Context:
Let $ABCDEF$ be a regular hexagon with side 1. Point $X, Y$ are on sides $CD$ and $DE$ respectively, such that the perimeter of $DXY$ is $2$. Determine $\angle XAY$.
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Context:
设 V 是 10 维列向量构成的内积空间，其中对 v=(v_1,v_2,\cdots,v_{10})^T, w=(w_1,w_2,\cdots,w_{10})^T，v,w 的内积 \langle v,w\rangle=\ds{i=1}{10}v_iw_i.对 u\in V，定义 V 上的线性变换 P_u 如下：P_u:V\rightarrow V,\quad x\mapsto x-\df{2\langle x,u\rangle u}{\langle u,u\rangle}.给定 v,w\in V，满足0<\langle v,w\rangle<\sqrt{\langle v,v\rangle\langle w,w\rangle}.记 Q=P_v\circ P_w，则所有满足 P\circ Q=Q\circ P 的线性变换 P:V\rightarrow V 构成的线性空间的维数是__________.
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Context:
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. If the area is $x\pi$, what is the value of $x$?
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Context:
Let  $ABCD$  be a square with side length  $1$ . How many points  $P$  inside the square (not on its sides) have the property that the square can be cut into  $10$  triangles of equal area such that all of them have  $P$  as a vertex?
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Context:
Given that $11^{-1} \equiv 3 \pmod{31}$, find $20^{-1} \pmod{31}$. Provide the result as a residue modulo 31 (a number between 0 and 30, inclusive).
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Context:
任意＿个自然数，就可以保证其中必有四个数的和是4的倍数.
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Context:
Let $n$ be the smallest positive integer with exactly 2015 positive factors. What is the sum of the (not necessarily distinct) prime factors of $n$?
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Context:
Given two fuzzy topological spaces $(X, \tau_1)$ and $(Y, \tau_2)$, where $X=\{a\}$, $Y=\{x, y\}$, $\tau_1 =\{0_X, 1_X, \{(a, 0.3)\}\}$, $\tau_2 =\{0_Y, 1_Y, \{(x, 0.2), (y, 0.2)\}\}$, and a fuzzy function $f:X \rightarrow Y$ such that $f(a)=x$, find the inverse of the fuzzy set $\{(x, 0.5), (y, 0.3)\}$ in $X$. What is the inverse fuzzy set in $X$?
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Context:
Evaluate the following integral considering the parity of the integrand:
$$\hat g(n) = \frac{1}{2\pi}\int_0^{2\pi} xe^{-inx} dx$$
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Context:
By writing 6 different numbers, none of which is the number 1, in ascending order and multiplying them, Olya obtained the result 135135. Write down the numbers that Olya multiplied.
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Context:
$O K R A$ is a trapezoid with $O K$ parallel to $R A$. If $O K=12$ and $R A$ is a positive integer, how many integer values can be taken on by the length of the segment in the trapezoid, parallel to $O K$, through the intersection of the diagonals?
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Context:
Find the number of ordered pairs of integers $(x, y)$ such that $$\frac{x^2}{y}- \frac{y^2}{x}= 3 \left( 2+ \frac{1}{xy}\right)$$
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Context:
A block wall 100 feet long and 7 feet high will be constructed using blocks that are 1 foot high and either 2 feet long or 1 foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall?
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Context:
How many of the integers between 25 and a certain number are even if the total number of even integers is 10?
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Context:
Let $M$ be the maximum possible value of $x_1x_2 + x_2x_3 + \cdots + x_5x_1$ where $x_1, x_2, \cdots, x_5$ is a permutation of $(1, 2, 3, 4, 5)$. Let $N$ be the number of permutations for which this maximum is attained. Evaluate $M + N$.
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Context:
Determine the points in \(\mathbb{R}^3\) where the distribution \(D = \langle X_1, X_2 \rangle\) has integral manifolds, given the vector fields: \[X_1 = \frac{\partial}{\partial x} + z \frac{\partial}{\partial y}, \quad X_2 = \frac{\partial}{\partial y} + z \frac{\partial}{\partial z}.\]
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Context:
Sides of a triangle form an arithmetic sequence with common difference  $2$ , and its area is  $6 \text{ cm }^2$ . Find its
sides.
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Context:
For which values of \(\alpha \geq 1\) can Hephaestus guarantee victory in a finite number of turns, regardless of Poseidon's initial choice of flooded cells, in the game described?
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Context:
The sides of  $\triangle ABC$  measure 11,20, and 21 units. We fold it along  $PQ,QR,RP$  where  $P,Q,R$  are the midpoints of its sides until  $A,B,C$  coincide. What is the volume of the resulting tetrahedron?
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Context:
Given that in the expansion of $(1-2x)^n$, only the coefficient of the third term is the largest, find the sum of the coefficients of the expansion.
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Context:
Arrange for five students, namely A, B, C, D, and E, to participate in the volunteer activities at the Shanghai World Expo, with each person undertaking one of the following tasks: translation, tour guiding, etiquette, or driving. Each task must have at least one participant. Students A, B, and C cannot drive but can undertake the other three tasks, while students D and E are capable of handling all four tasks. Calculate the number of different arrangement possibilities.
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Context:
You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1 , the value of each blue card is equal to twice the number of red cards, and the value of each yellow card is equal to three times the number of blue cards. What is the maximum score you can get with fifteen cards?
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Context:
椭圆 $\frac{x^{2}}{4}+\frac{y^{2}}{3}=\lambda, F$ 为左焦点, $A, B$ 为椭圆上两点且 $|F A|=5,|F B|=8$, 求直线 $A B$ 的斜率 $k$ 的范围。请给出该范围上下限绝对值之和的近似整数值
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Context:
Flora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance m with probability $\frac{1}{2^{m}}$. What is the probability that Flora will eventually land at 10?
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Context:
In a sequence of numbers, a term is called "golden" if it is divisible by the term immediately before it. What is the maximum possible number of golden terms in a permutation of $1, 2, 3, \ldots, 2021$?
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Context:
Consider the curve $C: y^2z = x^3 - xz^2$ in the projective plane $\mathbb{P}^2$ and the point $P = (0:0:1)$. Define the rational map $\phi: \mathbb{P}^2 \dashrightarrow \mathbb{P}^1$ by $\phi(x:y:z) = (x:y)$. Determine the value of $\psi(P)$, where $\psi = \phi|_C: C \to \mathbb{P}^1$.
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Context:
Determine the order of growth of the complex polynomial \( p(z) = a_nz^n + \ldots + a_1z + a_0 \).
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Context:
Bingo has three dogs. On average, they walk a total of 100 miles a week. The first dog walks an average of 2 miles a day with a constant speed of 3 miles per hour. The second dog's daily walks depend on the last digit of the day of the month: 1 mile if it is odd, and 3 miles if it is even, with a varying speed between 4 to 6 miles per hour. The third dog has rest days on weekends and its daily walks are impacted by seasonal weather changes (assume 3 miles during summer, 2 miles during spring and fall, and 1 mile during winter). Calculate the average daily distance the third dog walks in summer, considering a 30-day month, and taking into account its rest days.
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Context:
If 700 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers?
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Context:
Given \(\frac{1 - |\cos x|}{1 + |\cos x|} = \sin x\), find \(x\).
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\boxed{k\pi, 2k\pi + \frac{\pi}{2} (k \in \mathbb{Z})}


