal. What is the maximum possible number of parts he could get?
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Context:
实系数方程 $x^{4}+a x^{3}+b x^{2}+c x+d=0$ 的根都不是实数，其中两个根的和为 $2+i$，另两根的积为 $5+6i$，求 $b$ 的值。
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Context:
Let $m \in \mathbb{R}$. A moving line passing through a fixed point $A$ is given by $x+my=0$, and a line passing through a fixed point $B$ is given by $mx-y-m+3=0$. These two lines intersect at point $P(x, y)$. Find the maximum value of $|PA|+|PB|$.
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Context:
Given the conditions $p$: {$x ||x-a| < 3 $}, and $q$: {$x |x^{2}-2x-3 < 0 $}, and $\neg p$ is a necessary but not sufficient condition for $\neg q$, find the range of values for $a$.
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Context:
From 1812 identical squares with a side length of 1 mm, a rectangular frame was made for a group photo. The boundaries of the photo coincide with the inner boundaries of the frame. Then the photo was cut along the millimeter grid line into two rectangular parts. Now, two frames were needed, which used up 2018 of the same squares. Find the dimensions of the original photo.
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Context:
Naeun, Leejun, Yoonseo, Seohyun, and Soyul are interviewed in order. When Yoonseo and Seohyeon switched, who will be the 4th person being interviewed?
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Context:
Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\heartsuit$ and the second card dealt is a $\clubsuit$?
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Context:
Generalize the cube-octahedron-cube problem to an n-dimensional hypercube. What is the ratio of the side lengths of the innermost and outermost hypercubes in terms of n?
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Context:
Calculate: $\frac{145}{273} \times 2 \frac{173}{245} \div 21 \frac{13}{15}=$
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Context:
In a plane, there is a square \(ABCD\) with side length 1 and a point \(X\). It is known that \(XA = \sqrt{5}\) and \(XC = \sqrt{7}\). What is the length of \(XB\)?
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Context:
Let \( N \) be the set of all positive integers, and let \( f \) be a function from \( N \) to \( N \) such that for any \( s \) and \( t \) in \( N \), the following condition is satisfied:
$$
f\left(t^{2} f(s)\right)=s(f(t))^{2}.
$$

Determine the minimum possible value of \( f(1998) \) among all such functions \( f \).
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Context:
Xiaoming has 4 identical coins, each with a head and a tail. He wants to stack the 4 coins in such a way that the heads of adjacent coins do not face each other. Find the number of different stacking methods. Express your answer as a whole number.
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Context:
Let $x$ and $y$ be real numbers such that $x + y = 3.$  Find the maximum value of
\[x^4 y + x^3 y + x^2 y + xy + xy^2 + xy^3 + xy^4.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Find the largest integer $k$ such that:
\[
k \leq \sqrt{2} + \sqrt[3]{\frac{3}{2}} + \sqrt[4]{\frac{4}{3}} + \sqrt[5]{\frac{5}{4}} + \cdots + \sqrt[2015]{\frac{2015}{2014}}.
\]
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Context:
Two trains of different lengths run at the speed of 60 kmph and 48 kmph in opposite directions in parallel tracks. The time they take to cross each other is 9.99920006399488 seconds. If the other train is 160 m long, how long is the first train?
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Context:
袋子 $A$ 中装有 2 张 10 元纸币和 3 张 1 元纸币，袋子 $B$ 中装有 4 张 5 元纸币和 3 张 1元纸币。现随机从两个袋子中各取出两张纸币，则 $A$ 中剩下的纸币面值之和大于 $B$ 中剩下的纸币面值之和的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(f(x)\) be an odd function defined on \(\mathbb{R}\), and \(f(2)=0\). When \(x > 0\), it always holds that \(xf''(x)-f(x) < 0\). Find the solution set of the inequality \(x^{2}f(x) > 0\). Express your answer in interval notation.
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Context:
3. What is the minimum number of 3-cell corners that need to be painted in a $5 \times 5$ square so that no more corners can be painted? (Painted corners should not overlap.)
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Context:
Three equally spaced parallel lines intersect a circle, creating three chords of lengths 38, 38, and 34. What is the distance between two adjacent parallel lines?
Please reason step by step, and put your final answer within \boxed{}.