Context:
Of the goose eggs laid at a certain pond, 1/4 hatched and 4/5 of the geese that hatched from those eggs survived the first month. Of the geese that survived the first month, 3/5 did not survive the first year. If no more than one goose hatched from each egg, how many geese survived the first year?
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Context:
把一个半径为1.1的圆形纸片放到直角坐标平面上，纸片最少要盖住＿个整点.
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Context:
The volume of a parallelepiped is \( V \). Find the volume of the polyhedron whose vertices are the centers of the faces of this parallelepiped.
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Context:
Points \(P\) and \(Q\) are two distinct points in the \(xy\)-plane. Find the number of different places in the \(xy\)-plane where a third point, \(R\), can be placed so that \(PQ = QR = PR\). Express your answer as a single integer.
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Context:
The sum of the largest number and the smallest number of a triple of positive integers $(x,y,z)$ is the power of the triple. Compute the sum of powers of all triples $(x,y,z)$ where $x,y,z \leq 9$.
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Context:
The distance between points A and B is 1000 meters. Person A departs from point A and takes 1 hour to reach point B. Person B departs from point B, 20 minutes after person A leaves, and takes 40 minutes to reach point A. How many meters from point A do person A and person B meet?
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Context:
Analyze the recurrence $$T(n)=T\left(n-\frac{n}{\lg n}\right)+ O(n)$$ and determine its time complexity.
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Context:
Five equilateral triangles of side length $2\sqrt{3}$ are arranged on the same side of a straight line, such that the midpoint of the base of each triangle is the vertex of the next triangle. Calculate the total area covered by these five triangles in the plane. Express your answer as a product of a number and $\sqrt{3}$.
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Context:
Calculate the sum of the sequence $2 - 6 + 10 - 14 + 18 - \cdots - 98 + 102$.
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Context:
Find the minimum value of the expression \( \frac{(1 + x + x^2)(1 + y + y^2)}{xy} \) given that \( x \neq 0 \).
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Context:
1. Petya came to the bus stop of the bus going to school with stops equally spaced from each other, and, not seeing the bus on the road, decided to run and get on the bus at the next stops along the way to school. Petya ran in such a way that at any moment he could notice the appearance of the bus on the road behind him. Seeing the bus, Petya can turn back or maintain his direction of movement. It is known that the speed of the bus is 5 times greater than Petya's running speed, and he can see the bus at a distance of no more than 0.6 km. Find the greatest value of the distance between stops, such that regardless of whether Petya turns back upon discovering the bus or not, he will be able to get on the bus at a stop. (the time the bus spends at the stop is not considered)
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Context:
Determine the limiting distribution of \( h(X_n) = \sqrt{(\bar{X}_n - 1)^2} \) where \( X_1, \ldots, X_n, \ldots \) are Poisson random variables with parameter \( \theta \).
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Context:
Consider a 2D grid with \( n \) rows and \( m \) columns. You can fill this grid with numbers ranging from 1 to \( k \) (inclusive). The condition is that for each number \( r \) such that \( 1 \leq r \leq k \), no two rows can have exactly the same number of the number \( r \). Determine the number of ways to fill the grid under these conditions.
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Context:
Ten people are sitting around a round table. Each of them is either a knight, who always tells the truth, or a liar, who always lies. Each person was given a token. Then each of the people passed their token to one of their two neighbors. After this, 5 people said, “I have one token,” and the remaining 5 people said, “I have no tokens.” What is the maximum number of knights that could be among these 10 people?
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Context:
In an equilateral triangle ABC, three rods of lengths 3, 4, and 5 units are placed such that they intersect at a common point O, with the other ends at vertices A, B, and C, respectively. Given that BO = 3 units and CO = 4 units, find the angle ∠BOC.
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Context:
from an island, it is possible to reach the mainland by either ferry p, ferry q, or ferry r. Ferry p travels for 3 hours at 6 kilometers per hour, while ferry q takes a route that is three times longer than ferry p. If ferry p is slower than ferry q by 3 kilometers per hour, and ferry r takes a route that is 5 hours long and is half the speed of ferry q, find the difference in journey time between the slowest and fastest ferries.
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Context:
Suppose  $ 2015= a_1 <a_2 < a_3<\cdots <a_k $  be a finite sequence of positive integers, and for all  $ m, n \in \mathbb{N} $  and  $1\le m,n \le k $ ,   $$  a_m+a_n\ge a_{m+n}+|m-n|  $$   Determine the largest possible value  $ k $  can obtain.
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Context:
Find the largest integer $n \ge 3$ for which there exists an $n$-digit number $\overline{a_1a_2a_3\ldots a_n}$ with non-zero digits $a_1$, $a_2$, and $a_n$, such that the number is divisible by $\overline{a_2a_3\ldots a_n}$.
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Context:
A rope of length 10 *m* is tied tautly from the top of a flagpole to the ground 6 *m* away from the base of the pole. An ant crawls up the rope and its shadow moves at a rate of 30 *cm/min*. How many meters above the ground is the ant after 5 minutes? (This takes place on the summer solstice on the Tropic of Cancer so that the sun is directly overhead.)
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Context:
Let $S$ be the region enclosed by the curves $C_1: y = ax^2$ and $C_2: y = -ax^2 + 2abx$, where $a$ and $b$ are positive constants. Calculate the ratio of the volume $V_x$ of the solid formed by revolving $S$ around the $x$-axis to the volume $V_y$ of the solid formed by revolving $S$ around the $y$-axis. Find $\frac{V_x}{V_y}$. \( \frac{V_x}{V_y} \)
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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:
Calculate the number of elements of order 24 in the group \( C_{24} \times C_{40} \).
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Context:
For each natural number \( k \geqslant 2 \), a sequence \(\{a_{n}\}\) is defined by:
\[ a_{0} = k, \quad a_{n} = d(a_{n-1}), \quad n = 1, 2, 3, \cdots \]
where \( d(a) \) denotes the number of positive integer divisors of \( a \). Find all values of \( k \) such that the sequence \(\{a_{n}\}\) does not contain any perfect squares.
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Context:
The ratio of the two base angles in an isosceles triangle is 2:1. Find the measure of the vertex angle of this isosceles triangle.
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Context:
Suppose that a polynomial of the form $p(x) = x^{2010} \pm x^{2009} \pm \cdots \pm x \pm 1$ has no real roots. What is the maximum possible number of coefficients of $-1$ in $p$?
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Context:
Let $ABCD$ be a cyclic quadrilateral with $AB = 5$, $BC = 10$, $CD = 11$, and $DA = 14$. The value of $AC + BD$ can be written as $\tfrac{n}{\sqrt{pq}}$, where $n$ is a positive integer and $p$ and $q$ are distinct primes. Find $n + p + q$.
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Context:
Find, as a function of $\, n, \,$ the sum of the digits of \[9 \times 99 \times 9999 \times \cdots \times \left( 10^{2^n} - 1 \right),\] where each factor has twice as many digits as the previous one.
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Context:
In a $7 \times 7$ grid, 19 squares are colored red. A row or column is considered to be red if it contains at least 4 red squares. What is the maximum number of red rows and columns in this grid?
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Context:
A 2-kg rock is suspended by a massless string from one end of a uniform 1-meter measuring stick. The measuring stick is balanced by a support force at the 0.20-meter mark. Calculate the mass of the measuring stick. Express your answer in kg, rounded to two decimal places.
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Context:
How many zeroes does the function $f(z) = e^z + z$ have in the square $[-10,10] \times [-6 \pi i, 6 \pi i]$?
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Context:
Given a stationary stochastic process with a spectral density \( S_{XX}(\omega) = 1 - \frac{|\omega|}{8 \pi} \), find the mean square value of the process.
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Context:
At the bake sale, Tamara makes $32 from the brownies. She made 2 pans of brownies which were all sold. The brownies were cut into a certain number of big square pieces, and each brownie cost $2. How many big square pieces were the brownies cut into?
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Context:
A classroom has  $30$  seats arranged into  $5$  rows of  $6$  seats. Thirty students of distinct heights come to class every day, each sitting in a random seat. The teacher stands in front of all the rows, and if any student seated in front of you (in the same column) is taller than you, then the teacher cannot notice that you are playing games on your phone. What is the expected number of students who can safely play games on their phone?
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Context:
Two good trains, each 250 m long, are running in opposite directions on parallel tracks. One train has a certain speed, and the other train has a speed of 30 km/hr. The time taken by the slower train to pass the driver of the faster one is 23.998080153587715 seconds. What is the speed of the faster train?
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Context:
On the sides of a right triangle, regular nonagons are constructed outside. These nonagons are built on one of the catheti and on the hypotenuse, with areas equal to $1602 \, \text{cm}^2$ and $2019 \, \text{cm}^2$, respectively. What is the area of the nonagon that is constructed on the other cathetus of this triangle?
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Context:
The complex number $z = re^{i \theta}$ satisfies
\[\left| 2z + \frac{1}{z} \right| = 1.\]Find the minimum value of $\sin^2 \theta.$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:
Given a table with numbers $1, 2, \ldots, 18$, determine the minimal number of numbers that must be erased so that the sum of any two remaining numbers is not a perfect square of a positive integer.
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Context:
78. The Walking Passenger. The train is moving at a speed of 60 km/h. A passenger is walking from the end of the train to its beginning through the passageways between the cars at a speed of 3 km/h. What is the speed of the passenger relative to the railway track?