Context:
11. Among the positive integers $1,2, \cdots, 20210418$, how many numbers have a digit sum of 8?
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Context:
On a circular railway, there are \( n \) stations. At times, the station attendants communicate with each other via radio. At any given moment, the communication session is held by only two people. Over the course of a day, there was exactly one radio session between each pair of stations. For each station (considering only the sessions it engaged in), it turned out that it communicated with other stations alternately in the order of their arrangement on the railway (either clockwise or counterclockwise, with different directions for different stations), starting with one neighboring station and ending with the other. What could \( n \) be equal to?
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Context:
Let $a, b, c, d, e$ be nonnegative integers such that $625 a+250 b+100 c+40 d+16 e=15^{3}$. What is the maximum possible value of $a+b+c+d+e$ ?
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Context:
Given the ellipse $\frac{x^2}{3} + y^2 = 1$, let line $l$ intersect the ellipse at points $A$ and $B$. The distance from the coordinate origin $O$ to line $l$ is $\frac{\sqrt{3}}{2}$. Determine the maximum area of $\triangle AOB$.
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Context:
How many polynomials of degree exactly $5$ with real coefficients map the set \( \{1, 2, 3, 4, 5, 6\} \) to a permutation of itself?
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Context:
When a swing is stationary, with the footboard one foot off the ground, pushing it forward two steps (in ancient times, one step was considered as five feet) equals 10 feet, making the footboard of the swing the same height as a person who is five feet tall, determine the length of the rope when pulled straight at this point.
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Context:
Given four points in space that are not coplanar, how many planes are there from which all four points are equidistant?
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Context:
已知边长为 $a$ 的正方形 $A B C D$ 的顶点 $A$ 在平面 $\beta$ 内，其余顶点均在平面 $\beta$ 的同侧，且 $B 、 D$ 两点到平面 $\beta$ 的距离分别为 $1 、 2$ 。若平面 $A B C D$ 与平面 $\beta$ 的夹角为 $30^{\circ}$, 则 $a=$
$\qquad$.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:
A number is supposed to be multiplied by 9 and then squared, but instead it is divided by 9 and then the square root is taken. What is the percentage of error obtained in the final result?
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Context:
We divided a chord $AB$ of a circle, which has the length of the radius, and the shorter arc $AB$ into three equal parts each. The line connecting the division points closer to $A$ intersects the line connecting the division points closer to $B$ at point $P$. What is the measure of angle $APB$?
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Context:
Given a regular tetrahedron $\mathrm{ABCD}$ with a point $\mathrm{P}$ inside it such that $P A=P B=\sqrt{11}$ and $P C=P D=\sqrt{17}$. What is the side length of the tetrahedron $\mathrm{ABCD}$?
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Context:
Let x1, x2, x3,..., x9 be distinct positive integers such that x1 < x2 <... < x9 and their sum is 220. When the value of x1 + x2 +... + x5 is maximized, find the minimum value of x9 - x1. Express your answer as a single integer.
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Context:
Compute the norm of the functional $f \in (W^{2,1}[0,1])^*$ defined by
$$f(u) = \int_0^{1/2} u(t)\,dt.$$
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Context:
There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is the value of $N$?
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Context:
John and Ingrid pay 30% and 40% tax annually, respectively. However, John is eligible for a $2000 tax reduction and Ingrid has an exemption of 15% on her total income before taxation. If John's annual income is $58000 and Ingrid's is $72000, what is their combined tax rate after considering these deductions and exemptions?
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Context:
Suppose you have $27$ identical unit cubes colored such that $3$ faces adjacent to a vertex are red and the other $3$ are colored blue. Suppose further that you assemble these $27$ cubes randomly into a larger cube with $3$ cubes to an edge (in particular, the orientation of each cube is random). The probability that the entire cube is one solid color can be written as $\frac{1}{2^n}$ for some positive integer $n$. Find $n$.
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Context:
The sequence $\left(z_{n}\right)$ of complex numbers satisfies the following properties: $z_{1}$ and $z_{2}$ are not real. $z_{n+2}=z_{n+1}^{2} z_{n}$ for all integers $n \geq 1$. $\frac{z_{n+3}}{z_{n}^{2}}$ is real for all integers $n \geq 1$. $\left|\frac{z_{3}}{z_{4}}\right|=\left|\frac{z_{4}}{z_{5}}\right|=2$ Find the product of all possible values of $z_{1}$.
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Context:
Given a positive integer $n (n>2004)$, we put 1, 2, 3, …,$n^2$ into squares of an $n\times n$ chessboard with one number in a square. A square is called a “good square” if the square satisfies following conditions:
1) There are at least 2004 squares that are in the same row with the square such that any number within these 2004 squares is less than the number within the square.
2) There are at least 2004 squares that are in the same column with the square such that any number within these 2004 squares is less than the number within the square.
Find the maximum value of the number of the “good square”.
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Context:
A store has buttons in six different colors. What is the minimum number of buttons that need to be bought in order to sew them in a row such that for any two different colors in the row, there are two adjacent buttons of those colors?
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Context:
If the equation $\left(a-3\right)x^{2}-4x+1=0$ with respect to $x$ is a quadratic equation, then the range of values for $a$ is ______.
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Context:
In how many different ways can the faces of a cube be painted using six given colors (each face must be painted entirely with one color), considering that only those colorings are different which cannot be made identical by rotating the cube?
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Context:
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?
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Context:
Two hexagons are attached to form a new polygon \( P \). Compute the minimum number of sides that \( P \) can have.
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Context:
Given the inequality $(x-a+2)(x^2-ax-2) \geq 0$ for $x > 0$, find the exact value of the real number $a$.
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Context:
In a city, there are $2019$ metro stations. Some pairs of stations are connected by tunnels, and from any station, you can reach any other station through these tunnels. The mayor ordered the organization of several metro lines: each line should include several different stations connected in series by tunnels (several lines can pass through the same tunnel), and each station must be on at least one line. To save money, no more than $k$ lines should be made. It turned out that the mayor's order is not feasible. What is the largest value of $k$ for which this could happen?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the infinite geometric series:
$$\frac{5}{3} - \frac{5}{4} + \frac{25}{48} - \frac{125}{384} + \dots$$
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Context:
In a right square prism $A B C D A_{1} B_{1} C_{1} D_{1}$, the height is half the length of the side of the base. Find the maximum value of the angle $A_{1} M C_{1}$, where $M$ is a point on the edge $A B$.
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Context:
On an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the same route, the reading was $440$ miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was $15$ inches. If the answer is $x$ inches, what is the value of $100x$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the expected value of the number of maximal decrescent blocks in a random permutation of $\{1, \dots, n\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
At 7:00, five sheep, designated as A, B, C, D, and E, have distances to Wolf Castle forming an arithmetic sequence with a common difference of 20 meters. At 8:00, these same five sheep have distances to Wolf Castle forming another arithmetic sequence, but with a common difference of 30 meters, and their order has changed to B, E, C, A, D. Find how many more meters the fastest sheep can run per hour compared to the slowest sheep.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and $f(2) = 0$. If for $x > 0$, $xf'(x) + f(x) > 0$, then the solution set of the inequality $xf(x) > 0$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A company manufactures a brand of light bulb with a lifetime in months that is normally distributed with mean 3 and variance 1. A consumer buys a number of these bulbs with the intention of replacing them successively as they burn out. The light bulbs have independent lifetimes. What is the smallest number of bulbs to be purchased so that the succession of light bulbs lasts for at least 40 months with probability at least 0.9772?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, segment $AD$ is the angle bisector, $AD = l$, $AB = c$, and $AC = b$. Find angle $A$.
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Context:
Given four points \( A, B, C, D \) on the surface of a unit sphere, such that \( AB = AC = AD \) and \( BC = CD = DB \), find the maximum volume of the tetrahedron \( ABCD \).
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Context:
of the 80 people in a room , 2 / 5 are women . if 1 / 2 of the people are married , what is the maximum number of women in the room who could be unmarried ?
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Context:
In an exam, 153 people scored no more than 30 points, with an average score of 24 points. 59 people scored no less than 80 points, with an average score of 92 points. The average score of those who scored more than 30 points is 62 points. The average score of those who scored less than 80 points is 54 points. How many people participated in this exam?
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Context:
Consider two sequences \(a_n\) and \(b_n\) where \(a_n, b_n > 0\). If \(a_n \sim b_n\) and both \(\sum a_n\) and \(\sum b_n\) diverge, determine whether it implies that \(\lim_{N\rightarrow\infty} \frac{\sum_{n\leq N} a_n}{\sum_{n\leq N} b_n} = 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In Ms. Smith's class, each student averages one day absent out of thirty. What is the probability that out of any two students chosen at random, one student will be absent while the other is present? Express your answer as a percent rounded to the nearest tenth.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine if $x = 2$ is the only real solution to the equation $a^x + b^x = c^x$ when $(a, b, c)$ is a Pythagorean triplet.
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Context:
Given a finitely generated module $M$ over a commutative ring $A$, suppose there exist a positive integer $n$ and homomorphisms $\pi: A^n \rightarrow M$ (surjective) and $\phi: A^n \rightarrow M$ (injective). Does it follow that $M \cong A^n$?
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Context:
Determine whether the limit of the series \( \sum_{k \ge 0} \frac{\Gamma(j)}{\Gamma(j+k/2)}(-t)^k \) approaches zero as \( t \to \infty \), where \( j \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A hollow glass sphere with uniform wall thickness and an outer diameter of $16 \mathrm{~cm}$ floats in water in such a way that $\frac{3}{8}$ of its surface remains dry. What is the wall thickness, given that the specific gravity of the glass is $s = 2.523$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The length of a rectangle is 5 less than 3 times its width. Find the dimensions of the rectangle if the area is 782 m².
Please reason step by step, and put your final answer within \boxed{}.

\boxed{46 \text{ m} \times 17 \text{ m}}


Context:
Consider all possible quadrilaterals $ABCD$ that have the following properties; $ABCD$ has integer side lengths with $AB\parallel CD$, the distance between $\overline{AB}$ and $\overline{CD}$ is $20$, and $AB = 18$. What is the maximum area among all these quadrilaterals, minus the minimum area?

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


Context:
If altitude $CD$ is $\sqrt3$ centimeters, what is the number of square centimeters in the area of $\Delta ABC$?

[asy] import olympiad; pair A,B,C,D; A = (0,sqrt(3)); B = (1,0);
C = foot(A,B,-B); D = foot(C,A,B); draw(A--B--C--A); draw(C--D,dashed);
label("$30^{\circ}$",A-(0.05,0.4),E);
label("$A$",A,N);label("$B$",B,E);label("$C$",C,W);label("$D$",D,NE);
draw((0,.1)--(.1,.1)--(.1,0)); draw(D + .1*dir(210)--D + sqrt(2)*.1*dir(165)--D+.1*dir(120));
[/asy]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a quadrilateral with $\overline{AB}\parallel\overline{CD}$, $AB=16$, $CD=12$, and $BC<AD$. A circle with diameter $12$ is inside of $ABCD$ and tangent to all four sides. Find $BC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Of the four points $(2,2)$, $(9,11)$, $(5,7)$, and $(11,17)$, three lie on the same line. Which point is not on the line?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of zeros of the polynomial \( p(z) = 1 + 2z^4 + \frac{7}{10}z^{10} \) that lie within the unit disc, and specify their multiplicities.
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Context:
Quadrilateral ABCD is a cyclic quadrilateral inscribed in circle O with diameter BC. Point D is the midpoint of arc AC. Given that AB = 2 and AD = 2√3, find the area of circle O. Express your answer in terms of π as a product of a numerical value and π.
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Context:
Given a quadratic polynomial \(P(x,y)=ax^2+bxy+cy^2+ex+fy+g\) with integer coefficients, where \((a,b,c)=1\), determine if there exists an equivalent polynomial \(P'(x,y)=a'x^2+b'xy+c'y^2+e'x+f'y+g'\) such that \((b',a'c')=1\). Answer 'yes' or 'no'.
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Context:
In a certain company, there are 100 shareholders, and any group of 66 of them collectively owns at least 50% of the company's shares. What is the maximum percentage of all shares that one shareholder can own? (The percentage of the company's shares owned by a shareholder can be a non-integer value.)
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Context:
Kelvin the Frog writes 2020 words on a blackboard, with each word chosen uniformly randomly from the set $\{\verb|happy|, \verb|boom|, \verb|swamp|\}$. A multiset of seven words is merry if its elements can spell $``\verb|happy happy boom boom swamp swamp swamp|."$ For example, the eight words
\[\verb|swamp|, \verb|happy|, \verb|boom|, \verb|swamp|, \verb|swamp|, \verb|boom|, \verb|swamp|, \verb|happy|\]contain four merry multisets. Determine the expected number of merry multisets contained in the words on the blackboard.

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Context:
Jeremy wrote all the three-digit integers from 100 to 999 on a blackboard. Then Allison erased each of the 2700 digits Jeremy wrote and replaced each digit with the square of that digit. Thus, Allison replaced every 1 with a 1, every 2 with a 4, every 3 with a 9, every 4 with a 16, and so forth. The proportion of all the digits Allison wrote that were ones is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
There are many CDs in the store. The rock and roll CDs are $7 each, the pop CDs are $12 each, the dance CDs are $5 each, and the country CDs are $9 each. If Julia buys 3 or more of each genre, she gets a 15% discount on that genre. She wants to buy 5 rock and roll CDs, 3 pop CDs, 6 dance CDs, and 4 country CDs, but some genres have limited availability: there are only 4 rock and roll CDs left, and only 5 dance CDs left. Julia only has 80 dollars. How much money is she short?
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Context:
Three people, A, B, and C, visit three tourist spots, with each person visiting only one spot. Let event $A$ be "the three people visit different spots," and event $B$ be "person A visits a spot alone." Then, the probability $P(A|B)=$ ______.
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Context:
Calculate the number of different spanning trees for the simple graph $G_{k,n}=([n],E)$, where $ij\in E\Leftrightarrow 0 <|i-j|\leq k$ for $i\neq j\in [n]$. Specifically, find the number of spanning trees when $k=3$ and $n=5$. Use Kirchoff's Matrix Tree theorem to verify your result.
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Context:
At an international table tennis championship, 200 participants arrived. The tournament follows a single-elimination format, meaning that in each match two players compete, the loser is eliminated from the championship, and the winner remains. Find the maximum possible number of participants who have won at least three matches.
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Context:
Find all positive integers \( a,b,c \) and prime \( p \) satisfying that
 \( {2}^{a}{p}^{b} = {\left( p + 2\right) }^{c} + 1. \)
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Context:
Can the Gaussian integral \( \int_{-\infty}^{\infty} e^{-x^2}\,dx \) be evaluated using contour integration techniques from complex analysis?
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Context:
A man named Juan has three rectangular solids, each having volume 128. Two of the faces of one solid have areas 4 and 32. Two faces of another solid have areas 64 and 16. Finally, two faces of the last solid have areas 8 and 32. What is the minimum possible exposed surface area of the tallest tower Juan can construct by stacking his solids one on top of the other, face to face? (Assume that the base of the tower is not exposed).
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Context:
Let $P_n$ be the number of permutations $\pi$ of $\{1,2,\dots,n\}$ such that\[|i-j|=1\text{ implies }|\pi(i)-\pi(j)|\le 2\]for all $i,j$ in $\{1,2,\dots,n\}.$  for $n\ge 2,$ the quantity\[P_{n+5}-P_{n+4}-P_{n+3}+P_n\]does not depend on $n,$  find its value.
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Context:
In triangle $ABC$, the angle bisector $AM$ and the median $BN$ intersect at point $O$. It turns out that the areas of triangles $ABM$ and $MNC$ are equal. Find $\angle MON$.
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Context:
Given $n=2020$, sort the 6 values: $$n^{n^2},\, 2^{2^{2^n}},\, n^{2^n},\, 2^{2^{n^2}},\, 2^{n^n},\, \text{and}\, 2^{n^{2^2}}$$ from least to greatest. Provide your answer as a 6-digit permutation of the string "123456", where the number $i$ corresponds to the $i$-th expression in the list, from left to right.
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Context:
A retailer sold a watch at a loss of 10% in USD. On the same day, they sold an identical watch for a gain of 4% in EUR. If they had sold the first watch for 200 USD more, there would have been a gain of 4% in USD as well. On that particular day, the exchange rate was changing: the USD was depreciating at a rate of 5% in a 24-hour period, while the EUR was appreciating at a rate of 3% during the same time frame. Assume the watches were sold at exactly noon, halfway through the 24-hour period, and that the exchange rate changes are linear. Calculate the original combined cost price of both watches in USD and EUR at the time of purchase according to these parameters.
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Context:
The current birth rate per certain number of people is unknown, whereas the corresponding death rate is 16 per the same number of people. If the net growth rate in terms of population increase is 1.2 percent, and there are initially 3000 persons, what is the birth rate per certain number of people?
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Context:
In the convex polygon $M N K L P$, the segment $N L$ bisects each of the angles $K N P$ and $K L M$, and the segment $K P$ bisects each of the angles $M K L$ and $N P L$. The diagonal $N P$ intersects the diagonals $M K$ and $M L$ at points $F$ and $E$, respectively. Determine if $K F = L E$.
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Context:
 Assuming that each distinct arrangement of the letters in $DISCUSSIONS is equally likely to occur, what is the probability that a random arrangement of the letters in $DISCUSSIONS has all the $S$’s together?

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Context:
Teddy owns a square patch of desert. He builds a dam in a straight line across the square, splitting the square into two trapezoids. The perimeters of the trapezoids are$ 64$ miles and $76$ miles, and their areas differ by $135$ square miles. Find, in miles, the length of the segment that divides them.