We are not going to engage in sophistry here, like Zeno's paradox of the flying arrow, or Einstein's theory of relativity, but are talking about movement in the ordinary sense of the word relative to the railway track.
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Context:
Find the diagonals of the quadrilateral formed by the angle bisectors of the internal angles of a rectangle with sides 1 and 3.
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Context:
A \(3 \times 3\) square is partitioned into \(9\) unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated \(90\,^{\circ}\) clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. Find the probability that the grid is now entirely black. The original answer is in \(\frac{m}{n}\) format, please give the value of m + n.
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Context:
Alice is planning a trip from the Bay Area to one of $5$ possible destinations, each serviced by only $1$ airport. She wants to book two flights: one to her destination and one returning. There are $3$ airports within the Bay Area from which she may leave and to which she may return. In how many ways can she plan her flight itinerary?
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Context:
Given that the real numbers $a$, $b$, and $c$ satisfy $a \geq 1$, $b \geq 1$, $c \geq 1$, and $abc = 10$, and that $a^{\log_2{a}} \cdot b^{\log_2{b}} \cdot c^{\log_2{c}} \geq 10$, find the value of $a + b + c$.
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Context:
Calculate the integral $\int_M(1-x_3)^2d\sigma$ over the closed upper hemisphere $M=\{(x_1,x_2,x_3) \in \mathbb{S}^2 | x_3 \ge 0\}$ in $\mathbb{R}^3$, where $d\sigma$ denotes the standard spherical measure. Can this be done without using spherical coordinates?
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Context:
Let $H$ be a convex, equilateral heptagon whose angles measure (in degrees) $168^\circ$, $108^\circ$, $108^\circ$, $168^\circ$, $x^\circ$, $y^\circ$, and $z^\circ$ in clockwise order. Compute the number $y$.
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Context:
Vladimir chose three digits \(a, b,\) and \(c\) such that \(a > b > c > 0\), and with them, he formed the numbers \(abc\), \(cba\), and \(cab\). Note that \(abc\) is not the product of \(a, b\), and \(c\), but rather the number composed of the digits \(a, b\), and \(c\). For example, if \(a = 1, b = 2\), and \(c = 3\), \(abc\) would be the number 123.

After choosing these three digits \(a, b\), and \(c\), Vladimir realized that one of the numbers formed was equal to the sum of the other two. Find the numbers formed by Vladimir.
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Context:
平面上任意给定5个点，其中任三点不共线，则在以它们为顶点的三角形中，至多有＿个锐角三角形.
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Context:
Let  $P(x), Q(x), $  and  $R(x)$  be three monic quadratic polynomials with only real roots, satisfying   $$ P(Q(x))=(x-1)(x-3)(x-5)(x-7) $$    $$ Q(R(x))=(x-2)(x-4)(x-6)(x-8) $$      for all real numbers  $x.$  What is  $P(0)+Q(0)+R(0)?$
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Context:
How many nonzero coefficients can a polynomial  $ P(x)$  have if its coefficients are integers and  $ |P(z)| \le 2$  for any complex number  $ z$  of unit length?
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Context:
In the figure, there are two common tangents. These common tangents meet the circles at points \( A, B, C \), and \( D \). If \( AC = 9 \, \text{cm} \), \( BD = 3 \, \text{cm} \), \( \angle BAC = 60^\circ \), and \( AB = s \, \text{cm} \), find the value of \( s \).
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Context:
当a为何值时，不等式$\log _ { \frac { 1 } { a } } ( \sqrt { x ^ { 2 } + a x + 5 } + 1 ) \cdot \log _ { 5 } ( x ^ { 2 } + a x +$ $6 ) + \log _ { a } 3 \geq 0$有且只有一个解？
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Context:
Find the number of moles of HCl formed on combining 2 moles of C2H6 and 2 moles of Cl2. How many moles of HCl are formed?
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Context:
Each of the integers 1 to 7 is to be written, one in each circle in the diagram. The sum of the three integers in any straight line is to be the same. Find the number of different ways the centre circle can be filled. Express your answer as a whole number.
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Context:
Compute the $\textit{number}$ of ordered quadruples $(w,x,y,z)$ of complex numbers (not necessarily nonreal) such that the following system is satisfied:
\begin{align*}
wxyz &= 1\\
wxy^2 + wx^2z + w^2yz + xyz^2 &=2\\
wx^2y + w^2y^2 + w^2xz + xy^2z + x^2z^2 + ywz^2 &= -3 \\
w^2xy + x^2yz + wy^2z + wxz^2 &= -1\end{align*}
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC,$ $D,$ $E,$ and $F$ are points on sides $\overline{BC},$ $\overline{AC},$ and $\overline{AB},$ respectively, so that $BD:DC = CE:EA = AF:FB = 1:2.$

[asy]
unitsize(0.8 cm);

pair A, B, C, D, E, F, P, Q, R;

A = (2,5);
B = (0,0);
C = (7,0);
D = interp(B,C,1/3);
E = interp(C,A,1/3);
F = interp(A,B,1/3);
P = extension(A,D,C,F);
Q = extension(A,D,B,E);
R = extension(B,E,C,F);

fill(P--Q--R--cycle,gray(0.7));
draw(A--B--C--cycle);
draw(A--D);
draw(B--E);
draw(C--F);

label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, SE);
label("$D$", D, S);
label("$E$", E, NE);
label("$F$", F, W);
label("$P$", P, NE);
label("$Q$", Q, NW);
label("$R$", R, S);
[/asy]

Line segments $\overline{AD},$ $\overline{BE},$ and $\overline{CF}$ intersect at $P,$ $Q,$ and $R,$ as shown above.  Compute $\frac{[PQR]}{[ABC]}.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two squares, each with side length 5 cm, overlap as shown. The shape of their overlap is a square, which has an area of 4 cm². Find the perimeter, in centimeters, of the shaded figure. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the chromatic number of the graph obtained by removing $k$ vertex-disjoint edges from the complete bipartite graph $K_{m,n}$, where $k \leq \min\{m,n\}$. Express your answer in terms of $m$, $n$, and $k$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
The numbers $\sqrt{3v-2}$, $\sqrt{3v+1}$, and $2\sqrt{v}$ are the side lengths of a triangle. What is the measure of the largest angle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x,$ $y,$ and $z$ be real numbers such that $x + y + z = 7$ and $x, y, z \geq 2.$ Find the maximum value of
\[\sqrt{2x + 3} + \sqrt{2y + 3} + \sqrt{2z + 3}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest real number \( k \) such that the inequality
\[
(a+b+c)^{3} \geq \frac{5}{2}\left(a^{3}+b^{3}+c^{3}\right)+k a b c
\]
holds for any side lengths \( a \), \( b \), and \( c \) of a triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of distinguishable groupings into which you can place $3$ indistinguishable red balls and $3$ indistinguishable blue balls. Here the groupings $RR-BR-B-B$ and $B-RB-B-RR$ are indistinguishable because the groupings are merely rearranged, but $RRB-BR-B$ is distinguishable from $RBB-BR-R$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the number \(2016 * * * * 02 *\), each of the 5 asterisks needs to be replaced by any of the digits \(0, 2, 4, 7, 8, 9\) (digits can repeat) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\{a\_n\}$ be a geometric sequence with a common ratio of $q$ ($|q| > 1$). Define $b\_n = a\_n + 1 (n=1,2,...)$. If the sequence $\{b\_n\}$ has four consecutive terms in the set $\{-53, -23, 19, 37, 82\}$, find the value of $6q$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n \to \infty}\displaystyle\sum_{r=1}^{n}\dfrac{r^3}{n^4 + n^2 + r^2}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the line $l$ passing through $M(-2,0)$ intersects the ellipse $x^{2}+2y^{2}=2$ at points $P_{1}$ and $P_{2}$, and the midpoint of segment $P_{1}P_{2}$ is $P$. If the slope of line $l$ is $k_{1}$ ($k_{1}\neq 0$) and the slope of line $OP$ (where $O$ is the origin) is $k_{2}$, find the value of $k_{1}k_{2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using  $600$  cards,  $200$  of them having written the number  $5$ ,  $200$  having a  $2$ , and the other  $200$  having a  $1$ , a student wants to create groups of cards such that the sum of the card numbers in each group is  $9$ . What is the maximum amount of groups that the student may create?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f(x) =x^4+14x^3+52x^2+56x+16.$  Let $z_1, z_2, z_3, z_4$ be the four roots of $f$.  Find the smallest possible value of $|z_az_b+z_cz_d|$ where $\{a,b,c,d\}=\{1,2,3,4\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a number, we calculate its square and add $1$ to the sum of the digits in this square, obtaining a new number. If we start with the number $7$ we will obtain, in the first step, the number $1+(4+9)=14$, since $7^2 = 49$. What number will we obtain in the $1999$th step?
Please reason step by step, and put your final answer within \boxed{}.


Context:
9. Given that the increasing function $f(x)$ is defined on $(0,+\infty)$, and for any positive number $x$ it satisfies $f(x)$ $f\left[f(x)+\frac{1}{x}\right]=1$, find the value of $f(1)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Barney Schwinn notices that the odometer on his bicycle reads $1441$, a palindrome, because it reads the same forward and backward. After riding more hours that day and $6$ the next, he notices that the odometer shows another palindrome, $1661$. What was his average speed in miles per hour?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In acute $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $a+2b=4$ and $a\sin A+4b\sin B=6a\sin B\sin C$, find the minimum value of $c^{2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $k$  and  $n$  be positive integers. Consider an array of  $2\left(2^n-1\right)$  rows by  $k$  columns. A  $2$ -coloring of the elements of the array is said to be *acceptable* if any two columns agree on less than  $2^n-1$  entries on the same row.