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Context:
Given that $4^{-1} \equiv 57 \pmod{119}$, find $64^{-1} \pmod{119}$, as a residue modulo 119. (Give an answer between 0 and 118, inclusive.)
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Context:
18. A circle with center $C$ and radius $r$ intersects the square $E F G H$ at $H$ and at $M$, the midpoint of $E F$. If $C, E$ and $F$ are collinear and $E$ lies between $C$ and $F$, what is the area of the region outside the circle and inside the square in terms of $r$ ?
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Context:
Determine whether there is a closed-form solution for the nonlinear ordinary differential equation \( y'' = y^3 - y \) with the initial conditions \( y(0) = 1 \) and \( y'(0) = 0 \).
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Context:
Determine whether the sequence \( y_{n} = n\ln(n) - n - (\ln(n))^{2} \) converges or diverges.
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Context:
Determine whether the sequence \( a_{n} = \frac{1}{\sqrt{n^2+n}} + \frac{1}{\sqrt{n^2+n+1}} + \ldots + \frac{1}{\sqrt{n^2+2n-1}} \) converges to a limit.
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Context:
A Gauss brand light bulb will work for 24999 hours. If it is used for exactly 2 hours every day starting on a Monday, on what day of the week will it stop working? Express your answer as the name of the day of the week.
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Context:
Consider the function \( f(x) = \frac{1}{3}(x^2 + x + 1) \). Define \( f_0(x) = x \) and for all \( k \ge 1 \), \( f_k(x) = f(f_{k-1}(x)) \). Let \( a_k = f_k(0) - f_{k-1}(0) \). Determine whether the series \( \sum\limits_{k=1}^{\infty} k a_k \) is infinite.
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Context:
Let $G$ be an infinite subgroup of $(\mathbb{C} \setminus \{0\}, \cdot)$ such that $G$ is compact as a subset of $\mathbb{C}$. Determine whether it is true that $G = S^1$.
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Context:
Let $a\in\mathbb R$, define a sequence $a_n$ as the distance from $a$ to the nearest number of the form $\frac{m}{2^n}$, where $m\in\mathbb Z$. Find the maximal value of the sum $\sum\limits_{n=0}^{\infty}a_n$.
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Context:
From among $2^{1/2},$ $3^{1/3},$ $8^{1/8},$ $9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are?
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Context:
The graphs of the equations
$y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k,$
are drawn in the coordinate plane for $k=-10,-9,-8,\ldots,9,10.\,$ These $63$ lines cut part of the plane into equilateral triangles of side length $\tfrac{2}{\sqrt{3}}.\,$ How many such triangles are formed?
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Context:
Buses leave Moscow for Voronezh every hour, at 00 minutes. Buses leave Voronezh for Moscow every hour, at 30 minutes. The trip between cities takes 8 hours. How many buses from Voronezh will a bus leaving Moscow meet on its way?
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Context:
How many positive integers less than $250$ are multiples of $5$, but not multiples of $10$?
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Context:
Determine if the expression \( \frac{d}{dy}\bigg|_{y = x}\big[[f'(y) - f'(x)]^2\big] \) is defined, given that \( f : S \subseteq \mathbb{R} \to \mathbb{R} \) is differentiable on \( S \) and \( f' \) is also differentiable on \( S \).
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Context:
Ashraful has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Ashraful have?

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Context:
A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?
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Context:
What is the value of

$$\log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27?$$

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Context:
If E is a set with infinite measure and c is a positive constant function defined on E, what is the value of the integral of c over E in terms of c and the measure M(E)? Specifically, how should this be interpreted when M(E) is infinite?
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Context:
$\triangle ABC$ has positive integer side lengths of $x$,$y$, and $17$. The angle bisector of $\angle BAC$ hits $BC$ at $D$. If $\angle C=90^\circ$, and the maximum value of $\frac{[ABD]}{[ACD]}=\frac{m}{n}$ where $m$ and $n$ are relatively prime positive intgers, find $m+n$. (Note $[ABC]$ denotes the area of $\triangle ABC$).

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Context:
Let \( H \) be a Hilbert space with an orthonormal basis \((e_n)_{n \in \mathbb{N}}\) and \( F \) a linear operator such that \( \langle e_k, F e_n \rangle = \phi(n,k) \). Find an estimate for the operator norm \( \lVert F \rVert \) in terms of \( \phi(n,k) \). Apply this estimate to the case where \( \phi(n,k) = \frac{1}{n+k} \).
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Context:
Determine whether the ideal \((I, J)\) is equal to the ideal \((I, J, IJ)\), where \(IJ\) is the product of the ideals \(I\) and \(J\).
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Context:
Consider a probability measure without atoms. Does the probability of a sequence of nested sets approaching an empty set tend to 0?
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Context:
6. The first circle with center at point $O$ is inscribed in triangle $A B C$. Points $A$ and $B$ lie on the second circle with the same center $O$. Line $A C$ intersects the second circle at point $D$ $(D \neq A)$, and line $B C$ intersects the second circle at point $E(B \neq E)$. It is known that angle $A B C$ is equal to angle $C A E$. Find the cosine of angle $B A C$.
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Context:
Evaluate whether the expression \( \epsilon^{\rho\sigma\mu\nu}\epsilon_{\rho\sigma\mu\nu} \) is zero, given that \( \epsilon^{\rho\sigma\mu\nu}\epsilon_{\mu\nu\rho'\sigma'} = -2(\delta^{\rho}_{\rho'}\delta^{\sigma}_{\sigma'}-\delta^\rho_{\sigma'}\delta^{\sigma}_{\rho'}) \).
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Context:
In a cross-country race, each team has 5 players. If a player earns $n$ points for their team by placing $n$-th, the team with fewer points wins. Without any ties, what is the number of possible scores for the winning team?
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Context:
At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of \$50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the \$150 regular price did he save?

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Context:
Find all values of $k$ for which the system
\begin{align*}
x + ky - z &= 0, \\
kx - y - z &= 0, \\
x + y - kz &= 0
\end{align*}has a non-trivial solution.  (In other words, find all values of $k$ for which the system has a solution other than $(x,y,z) = (0,0,0).$)
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Context:
Consider the series \(\sum_{i=1}^\infty c_i^{(0)}\) which is a positive convergent series. Define the series \(\sum_{i=1}^\infty c_i^{(1)} = \sum_{i=1}^\infty \ln(1+c_i^{(0)})\). Generally, let \(c_i^{(N+1)} = \ln(1+c_i^{(N)})\) and define the sum \(S^{(N)} = \sum_{i=1}^\infty c_i^{(N)}\). Determine the nature of the limit \(\lim_{N\rightarrow\infty} S^{(N)}\). Is this limit zero or nonzero?
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Context:
Can the squares of a $1990 \times 1990$ chessboard be colored black or white so that half the squares in each row and column are black and cells symmetric with respect to the center are of opposite color?
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Context:
Determine the asymptotic behavior of the series \( \sum_{k=1}^n \frac{1}{\sqrt{k}} \) and compare it to the series \( \sum_{k=1}^n \frac{1}{k} \), which is approximately \( \log(n) \).
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Context:
Suppose you have a $k \times n$ matrix $A$ where $k \geq n$ and $\mathrm{rank}(A) = m$ with $m \leq n \leq k$. Let $r_1, \dots, r_k$ be the rows of $A$. Consider the $n \times n$ matrix $B = r_1^Tr_1 + \cdots + r_k^Tr_k$. Determine if $\mathrm{rank}(B) = m$.
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Context:
Evaluate the line integral over the curve defined by the intersection of the cone \(z^2 = x^2 + y^2\) and the cylinder \(x^2 + y^2 = 1\).
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Context:
For any integer $r \geq 1$, determine the smallest integer $h(r) \geq 1$ such that for any partition of the set $\{1, 2, \cdots, h(r)\}$ into $r$ classes, there are integers $a \geq  0 \ ; 1 \leq x \leq y$, such that $a + x, a + y, a + x + y$ belong to the same class.
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Context:
Let the roots of
\[z^3 = 2 + 2i\]be $a_1 + ib_1,$ $a_2 + ib_2,$ and $a_3 + ib_3.$  Compute $a_1 a_2 a_3.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
A friendly football match lasts 90 minutes. In this problem, we consider one of the teams, coached by Sir Alex, which plays with 11 players at all times.
 Sir Alex wants for each of his players to play the same integer number of minutes, but each player has to play less than 60 minutes in total. What is the minimum number of players required?
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Context:
Five fair six-sided dice are rolled. What is the probability that at least three of the five dice show the same value?
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Context:
Two trains of different lengths are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. The trains will be clear of each other in 12.998960083193344 seconds from the moment they meet. If the second train is 160 m long, how long is the first train?
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Context:
Folklore
In Chicago, there are 36 gangsters, some of whom are at odds with each other. Each gangster is a member of several gangs, and there are no two gangs with the same membership. It turned out that gangsters who are in the same gang do not feud, but if a gangster is not a member of a certain gang, then he feuds with at least one of its members. What is the maximum number of gangs that could exist in Chicago?
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Context:
27 identical dice were glued together to form a $3 \times 3 \times 3$ cube in such a way that any two adjacent small dice have the same number of dots on the touching faces. How many dots are there on the surface of the large cube?
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Context:
Given that \( A, B, C, D \) are four non-coplanar points on the surface of a sphere, and \( AB = BC = AD = 1 \), \( BD = AC = \sqrt{2} \), and \( BC \perp AD \), find the volume of the sphere.
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Context:
Suppose that $f(x)=x^2-10x+21$. Find all distinct real roots of $f(f(x)+7)$.
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Context:
Calculate the expected number of tosses to observe the first Head when rolling a fair six-sided die. If the die lands on 1 or 2, toss a fair coin (with probability 0.5 of showing Heads) until a Head appears; otherwise, if the die lands on 3 or more, toss a biased coin (with probability 0.25 of showing Heads) until a Head appears. Express the answer correct to two decimal places.
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Context:
Calculate the effect changes in dimension of a rectangle will have on its area, if length is increased by 40% and its breadth is decreased by some percentage. The effect is 5%. What is the percentage decrease in breadth?
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Context:
Evaluate the limit: $$\lim_{n \to 0} \left[\frac{(2020+n)!}{2020!}\right]^{1/n}$$ and provide a numerical approximation for the limit.
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Context:
Determine the domain of the function $f(\lg \frac {x^{2}+x}{2})$, given that the domain of the function $f(x)$ is $(0,1]$. Express your answer in interval notation.
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Context:
在一个圆周上有 12 个红点. 求最小的正整数 n，使得存在以红点为顶点的 n 个三角形，满足以红点为端点的每条弦都是其中某个三角形的一条边.
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Context:
Gary purchased a large beverage, but only drank m/n of it, where m and n are relatively prime positive integers. If he had purchased half as much and drunk twice as much, he would have wasted only 2/9 as much beverage. Find m+n.

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Context:
In triangle $ \triangle ABC $, the sides opposite angles A, B, C are respectively $ a, b, c $, with $ A = \frac{\pi}{4} $, $ \sin A + \sin(B - C) = 2\sqrt{2}\sin 2C $ and the area of $ \triangle ABC $ is 1. Find the length of side $ BC $.
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Context:
Consider the system
$$
x + y = z + u,
$$
$$
2xy = zu.
$$

Find the greatest value of the real constant $ m $ such that
$ m \leq \frac{x}{y} $
for any positive integer solution $(x, y, z, u)$ of the system, with $ x \geq y $.

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Context:
Determine whether the random variable \( Y = aX - \lfloor aX \rfloor \), where \( X \sim U[0,1) \) and \( a \geq 2 \) is an integer, follows a uniform distribution \( U[0,1) \).
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Context:
Vertices $A$ and $B$ of the prism $A B C A 1 B 1 C 1$ lie on the axis of the cylinder, while the other vertices lie on the lateral surface of the cylinder. Find the dihedral angle in this prism with edge $A B$.
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Context:
There are five courses at my school. 243 take algebra. 323 take language arts. 143 take scocial studies. 241 take biology. 300 take history. 213 take algebra and language arts. 264 take algebra and social studies. 144 take algebra and biology. 121 take algebra and history. 111 take language arts and social studies. 90 take language atrts and biology. 80 take language arts and history. 60 take social studies and biology. 70 take social studies and history. 60 take biology and history. 50 take algebra, language arts and social studies. 50 take algebra, language arts and biology. 50 take algebra, language arts and history. 50 take algebra, social studies and biology. 50 take algebra, social studies and history. 50 take algebra, biology and history. 50 take language arts, social studies and biology. 50 take language arts, social studies and history. 50 take language arts, biology and history. 50 take social studies, biology and history. 20 take algebra, language arts, social studies and biology. 15 take algebra, language arts, social studies and history. 15 take algebra, language arts, biology and history. 10 take algebra, social studies, biology and history. 10 take language arts, social studies, biology and history. 5 take all three. None take none.

How many people are in my school?

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Context:
For a positive sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, and there exists a positive number $t$, such that for any positive integer $n$, $$\sqrt {tS_{n}}= \frac {t+a_{n}}{2}$$ holds. If $$\lim_{n\rightarrow +\infty } \frac { \sqrt {S_{n}}}{a_{n}}<t$$, then the range of $t$ is \_\_\_\_\_\_\_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left( \sqrt[3]{\frac {1}{4}}, +\infty \right)}


Context:
In a probability space \((\Omega, \mathcal{A}, P)\), consider a sequence of identically independently distributed (i.i.d.) random variables \((X_n)_{n \in \mathbb{N}}\), which are not integrable. Define \(C_n = \{|X_n| \geq nK\}\) for each \(n \in \mathbb{N}\), where \(K > 0\). Determine whether it is true that \(P\left(\limsup_{n \to \infty} C_n \right) = 1\).
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Context:
Determine which of the following sets is recursively enumerable: \( L_1 = \{ M \mid \text{Turing Machine } M \text{ terminates for at least 637 inputs} \} \) and \( L_2 = \{ M \mid \text{Turing Machine } M \text{ terminates for at most 636 inputs} \} \).
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Context:
Determine whether the function \( f(x,y) = \begin{cases} \frac{x^2y^{\frac{4}{3}}}{x^4+y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} \) is continuous at the point \((0,0)\). Specifically, evaluate whether \( \lim_{(x,y)\to(0,0)} f(x,y) = 0 \).
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Context:
In a trapezoid, the lengths of the parallel sides are \(a\) and \(b\), and the height is \(m\). What is the length of the line parallel to \(a\) and \(b\) that divides the trapezoid into two parts whose areas are in the ratio \(p: q\)?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\sqrt{\frac{pa^2 + qb^2}{p + q}}}


Context:
Determine the radius of convergence of the power series expansion of the function $c(\zeta)$ around the origin, given its implicit equation.
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Context:
If \( \frac{\partial u}{\partial v} = a \), is it true that \( \frac{\partial v}{\partial u} = \frac{1}{a} \)?
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Context:
$\underline{\text { Folklore }}$

Among the actors of Karabas Barabas theater, a chess tournament was held. Each participant played exactly one game with each of the others. One solido was given for a win, half a solido for a draw, and nothing for a loss. It turned out that among any three participants, there would be a chess player who earned exactly 1.5 solido in games with the other two. What is the maximum number of actors that could have participated in such a tournament?
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Context:
There are $10000$ trees in a park, arranged in a square grid with $100$ rows and $100$ columns. Find the largest number of trees that can be cut down, so that sitting on any of the tree stumps one cannot see any other tree stump.

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Context:
Find the minimum length of cable required to connect an antenna to the points A, B, C, and D. The antenna must be placed on the same axis as point A, and its position can be between A and B, or behind either A or B. Use the relationship:

\[ D_1 + D_2 = \sqrt{2^2 + X^2} + \sqrt{(12-x)^2+1^2} \]

where \(D_1\) and \(D_2\) are the distances from the antenna to points A and B, respectively. Determine the optimal position of the antenna to minimize the total cable length.
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Context:
For what value of $x$ does

$$\log_{\sqrt{2}}\sqrt{x}+\log_{2}{x}+\log_{4}{x^2}+\log_{8}{x^3}+\log_{16}{x^4}=40?$$

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Context:
$(\text{b})$ Do there exist 21 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\le p \le 13$?