Given  $n$ , determine the maximum value of  $k$  for an acceptable  $2$ -coloring to exist.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the polynomials $p(x)$ for which the function $\sin p(x)$ is periodic.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the equation: $(2x+1)^2=3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all roots of the polynomial $x^4+2x^3-7x^2-8x+12$. Enter your answer as a list of numbers separated by commas.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimal value of integer $ $ that guarantees: Among $ n$ sets, there exits at least three sets such that any of them does not include any other; or there exits at least three sets such that any two of them includes the other.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all polynomials \( P(x) \) with integer coefficients such that \( P(n) \) is a palindrome for every integer \( n \geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A box $2$ centimeters high, $3$ centimeters wide, and $5$ centimeters long can hold $40$ grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold $n$ grams of clay. What is $n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), and $P$ is a point on the ellipse, with $\overrightarrow{PF_{1}} \cdot (\overrightarrow{OF_{1}} + \overrightarrow{OP}) = 0$, if $|\overrightarrow{PF_{1}}| = \sqrt{2}|\overrightarrow{PF_{2}}|$, determine the eccentricity of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The area of the largest square that can be inscribed in a regular hexagon with side length $1$ can be expressed as $a - b\sqrt{c}$, where $c$ is not divisible by the square of any prime. Find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find \( \lim_{n \to \infty} a_n^n \) where \( a_n = \frac{(n!)^2}{(n+k)!(n-k)!} \) for each \( k \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x_1,x_2,\ldots,x_n$ be $n$ non-negative numbers ($n>2$) with a fixed sum $S$. What is the maximum of $x_1 x_2 + x_2 x_3 + \dots + x_n x_1$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$ \lim_{x\to 3} \frac {xe^x-3e^x-xe^3+3e^3}{\ln^2(x^2+x-11)} $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u-3w, v+4w)$ with $0\le u\le1$, $0\le v\le1,$ and $0\le w\le1$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the line integral \( \int_{dA} (x^2 \, dx + y^2 \, dy) \) over the region \( A = \{(x, y) \in \mathbb{R}^2 : -\frac{\pi}{2} \leq x \leq \frac{\pi}{2}, \ -1 \leq y \leq \cos(x)\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A right rectangular prism  $P$  (i.e., a rectangular parallelpiped) has sides of integral length  $a, b, c,$  with  $a\le b\le c.$   A plane parallel to one of the faces of  $P$  cuts  $P$  into two prisms, one of which is similar to  $P,$  and both of which have nonzero volume.  Given that  $b=1995,$  for how many ordered triples  $(a, b, c)$  does such a plane exist?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A bowl contained  $10\%$  blue candies and  $25\%$  red candies.  A bag containing three quarters red candies and one quarter blue candies was added to the bowl.  Now the bowl is  $16\%$  blue candies.  What percentage of the candies in the bowl are now red?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the Cauchy Principal Value of the integral \( \int_{0}^{2\pi}\frac{d\theta}{1+2\cos(\theta)} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut. What is the smallest possible volume of the largest of the three cuboids?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a_0=1, a_1=2$, and $a_n=4 a_{n-1}-a_{n-2}$ for $n \geq 2$. Find an odd prime factor of $a_{2015}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A total of  $731$  objects are put into  $n$  nonempty bags where  $n$  is a positive integer. These bags can be distributed into  $17$  red boxes and also into  $43$  blue boxes so that each red and each blue box contain  $43$  and  $17$  objects, respectively. Find the minimum value of  $n$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Unit circles $a, b, c$ satisfy $d(a, b) = 1$, $d(b, c) = 2$, and $d(c, a) = 3,$ where $d(x, y)$ is defined to be the minimum distance between any two points on circles $x$ and $y$. Find the radius of the smallest circle entirely containing $a$, $b$, and $c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Joan's high school played 1,200 baseball games this year. Out of them, 30% were played at night, 20% were played during holidays, and 25% were played during her away days due to family visits. Additionally, Joan was sick and missed 50 games throughout the year. She attended 50% of the night games, 75% of the holiday games, and 60% of the away day games. How many baseball games did Joan miss?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A, E, H, L, T$, and $V$ be chosen independently and at random from the set $\left\{0, \frac{1}{2}, 1\right\}$. Compute the probability that $\lfloor T \cdot H \cdot E\rfloor=L \cdot A \cdot V \cdot A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $(a_n)_{n \geq 0}$  be the sequence of integers defined recursively by  $a_0 = 0, a_1 = 1, a_{n+2} = 4a_{n+1} + a_n$  for  $n \geq 0.$  Find the common divisors of  $a_{1986}$  and  $a_{6891}.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest natural number that cannot be written in the form \(\frac{2^{a} - 2^{b}}{2^{c} - 2^{d}}\), where \(a\), \(b\), \(c\), and \(d\) are natural numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A bacterium develops on a $100 \times 100$ grid. It can contaminate a new cell if and only if two adjacent cells are already contaminated. What is the minimal number of initially contaminated cells required for the bacterium to be able to spread everywhere on the grid?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$.  What is $k$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For $n \ge 3,$ if $$\sum_{k=1}^{n}x_k^m = y_m = (m+1)(n+1)-2, \quad \forall 1 \leqslant m \leqslant n,$$ then find the value of $$\sum_{k=1}^{n}x_k^{n+1} = y_{n+1}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Kelvin the Frog was bored in math class one day, so he wrote all ordered triples $(a, b, c)$ of positive integers such that $abc=2310$ on a sheet of paper. Find the sum of all the integers he wrote down. In other words, compute $$\sum_{\stackrel{abc=2310}{a,b,c\in \mathbb{N}}} (a+b+c),$$ where $\mathbb{N}$ denotes the positive integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\omega = \cos\frac{2\pi}{7} + i \cdot \sin\frac{2\pi}{7},$ where $i = \sqrt{-1}.$ Find the value of the product\[\prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right).\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of \(2 \times 2\) matrices \(A\) with entries in \(\mathbf{Z}/3\mathbf{Z}\) such that \(A^{50} = I\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
When 1 / 20 % of 8000 is subtracted from some fraction of 8000, the difference is 796. What is the fraction that needs to be subtracted?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that \[\operatorname{lcm}(1024,2016)=\operatorname{lcm}(1024,2016,x_1,x_2,\ldots,x_n),\] with  $x_1$ ,  $x_2$ ,  $\cdots$ ,  $x_n$  are distinct postive integers. Find the maximum value of  $n$ .

*Proposed by Le Duc Minh*
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three boys \( B_{1}, B_{2}, B_{3} \) and three girls \( G_{1}, G_{2}, G_{3} \) are to be seated in a row according to the following rules:
1) A boy will not sit next to another boy and a girl will not sit next to another girl,
2) Boy \( B_{1} \) must sit next to girl \( G_{1} \).
If \( s \) is the number of different such seating arrangements, find the value of \( s \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $N \in \left \{10, 11, \ldots, 99\right \} $ be a two-digit positive integer. Compute the number of values of $N$ for which the last two digits in the decimal expansion of $N^{21}$ are the digits of $N$ in the same order.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For the multiplicative group Z*35, determine the number of generators and list them.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using each of the eight digits $1,3,4,5,6,7,8$ and $9$ exactly once, a three-digit number $A$, two two-digit numbers $B$ and $C$, $B<C$, and a one digit number $D$ are formed. The numbers are such that $A+D=B+C=143$. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all polynomial functions \( p(x) \) with complex coefficients such that \( p(x) \) maps real numbers to real numbers and non-real complex numbers to non-real complex numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function f(x) = x^3 - 3x + m, and three numbers a, b, c chosen from the interval [0, 2], there exists a triangle with side lengths f(a), f(b), f(c). Calculate the range of values for m. Express your answer as an inequality (e.g., m > value).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The polynomial $f(x)=x^{2007}+17x^{2006}+1$ has distinct zeroes $r_1,\ldots,r_{2007}$.  A polynomial $P$ of degree $2007$ has the property that
\[P\left(r_j+\dfrac{1}{r_j}\right)=0\]for $j=1,\ldots,2007$.  Determine the value of $\frac{P(1)}{P(-1)}$.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A  $6 \times 6$  board is given such that each unit square is either red or green. It is known that there are no  $4$  adjacent unit squares of the same color in a horizontal, vertical, or diagonal line. A  $2 \times 2$  subsquare of the board is *chesslike* if it has one red and one green diagonal. Find the maximal possible number of chesslike squares on the board.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Annie spends 2 hours a week on chess club, 8 hours a week on drama club, and 3 hours a week on glee club. If there are 12 weeks in each semester and Annie takes the first two weeks off sick, how many hours of extracurriculars does she do before midterms?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The bisector $DE$ of triangle $ADC$ is extended to its intersection with the circumscribed circle at point $B$. Given that the length $BD = 1$ and $\sin \angle ADC = a$. Find the area of the quadrilateral $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a room illuminated by two light bulbs, each bulb has an exponential lifetime with a parameter of \( \lambda = 1 \). The bulbs are replaced only when both have burned out. What fraction of the time is the room half-lit, meaning only one bulb is functioning?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A farmer wants to start growing watermelons. He wants to sell at least 10,000 watermelons each year. Watermelons are grown from seeds (one seed grows into one watermelon). Each watermelon can produce 250 good seeds, which can be planted the next year, but in that case, the watermelon cannot be sold. What is the minimum number of watermelon seeds the farmer must buy initially to start growing watermelons and never have to buy seeds again?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On March 1st the sun sets at 6 PM. Every day after the sun sets 1.2 minutes later. It is a certain time and 40 days after March 1st. The sun will set in 38 minutes. What is the current time?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the values of  $a\in [0,\infty)$  for which there exist continuous functions  $f:\mathbb{R}\rightarrow \mathbb{R}$ , such that  $f(f(x))=(x-a)^2,\ (\forall)x\in \mathbb{R}$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Person A and person B are playing a game with the following rules: In odd-numbered rounds, the probability of A winning is $\frac{3}{4}$; in even-numbered rounds, the probability of B winning is $\frac{3}{4}$. There are no ties in any round. The game ends when one person has won 2 more rounds than the other. Determine the expected number of rounds played when the game ends.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The mean of the set of numbers $\{106, 102, 95, 103, 100, y, x\}$ is 104. What is the median of this set of seven numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of the poles of the modular lambda function in terms of the variable $\lambda$. Provide your answer as a single number or expression.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ten families have an average of 2 children per family. If exactly two of these families are childless, what is the average number of children in the families with children? Express your answer as a decimal to the nearest tenth.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume of the region bounded by the surfaces $x^2 + z = 1$ and $y^2 + z = 1$ in the first octant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $n,k$  be positive integers such that  $n>k$ . There is a square-shaped plot of land, which is divided into  $n\times n$  grid so that each cell has the same size. The land needs to be plowed by  $k$  tractors; each tractor will begin on the lower-left corner cell and keep moving to the cell sharing a common side until it reaches the upper-right corner cell. In addition, each tractor can only move in two directions: up and right. Determine the minimum possible number of unplowed cells.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the expression $x^2 + 9x + 20$ can be written as $(x + a)(x + b)$, and the expression $x^2 + 7x - 60$ can be written as $(x + b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
3. On each of the lines $y=1$ and $y=12$, there are 200 points with abscissas $1,2,3, \ldots, 200$. In how many ways can three points be chosen from the 400 marked points so that they form the vertices of a right triangle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
## Task Condition

Calculate the definite integral:

$$
\int_{\arccos (1 / \sqrt{10})}^{\arccos (1 / \sqrt{26})} \frac{12 d x}{(6+5 \tan x) \sin 2 x}
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$, together with its interior. For real $r\geq0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point in $B$. The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$, where $a,$ $b,$ $c,$ and $d$ are positive real numbers. What is $\frac{bc}{ad}?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
在平面直角坐标系 xOy 中，A 为椭圆 \df{x^2}{4}+y^2=1 上一点，M 为线段 OA 上的动点. 过 M 作直线与椭圆交于 P, Q 两点，若 \overrightarrow{PM}=2\overrightarrow{MQ}，求四边形 OPAQ 面积的最大值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \(\int_C \vec G \cdot \vec{dr}\) where \(\vec G(x,y,z) = (8xz + 5y)\vec i + (2yz + 5x)\vec j + (x^2 + 3z^2)\vec k\) and \(C\) is the curve consisting of the line segment from \((1,0,1)\) to \((3,2,0)\) followed by the parabola \(y=x^2, z=1\) from \((3,9,1)\) to \((0,0,1)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an election for the Peer Pressure High School student council president, there are 2019 voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice's boyfriend Bob votes for Alice as well. Then one by one, each of the remaining 2016 voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate's votes. For example, the first undecided voter David has a  $\tfrac{2}{3}$  probability of voting for Alice and a  $\tfrac{1}{3}$  probability of voting for Celia.

What is the probability that Alice wins the election (by having more votes than Celia)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
An arithmetic sequence has a common difference of 4 and each term is a real number. The square of the first term and the sum of the remaining terms do not exceed 100. What is the maximum number of terms that such a sequence can have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A positive integer $n$ is magical if $\lfloor\sqrt{\lceil\sqrt{n}\rceil}\rfloor=\lceil\sqrt{\lfloor\sqrt{n}\rfloor}\rceil$ where $\lfloor\cdot\rfloor$ and $\lceil\cdot\rceil$ represent the floor and ceiling function respectively. Find the number of magical integers between 1 and 10,000, inclusive.
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Context:
The angle between two tangents drawn from one point to a circle is $70^{\circ}$.

Find the angular measures of the arcs enclosed between the points of tangency.
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Context:
Given that the regular price for one backpack is $60, and Maria receives a 20% discount on the second backpack and a 30% discount on the third backpack, calculate the percentage of the $180 regular price she saved.
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Context:
Find the prime decomposition of 2 in the cyclotomic field \( \mathbb{Q} ( \xi_5 ) \).
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Context:
Several sets of prime numbers, such as $\{7,83,421,659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?
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Context:
Xiaohua's route from home to school consists of a flat road and a downhill road. Assuming he always maintains a speed of 60 meters per minute on the flat road, 80 meters per minute on the downhill road, and 40 meters per minute on the uphill road, it takes him 10 minutes to get from home to school and 15 minutes to get from school to home. How far is Xiaohua's home from his school?
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Context:
Let \(ABCDE\) be a convex pentagon such that \(AB = AE = CD = BC + DE = 1\) and \(\angle ABC = \angle DEA = 90^\circ\). Calculate the area of \(ABCDE\).
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Context:
In triangle $\triangle ABC$, point $D$ is a point on side $AC$, $BD=1$, $\sin \angle DBC=\frac{3}{5}$, $\cos \angle ABC=\frac{\sqrt{10}}{10}$. Find the minimum value of the area of $\triangle ABC$.
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Context:
There are  $2022$  signs arranged in a straight line. Mark tasks Auto to color each sign with either red or blue with the following condition: for any given sequence of length  $1011$  whose each term is either red or blue, Auto can always remove  $1011$  signs from the line so that the remaining  $1011$  signs match the given color sequence without changing the order. Determine the number of ways Auto can color the signs to satisfy Mark's condition.
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Context:
Last summer $30\%$ of the birds living on Town Lake were geese, $25\%$ were swans, $10\%$ were herons, and $35\%$ were ducks. What percent of the birds that were not swans were geese?
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Context:
In the expression \( \left( {\_ \_  \times  \_ \_ }\right)  + \left( {\_ \_ \_  \times  \_ \_ \_ }\right) \) each blank is to be filled in with one
 of the digits\(1,2,3,\) or 4, with each digit being used once. How many different values can
 be obtained?
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Context:
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be three vectors such that $\mathbf{a}$ is a unit vector, $\mathbf{b}$ and $\mathbf{c}$ have magnitudes of 2, and $\mathbf{a} \cdot \mathbf{b} = 0$, $\mathbf{b} \cdot \mathbf{c} = 4$. Given that
\[
\mathbf{a} = p(\mathbf{a} \times \mathbf{b}) + q(\mathbf{b} \times \mathbf{c}) + r(\mathbf{c} \times \mathbf{a}),
\]
and $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 2$, find the value of $p + q + r$.
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Context:
Let $\Gamma_1, \Gamma_2, \Gamma_3$ be three pairwise externally tangent circles with radii $1,2,3,$ respectively. A circle passes through the centers of $\Gamma_2$ and $\Gamma_3$ and is externally tangent to $\Gamma_1$ at a point $P.$ Suppose $A$ and $B$ are the centers of $\Gamma_2$ and $\Gamma_3,$ respectively. What is the value of $\frac{{PA}^2}{{PB}^2}?$