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Context:
一台压路机的滚筒长1.5米，直径是6米．如果它每分钟转100圈，那么这种压路机每小时可以压路面多少平方米．
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Context:
Determine whether the sequence \(\left(\dfrac{1}{n^3 \cos n}\right)\) converges or diverges. If it converges, find its limit.
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Context:
Determine the largest possible value of the modulus of the entire function $f(z)$ at $z = \ln(2)$, given that $f(z)$ is analytic for all $z \in \mathbb{C}$ and satisfies $|f(z)| \leq |e^z|$ along the contour $|z| = 1$. Can the Maximum Modulus Principle provide a better upper bound than $e$?
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Context:
Consider a random walk $X_t = \sum_{s=1}^t D_s$ with i.i.d. increments $D_t \in \mathbb{R}^n$, where $X$ is a martingale with $\mathbb{E}[D_t]=\vec{0} \in \mathbb{R}^n$. The support of $D_t$ is bounded, and $D_{t,i}$ has strictly positive variance for all $i \in \{1,\ldots,n\}$. Is it true that the probability that the random walk eventually pointwise falls below some threshold $k < 0$ equals $1$, i.e., $$ \mathbb{P}[ \exists t \text{ such that } X_{t,i} \leq k \text{ for all } i \in \{1,\ldots,n\} ] = 1 $$ for all $k<0$?
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Context:
Consider a random permutation of the set $\{1, 2, . . . , 2015\}$. In other words, for each $1 \le i \le 2015$, $i$ is sent to the element $a_i$ where $a_i \in \{1, 2, . . . , 2015\}$ and if $i \neq j$, then $a_i \neq a_j$. What is the expected number of ordered pairs $(a_i, a_j )$ with $i - j > 155$ and $a_i - a_j > 266$?
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Context:
Define a permutation $a_1a_2a_3a_4a_5a_6$ of the set $1, 2, 3, 4, 5, 6$ to be $\text{ factor-hating }$ if $\text{ gcd }(a_k, a_{k+1}) = 1$ for all $1 \leq k \leq 5$. Find the number of $\text{ factor-hating }$ permutations.

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Context:
Find a pair of coprime integers \(x\) and \(y\) such that \((p^y + 1) \bmod (p^x + 1) = 0\) for a given prime number \(p\). If such a pair exists, provide an example.
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Context:
Evaluate the limit of the expression obtained after substituting \( x = \frac{1}{t} \) in the original limit expression, given that \( \frac{1}{2} < \sigma < 1 \).
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Context:
Consider the linear operator $T: l_2 \rightarrow l_2$ defined by \[T(\bar{x}) = (3x_1 + 2x_2, x_3 - 4x_4, 3x_5 + 2x_6, ...)\] where $\bar{x} = (x_1, x_2, x_3, ...,)$ is an element of the space of square-summable sequences $l_2$. Determine whether $T$ is bounded and find bounds for its operator norm $\|T\|$. Provide your answer as a single value or expression for the norm.
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Context:
Determine if the residue theorem can be used to evaluate the integral \( I = \int_0^{\infty} \frac{dx}{x^6+1} \) using a pie-shaped contour subtending an angle of \( \frac{\pi}{3} \).
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Context:
7. In any $m$ consecutive natural numbers, if there must be a number whose sum of digits is a multiple of 6, then the smallest $m$ is

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Context:
Determine whether the function $f \equiv 1$ is the unique normalized solution to the equation \( f(x) = \frac{1}{x} \int_0^x f(t) \, dt \) for \( f \in L^1(0,1) \), where \( f \) is non-negative almost everywhere and satisfies \( \int_0^1 f(x) \, dx = 1 \).
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Context:
Consider a stochastic process defined as \(x(t) = a \sin(2\pi f_0 t) + b \cos(2\pi f_0 t)\), where \(a\) and \(b\) are independent Gaussian random variables with zero mean and variance \(\sigma^2\). Determine if \(x(t)\) is a Gaussian random process and find its mean function.
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Context:
Evaluate the improper integral \(\int_0^\infty \frac{\sin x(1-\cos x)}{x^2}dx\) and determine whether it converges.
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Context:
Given that the probability of winning each bet is \( p = \frac{1}{3} \), find the probability that the gambler first goes broke after 5 bets using the generating function derived in the original answer.
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Context:
A PE class has 12 students, 6 girls and 6 boys. The coach has 4 jerseys in each of 3 colors to mark 3 teams for a soccer tournament. If the coach wants at least one girl and at least one boy on each team, how many ways can he give out the jerseys? (Jerseys of the same color are indistinguishable.)
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Context:
## Task Condition

Calculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4_{\text {and }}}$ its height dropped from vertex $A_{4 \text { to the face }} A_{1} A_{2} A_{3}$.

$A_{1}(1 ; 2 ; 0)$

$A_{2}(3 ; 0 ;-3)$

$A_{3}(5 ; 2 ; 6)$

$A_{4}(8 ; 4 ;-9)$
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Context:
Evaluate the limit: $$\lim_{n\to\infty}n\cdot \sum_{m=1}^{\infty}\left(1-\frac{1}{m}\right)^n\cdot \frac{1}{m^2}.$$ Is this limit equal to 1 as $n$ approaches infinity?
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Context:
Triangle $\vartriangle ABC$ is inscribed in a circle $\omega$ of radius $12$ so that $\angle B = 68^o$ and $\angle C = 64^o$ . The perpendicular from $A$ to $BC$ intersects $\omega$ at $D$, and the angle bisector of $\angle B$ intersects $\omega$ at $E$. What is the value of $DE^2$?

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Context:
NT 3. Find the largest integer $k(k \geq 2)$, for which there exists an integer $n(n \geq k)$ such that from any collection of $n$ consecutive positive integers one can always choose $k$ numbers, which verify the following conditions:

1. each chosen number is not divisible by 6 , by 7 and by 8 ;
2. the positive difference of any two different chosen numbers is not divisible by at least one of the numbers 6,7 or 8 .

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Context:
If the radius of convergence of \(\sum_{n=1}^\infty a_n z^n\) is \(R_1\) and the radius of convergence of \(\sum_{n=1}^\infty b_n z^n\) is \(R_2\), where \(R_1 < R_2\), is the radius of convergence of \(\sum_{n=1}^\infty (a_n - b_n) z^n\) equal to \(R_1\)?
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Context:
Let $\langle x_n\rangle$ be a positive sequence of real numbers such that $\displaystyle\limsup_{n\to\infty}x_n<\infty$. Let $\langle a_n\rangle$ be a sequence of real numbers such that $\displaystyle\sum_1^{\infty}\frac1{1+a_n}=\infty$. Determine whether $\displaystyle\sum_1^{\infty}\frac1{1+x_na_n}=\infty$.
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Context:
A consumer report revealed the following information about three tubes of toothpaste. Bright is $60 \%$ more expensive than Fresh and has $25\%$ less volume than Glow. Glow is $25\%$ less expensive than Bright and has $33\frac{1}{3} \%$ more volume than Fresh. Fresh costs $\$1.00$ per unit of volume. What is the number of cents per unit of volume of Glow?
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Context:
Find the minimum number of colors necessary to color the integers from $1$ to $2007$ such that if distinct integers $a$, $b$, and $c$ are the same color, then $a \nmid b$ or $b \nmid c$.
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Context:
12. B6 (FIN 3) Four distinct circles \( C, C_{1}, C_{2}, C_{3} \) and a line \( L \) are given in the plane such that \( C \) and \( L \) are disjoint and each of the circles \( C_{1}, C_{2}, C_{3} \) touches the other two, as well as \( C \) and \( L \). Assuming the radius of \( C \) to be 1, determine the distance between its center and \( L \).
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Context:
Consider the system $x + y = z + u,$ $2xy = zu.$ Find the greatest value of the real constant $m$ such that $m \leq x/y$ for any positive integer solution $(x,y,z,u)$ of the system, with $x \geq y$.