[i]Proposed by Kyle Lee[/i]
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Context:
Let $A$ and $B$ be points on a circle $\mathcal{C}$ with center $O$ such that $\angle AOB = \frac{\pi}{2}$. Circles $\mathcal{C}_1$ and $\mathcal{C}_2$ are internally tangent to $\mathcal{C}$ at $A$ and $B$ respectively and are also externally tangent to one another. The circle $\mathcal{C}_3$ lies in the interior of $\angle AOB$ and is tangent externally to $\mathcal{C}_1$, $\mathcal{C}_2$ at $P$ and $R$ and internally tangent to $\mathcal{C}$ at $S$. Find the measure of $\angle PSR$. \( \text{Express your answer in radians.} \)
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Context:
求最大的正实数 k，使得对任意满足 a+b+c=1 的非负实数 a,b,c，均有\ds{}{}\df{a}{1+9bc+k(b-c)^2}\geq\df{1}{2}.
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Context:
The number $a=\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying
\[\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\]is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\}=x- \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p+q$?
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Context:
Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positive integers. What is $a+b?$
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Context:
Around a circle are written the natural numbers from 1 to \( N \) with \( N > 2 \), such that each number is written only once and each pair of neighboring numbers has at least one digit in common. Find the smallest \( N > 2 \) for which this is possible.
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Context:
In how many ways can the numbers $1,2, \ldots, 2002$ be placed at the vertices of a regular 2002-gon so that no two adjacent numbers differ by more than 2? (Rotations and reflections are considered distinct.)
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Context:
The largest prime factor of 101101101101 is a four-digit number $N$. Compute $N$.
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Context:
In Figure 1, $AEFD$ is a unit square. The ratio of the length of the rectangle $ABCD$ to its width is equal to the ratio of the length of the rectangle $BCFE$ to its width. If the length of $AB$ is $W$ units, find the value of $W$.
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Context:
In a tetrahedron \( ABCD \), \( AB = AC = AD = 5 \), \( BC = 3 \), \( CD = 4 \), \( DB = 5 \). Find the volume of this tetrahedron.
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Context:
A pyramid has a rectangular base, its side edges are equal to each other and to one type of base edge. By varying the other type of base edge, when will the volume of the pyramid be the largest?
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Context:
A vertex-induced subgraph is a subset of the vertices of a graph together with any edges whose endpoints are both in this subset. An undirected graph contains 10 nodes and $m$ edges, with no loops or multiple edges. What is the minimum possible value of $m$ such that this graph must contain a nonempty vertex-induced subgraph where all vertices have degree at least 5?
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Context:
Let $a,b$ be positive integers satisfying
$$\sqrt{\frac{ab}{2b^2-a}}=\frac{a+2b}{4b}$$.
Find $|10(a-5)(b-15)|+8$.
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Context:
Determine for how many positive integers $n \in \{1, 2, \ldots, 2022\}$ it holds that $402$ divides at least one of
\[ n^2 - 1, \quad n^3 - 1, \quad n^4 - 1 \]
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Context:
Given the ellipse $$C: \frac {x^{2}}{4}+ \frac {y^{2}}{b^{2}}=1(0＜b＜2)$$, a straight line with a slope angle of $$\frac {3π}{4}$$ intersects the ellipse C at points A and B. The midpoint of the line segment AB is M, and O is the coordinate origin. The angle between $$\overrightarrow {OM}$$ and $$\overrightarrow {MA}$$ is θ, and |tanθ|=3. Find the value of b.
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Context:
 Let $A$ be the the area enclosed by the relation
\[x^2+y^2\le2023.\]Let $B$ be the area enclosed by the relation
\[x^{2n}+y^{2n}\le\left(A\cdot\frac{7}{16\pi}\right)^{n/2}.\]Compute the limit of $B$ as $n\rightarrow\infty$ for $n\in\mathbb{N}$.
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Context:
Evaluate the limit: $$\lim_{x\to0^+}\frac{e^x+e^{-x}}{-\cos x}\cdot\frac{x^2+2R_1x+R_1^2}{2x+\bar R_1(x)}$$
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Context:
Determine if the limit \( \lim_{x \rightarrow a} \frac{f(x) - f(2a -x)}{2(x-a)} \) always exists, given that a function \( f(x) \) is continuous in a neighborhood of \( a \) and \( f'(a) \) exists.
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Context:
Example 6 Let $a=\sqrt{17}, b=\mathrm{i} \sqrt{19}$. Try to find the maximum value of $\frac{|a-z|}{|b-z|}$, where $z$ is any complex number with a modulus of 1.
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Context:
Simplify first, then evaluate $2\left(x^{2}-2x^{2}y\right)-\left[3\left(x^{2}-xy^{2}\right)-\left(x^{2}y-2xy^{2}+x^{2}\right)\]$, where $x=-\dfrac{1}{3}$ and $y=-2$.
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Context:
Given the equation $2(x^3+y^3+z^3) = 3(x+y+z)^2$, where $x, y, z$ are non-negative real numbers, find the minimum value of $x+y+z$.
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Context:
Find the constant $c$ such that the random variable \( Y = \frac{c(X_1 + X_2)}{\sqrt{(X_3)^2 + (X_4)^2}} \) follows a t-distribution, given that $X_1$, $X_2$, $X_3$, and $X_4$ are independent standard normally distributed random variables.
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Context:
Along the school corridor hangs a Christmas garland consisting of red and blue bulbs. Next to each red bulb, there must be a blue bulb. What is the maximum number of red bulbs that can be in this garland if there are a total of 50 bulbs?
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Context:
Given a regular 2007-gon. Find the minimal number $k$ such that: Among every $k$ vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon.
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Context:
Determine the number of complex numbers \( z \) with \( |z| < 1 \) that satisfy the equation \( az^4 + z^3 + bz^2 + cz + d = 0 \), given that \( a, b, c, \) and \( d \) are positive numbers such that \( a + b + c + d < 1 \).
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Context:
设集合 $S$ 是 $\{0,1, \cdots, 98\}$ 的 $m(m \geqslant 3)$元子集，满足对任意的 $x, y \in S$ ，均存在 $z \in$ $S$, 使得 $x+y \equiv 2 z(\bmod 99)$. 求 $m$ 的所有可能值的和.
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Context:
Mr. Johnson is organizing the school Christmas play and needs 100 volunteers to help with decorating the auditorium. 4 students from each of the school's 5 math classes have volunteered to help. 10 teachers have also volunteered, including 3 who are skilled in carpentry. Additionally, 15 parents have agreed to volunteer, but only 6 of them have experience with lighting and sound setup. To ensure a smooth process, Mr. Johnson needs 8 more volunteers with carpentry skills and 10 more volunteers with lighting and sound experience. How many more volunteers will Mr. Johnson need in total?
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Context:
Quadrilateral \(ABCD\) is such that \(\angle BAC = \angle CAD = 60^\circ\) and \(AB + AD = AC\). It is also known that \(\angle ACD = 23^\circ\). How many degrees is \(\angle ABC\)?
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Context:
Determine the values of the parameter \( t \) for which the linear transformation \( \psi: \mathbb{R}^4 \rightarrow \mathbb{R}^3 \) given by
\[ \psi([x_1,x_2,x_3,x_4])=[x_1+x_2+x_3+2x_4,x_1+tx_2+x_3+3x_4,2x_1+x_2+tx_3+3x_4] \]
is surjective.
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Context:
A triangle in the $x y$-plane is such that when projected onto the $x$-axis, $y$-axis, and the line $y=x$, the results are line segments whose endpoints are $(1,0)$ and $(5,0),(0,8)$ and $(0,13)$, and $(5,5)$ and $(7.5,7.5)$, respectively. What is the triangle's area?
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Context:
Given that
$\frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!}$
 find the greatest integer that is less than $\frac N{100}$.
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Context:
Given a figure  $F: x^2+\frac{y^2}{3}=1$  on the coordinate plane. Denote by  $S_n$  the area of the common part of the  $n+1' s$  figures formed by rotating  $F$  of  $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$  radians counterclockwise about the origin. Find  $\lim_{n\to\infty} S_n$ .
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Context:
A，B，C，D四点在同一圆周上，且BC=CD=4,E为AC与BD的交点，且AE=6，线段BE和DE的长都是整数，则BD的长等于多少？
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Context:
Given a "number array" as shown on the left, where each column forms an arithmetic sequence, and starting from the third row, each row forms a geometric sequence with a common ratio equal to that of every other row. The first column is $\frac{1}{4}$, $\frac{1}{2}$, $\frac{3}{4}$, ..., the second column is $\frac{1}{4}$, $\frac{1}{4}$, $\frac{3}{8}$, ..., and the third column is $\frac{1}{4}$, $\frac{1}{4}$, $\frac{3}{16}$, ... . Denote the element in the $i$-th row and $j$-th column as ${a_{ij}}\left( {i \geqslant j,i,j \in {N^ * }} \right)$, find the value of ${a_{83}}$.
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Context:
The country of HMMTLand has 8 cities. Its government decides to construct several two-way roads between pairs of distinct cities. After they finish construction, it turns out that each city can reach exactly 3 other cities via a single road, and from any pair of distinct cities, either exactly 0 or 2 other cities can be reached from both cities by a single road. Compute the number of ways HMMTLand could have constructed the roads.
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Context:
Kayla rolls four fair $6$-sided dice. What is the probability that at least one of the numbers Kayla rolls is greater than $4$ and at least two of the numbers she rolls are greater than $2$?
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Context:
求$\cos \frac { \pi } { 1 5 } \cdot \cos \frac { 2 \pi } { 1 5 } \cdot \cos \frac { 3 \pi } { 1 5 } \cdot \cdots \cdot \cos \frac { 1 4 \pi } { 1 5 }$的值.
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Context:
A 360 meter long train starts from rest and accelerates until it crosses a man standing on a platform in 20 seconds. The track has an inclination of 5 degrees to the horizontal and the train has an acceleration of 1 m/s^2. Neglecting air resistance, what is the final speed of the train when it completely passes by the man?
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Context:
Problem 6. (30 points) Ten balls of the same radius are arranged in the form of a triangular pyramid such that each ball touches at least three others. Find the radius of the sphere in which the pyramid of balls is inscribed, if the radius of the sphere inscribed in the center of the pyramid of balls, touching six identical balls, is $\sqrt{2}-1$.