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Context:
Is it possible to draw some number of diagonals in a convex hexagon so that every diagonal crosses EXACTLY three others in the interior of the hexagon? (Diagonals that touch at one of the corners of the hexagon DO NOT count as crossing.)
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Context:
An integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition.For example, 4 can be partitioned in five distinct ways:
$4$
$3 + 1$
$2 + 2$
$2 + 1 + 1$
$1 + 1 + 1 + 1$

The number of partitions of n is given by the partition function $p(n)$. So $p(4) = 5$.
Determine all the positive integers so that $p(n) + p(n + 4) = p(n + 2) + p(n + 3)$.
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Context:
Let $F$ be a free group with a basis $X$, and let $X' = \{xF' : x \in X\}$, where $F'$ is the commutator subgroup of $F$. Determine if $|X| = |X'|$.
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Context:
Is the unit circle the only infinite compact subgroup of the multiplicative group of non-zero complex numbers? Specifically, if $G$ is an infinite subgroup of $(\mathbb{C} \setminus \{0\}, \cdot)$ that is compact as a subset of $\mathbb{C}$, is it true that $G = S^1$?
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Context:
Verify whether the Fourier coefficient \(B_n\) is zero for all \(n\) when \(f(x) = 0.01\sin{3x}\) using the formula \(B_n=\frac{2}{\pi} \int_0^{\pi} f(x) \sin{(nx)} \, dx\).
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Context:
Let $I$ be the set of points $(x,y)$ in the Cartesian plane such that$$x>\left(\frac{y^4}{9}+2015\right)^{1/4}$$Let $f(r)$ denote the area of the intersection of $I$ and the disk $x^2+y^2\le r^2$ of radius $r>0$ centered at the origin $(0,0)$. Determine the minimum possible real number $L$ such that $f(r)<Lr^2$ for all $r>0$.
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Context:
Is the area of the fundamental parallelogram of the lattice generated by \((n, n\alpha \mod 1)\) always 1, even if \(\alpha = \sqrt{3}\) or another number?
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Context:
Let the series \(\sum_{k=0}^\infty a_k x^k\) have a radius of convergence 1. Assume further that \(\sum_{k=0}^\infty a_k = \infty\). Determine whether \(\lim_{x\to 1^-} \sum_{k=0}^\infty a_k x^k = \infty\).
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Context:
Determine the largest positive integer $n$ such that the following statement holds:
If $a_1,a_2,a_3,a_4,a_5,a_6$ are six distinct positive integers less than or equal to $n$, then there exist $3$ distinct positive integers ,from these six, say $a,b,c$ s.t. $ab>c,bc>a,ca>b$.
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Context:
Identify and define the left adjoint to the functor \( \delta: \text{Set} \to \text{Cat} \) that maps a set to a discrete category consisting of its elements.
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Context:
Each day Maria must work $8$ hours. This does not include the $45$ minutes she takes for lunch. If she begins working at $\text{7:25 A.M.}$ and takes her lunch break at noon, then her working day will end at

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Context:
Let  $H$  be a rectangle with angle between two diagonal  $\leq 45^{0}$ . Rotation  $H$  around the its center with angle  $0^{0}\leq x\leq 360^{0}$  we have rectangle  $H_{x}$ . Find  $x$  such that  $[H\cap H_{x}]$  minimum, where  $[S]$  is area of  $S$ .
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Context:
In a school, $60\%$ of the students are boys and $40\%$ are girls. Given that $40\%$ of the boys like math and $50\%$ of the people who like math are girls, what percentage of girls like math?

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Context:
安迪和鲍勃去食堂买零食。他们花了相同的金额。安迪买了一罐价值1美元的苏打水和两个价值每个2美元的汉堡。鲍勃点了两个价值3美元的三明治和一罐果汁。鲍勃的果汁花了多少钱？
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Context:
Determine whether the limit \( \lim_{n \to \infty} a_{n} \) exists for the sequence \( a_{1}, a_{2}, \ldots, a_{n}, \ldots \) defined by the recursion \( a_{n+1} = (2 a_{n})^{1 / a_{n}} \), where \( a_{1} \) is any positive number.
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Context:
For any nonnegative integer $n$, let $S(n)$ be the sum of the digits of $n$. Let $K$ be the number of nonnegative integers $n \le 10^{10}$ that satisfy the equation
 \[
        S(n) = (S(S(n)))^2.
 \]
Find the remainder when $K$ is divided by $1000$.
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Context:
Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
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Context:
Evaluate the convergence and compute the value of the integral: $$\int_{0}^{\infty} \frac{\ln(x)}{x^2+x+1} \, dx$$ without using advanced techniques such as the calculus of residues.
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Context:
Compute the differential \(d(x_1x_2)\) using the product rule for the stochastic differential equations \(dx_1 = a_1 dt + b_1 dw_1(t)\) and \(dx_2 = a_2 dt + b_2 dw_2(t)\), where \(w_1\) and \(w_2\) are independent Wiener processes. What is the term \((dx_1)(dx_2)\) when \(w_1\) and \(w_2\) are independent?
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Context:
Determine the signature of the quadratic form given by:
\[ f(x,y,z) = 3x^2 + 4y^2 + z^2 - 9(5xy + 3xz + yz) \]
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Context:
Given the sequence  $\{y_n\}_{n=1}^{\infty}$  defined by  $y_1=y_2=1$  and
\[y_{n+2} = (4k-5)y_{n+1}-y_n+4-2k, \qquad n\ge1\]
find all integers  $k$  such that every term of the sequence is a perfect square.
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Context:
Determine whether the integral \( \int_{0}^{\infty} \frac{1}{e^x + e^{-x}} \, dx \) converges or diverges.
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Context:
Let $a, b, c$ be positive integers such that \( \gcd(a, b) + \text{lcm}(a, b) = \gcd(a, c) + \text{lcm}(a, c). \) Determine whether it necessarily follows that \( b = c. \)
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Context:
Let $a$, $b$, and $c$ be three prime numbers such that $a + b = c$. If the average of two of the three primes is four less than four times the fourth power of the last, find the second-largest of the three primes.

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Context:
Determine the cardinality of $\mathbb{R}^\infty$. Is it the same as $\beth_1$?
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Context:
Given that a normal vector of the straight line $l$ is $\overrightarrow{n} = (1, -\sqrt{3})$, find the size of the inclination angle of this straight line.
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Context:
The hypotenuse of a right triangle is $10$ inches and the radius of the inscribed circle is $1$ inch. The perimeter of the triangle in inches is:
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Context:
Let $R$ be a commutative ring with unity and $I$ be an ideal of $R$. Determine whether $\text{Hom}_R(R/I, R/I) \cong R/I$ as $R$-modules.
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Context:
Suppose $H$ is an indefinite quaternion algebra over $\mathbb{Q}$. Determine whether there are infinitely many quadratic fields that can be embedded into $H$. Answer 'finite' or 'infinite'.
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Context:
Planes \(\alpha\) and \(\beta\) intersect. Points \(\mathrm{M}\) and \(\mathrm{N}\) do not lie on these planes. A perpendicular is dropped from point \(\mathrm{M}\) to plane \(\alpha\) and from point \(\mathrm{N}\) to plane \(\beta\), and these perpendiculars lie in the same plane. If a perpendicular is then dropped from point \(\mathrm{M}\) to plane \(\beta\) and from point \(\mathrm{N}\) to plane \(\alpha\), must these perpendiculars also lie in the same plane? Answer 'yes' or 'no'.
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Context:
A circular plot with a radius of 30 feet requires a light placed directly above its center at a height that maximizes illumination at the plot's edge. The intensity of illumination, $I$, is directly proportional to the cosine of the angle of incidence, $\theta$, and inversely proportional to the square of the distance from the light source. If $h$ is the height of the light source, express the intensity $I$ in terms of $h$ and determine the height $h$ that maximizes the intensity at the edge of the plot.
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Context:
Let $ABC$ be a triangle.  There exists a positive real number $k$, such that if the altitudes of triangle $ABC$ are extended past $A$, $B$, and $C$, to $A'$, $B'$, and $C'$, as shown, such that $AA' = kBC$, $BB' = kAC$, and $CC' = kAB$, then triangle $A'B'C'$ is equilateral.

[asy]
unitsize(0.6 cm);

pair[] A, B, C;
pair D, E, F;

A[0] = (2,4);
B[0] = (0,1);
C[0] = (5,0);
D = (A[0] + reflect(B[0],C[0])*(A[0]))/2;
E = (B[0] + reflect(C[0],A[0])*(B[0]))/2;
F = (C[0] + reflect(A[0],B[0])*(C[0]))/2;
A[1] = A[0] + (1/sqrt(3))*(rotate(90)*(C[0] - B[0]));
B[1] = B[0] + (1/sqrt(3))*(rotate(90)*(A[0] - C[0]));
C[1] = C[0] + (1/sqrt(3))*(rotate(90)*(B[0] - A[0]));

draw(A[0]--B[0]--C[0]--cycle);
draw(A[1]--D);
draw(B[1]--E);
draw(C[1]--F);

label("$A$", A[0], NW);
dot("$A'$", A[1], N);
label("$B$", B[0], S);
dot("$B'$", B[1], SW);
label("$C$", C[0], S);
dot("$C'$", C[1], SE);
[/asy]

Find $k$.
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Context:
4. Find all values of the parameter $b$, for each of which there exists a number $a$ such that the system

$$
\left\{\begin{array}{l}
y=-b-x^{2}, \\
x^{2}+y^{2}+8 a^{2}=4+4 a(x+y)
\end{array}\right.
$$

has at least one solution $(x ; y)$.
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Context:
Bobby received his weekly paycheck today. Apart from his usual salary of $450 per week, he also received a bonus this week, which varies depending on the company's performance. If the company's performance rate this week was 12%, he receives that percentage as a bonus. The actual paycheck is reduced because of taxes, insurance payments, and fees removed from his pay before his employer writes the check. The government removes 1/3 in federal taxes, 8% in state taxes, and 5% in local taxes from his total income (salary and bonus combined). The health insurance removes $50 per week, the life insurance removes $20 per week, and the city charges him a fee of $10 per week for parking. Additionally, Bobby contributes 3% of his salary to his retirement fund, and his employer matches 50% of his contribution. What is the final amount left in Bobby's paycheck, in dollars?
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Context:
已知 $x^{2}=4 y, M(2,2)$, 过 $M$ 点的直线交抛物线于 $A, B$ 两点, 过 $A, B$ 两点作抛物线的切线交与点 $P$, 求 $S_{\triangle A B P}$ 的最小值。请给出该最小值。
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Context:
6. Points $A_{1}, \ldots, A_{12}$ are the vertices of a regular 12-gon. How many different 11-segment open broken lines without self-intersections with vertices at these points exist? Broken lines that can be transformed into each other by rotation are considered the same.
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Context:
Given a general triangle \(ABC\) with points \(K, L, M, N, U\) on its sides:

- Point \(K\) is the midpoint of side \(AC\).
- Point \(U\) is the midpoint of side \(BC\).
- Points \(L\) and \(M\) lie on segments \(CK\) and \(CU\) respectively, such that \(LM \parallel KU\).
- Point \(N\) lies on segment \(AB\) such that \(|AN| : |AB| = 3 : 7\).
- The ratio of the areas of polygons \(UMLK\) and \(MLKNU\) is 3 : 7.

Determine the length ratio of segments \(LM\) and \(KU\).
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Context:
Determine if there exists a solution in positive integers for the system of equations: \[ \begin{array}{l} x_{1}^{2}+x_{2}^{2}+\cdots+x_{1985}^{2}=y^{3}, \\ x_{1}^{3}+x_{2}^{3}+\cdots+x_{1985}^{3}=z^{2} \end{array} \] where $x_{i} \neq x_{j}$ for $i \neq j$. Answer with 'yes' or 'no'.
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Context:
Suppose $\cos R = \frac{5}{13}$ in the triangle below. If $RS = 13$, find the length of $QS$.
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Context:
If $A$ is a matrix with entries in the quaternions and $q$ is a quaternion, determine whether $(qA)^T = q A^T$ or $(qA)^T = \overline{q}A^T$.
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Context:
Determine the values of \(\alpha\) for which the sequence \(f_n(x) = n^\alpha X_{B_{1/n}}\) converges pointwise in \(\mathbb{R}^3\). For those values of \(\alpha\), find the pointwise limit.
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Context:
Everyday Isaac wakes up at 7 in the morning and goes to sleep at a certain time in the night. Isaac sleeps for 8 hours in a day. At what time does Isaac go to sleep at night?
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Context:
Determine whether the limit \( \lim_{x \to y^-} \sum_{\{n : r_n \in (x, y)\}} \frac{1}{2^n} \) is equal to zero, where \( \{r_n\}_{n=1}^\infty \) is an enumeration of the rational numbers.
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Context:
A shop offers an initial discount of $25\%$ on all items. A week later, an additional discount of $15\%$ is applied to the already discounted prices. Finally, an extra $10\%$ discount is applied to the new sale price on a special sale day. The shop advertises that the cumulative discount equals $50\%$ off the original prices. Compute the actual total discount and find the percentage difference between the shop's claim and the true discount.
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Context:
Each artist in the creative collective "Patience and Labor" has their own working schedule. Six of them paint one picture every two days, another eight of them paint one picture every three days, and the rest never paint pictures. From September 22 to September 26, they painted a total of 30 pictures. How many pictures will they paint on September 27?
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Context:
Let $\{a_n\}$ be a sequence such that $a_n \geq 0$ for all $n$, $\{a_n\}$ is monotonically decreasing, and $\sum_{n=1}^\infty a_n$ converges. Determine if it is true that as $n\rightarrow\infty$, $$n\log n\;a_n\rightarrow 0$$
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Context:
Let \( M = \left\{(x, y) \mid \tan(\pi y) + \sin^2(\pi x) = 0 \right\} \) and \( N = \left\{(x, y) \mid x^2 + y^2 \leq 2 \right\} \). Calculate the number of points in the intersection \( M \cap N \). Express your answer as a single integer.
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Context:
Determine the domain of the function $g(z, \bar{z}) = f(x, y)$, where $z = x + iy$ and $\bar{z} = x - iy$. Is the domain $g: \mathbb{C} \to \mathbb{C}$ or $g: \mathbb{C}^2 \to \mathbb{C}$?
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Context:
## Problem B3

$\mathrm{k}$ is the positive root of the equation $\mathrm{x}^{2}-1998 \mathrm{x}-1=0$. Define the sequence $\mathrm{x}_{0}, \mathrm{x}_{1}, \mathrm{x}_{2}, \ldots$ by $\mathrm{x}_{0}$ $=1, \mathrm{x}_{\mathrm{n}+1}=\left[\mathrm{k} \mathrm{x}_{\mathrm{n}}\right]$. Find the remainder when $\mathrm{x}_{1998}$ is divided by 1998.

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Context:
Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. Find the least positive integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \cdots, T_n$ returns the point $(1,0)$ back to itself.
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Context:
Evaluate the limit \(\lim_{a\rightarrow \infty} \int_{-a}^a \cos(x+b)\,dx\) and determine the values of \(b\) for which it exists.
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Context:
The equation \[\frac{x}{x+1} + \frac{x}{x+2} = kx\]has exactly two complex roots. Find all possible complex values for $k.$

Enter all the possible values, separated by commas.
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