#
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Context:
Consider the sequence \( a_{1}, a_{2}, \ldots, a_{105} \) which is a permutation of the numbers \( 1, 2, \ldots, 105 \). This sequence satisfies the condition that for any \( m \in \{3, 5, 7\} \), and for all \( n \) such that \( 1 \leq n < n+m \leq 105 \), the difference \( a_{n+m} - a_{n} \) is divisible by \( m \). Determine the number of such distinct permutations.
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Context:
Evaluate the limit of the sequence as \( n \to \infty \):

\[ \lim_{n\to \infty} \frac{((n^3+2n^2)^{1/3}-n)\cdot((n^3+2n^2)^{1/3}+n)}{(2n^2+n^3)^{1/3}+n} \]
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Context:
1. [15] Four standard six-sided dice are rolled. Find the probability that, for each pair of dice, the product of the two numbers rolled on those dice is a multiple of 4 .
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Context:
While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$.
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Context:
I2.4 Determine the unit digit, $\delta$, of $\left(\left(\gamma^{\gamma}\right)^{\gamma}\right)^{\gamma}$.
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Context:
There are $13$ cities in a certain kingdom. Between some pairs of the cities, two-way direct bus, train, or plane connections are established. What is the least possible number of connections to be established so that, by choosing any two means of transportation, one can travel from any city to any other without using the third kind of vehicle?
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Context:
Given a positive integer $n$ greater than 2004, fill the numbers $1, 2, \cdots, n^2$ into an $n \times n$ chessboard (consisting of $n$ rows and $n$ columns of squares) such that each square contains exactly one number. If a number in a square is greater than the numbers in at least 2004 other squares in its row and at least 2004 other squares in its column, then this square is called a "super square." Determine the maximum number of "super squares" on the chessboard.
(Feng Yuefeng, problem contributor)
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Context:
On Saturdays and Sundays, Suzanne makes french toast for the family. She and her husband each have some whole slices and her daughters split 1 slice. She uses a thick sliced bread that comes 12 slices per loaf. Over 52 weeks, she needs 26 loaves of bread to make french toast. How many whole slices do Suzanne and her husband each have?
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Context:
Circles $C_{1}, C_{2}, C_{3}$ have radius 1 and centers $O, P, Q$ respectively. $C_{1}$ and $C_{2}$ intersect at $A, C_{2}$ and $C_{3}$ intersect at $B, C_{3}$ and $C_{1}$ intersect at $C$, in such a way that $\angle A P B=60^{\circ}, \angle B Q C=36^{\circ}$, and $\angle C O A=72^{\circ}$. Find angle $A B C$ (degrees).
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Context:
Find 10 distinct natural numbers with the property that their sum is divisible by each of them.
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Context:
Let $A=\{1,2,3,\cdots ,17\}$. A mapping $f:A\rightarrow A$ is defined as follows: $f^{[1]}(x)=f(x)$, $f^{[k+1]}(x)=f(f^{[k]}(x))$ for $k\in\mathbb{N}$. Suppose that $f$ is bijective and that there exists a natural number $M$ such that:

1. When $m<M$ and $1\le i\le 16$, we have $f^{[m]}(i+1)- f^{[m]}(i) \not=\pm 1\pmod{17}$ and  $f^{[m]}(1)- f^{[m]}(17) \not=\pm 1\pmod{17}$;
2. When $1\le i\le 16$, we have $f^{[M]}(i+1)- f^{[M]}(i)=\pm 1 \pmod{17}$ and $f^{[M]}(1)- f^{[M]}(17)=\pm 1\pmod{17}$.

Find the maximal value of $M$.
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Context:
Given the curve $y = x^n + 1$ ($n \in \mathbb{N}^*$) at the point $(1,1)$, let $x_n$ be the x-coordinate of the intersection point between the tangent line and the x-axis. Determine the value of $\log_{2010} x_1 + \log_{2010} x_2 + \dots + \log_{2010} x_{2009}$.
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Context:
Tom invites his parents, 3 siblings, 2 grandparents, and 4 cousins to his house. They each eat 4 times a day. For breakfast and lunch, there are 2 courses per meal, whereas for dinner and a late-night snack, there are 3 courses per meal. Additionally, everyone uses 2 plates per course. His parents and siblings stay for 6 days, while his grandparents stay for 4 days, and his cousins stay for 3 days. How many plates do Tom and his guests use during their respective stays?
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Context:
## Task 3 - 200923

Given two circles $k_{1}$ and $k_{2}$ with radii $r_{1}$ and $r_{2}$ respectively, where $r_{1}>r_{2}$.

Furthermore, it is assumed that both circles touch each other externally, meaning they have exactly one common internal tangent. This internal tangent intersects one of the common external tangents of both circles at $P$ and the other common tangent at $Q$.

Determine, under these conditions, the length $P Q$ from $r_{1}$ and $r_{2}$!
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Context:
7. On the sides $\mathrm{AB}$ and $\mathrm{AC}$ of the right triangle $\mathrm{ABC}\left(\angle B C A=90^{\circ}\right)$, right triangles АВТ and АСК are constructed externally such that $\angle A T B=\angle A K C=90^{\circ}$, $\angle A B T=\angle A C K=60^{\circ}$. On the side $\mathrm{BC}$, a point $\mathrm{M}$ is chosen such that $\mathrm{BM}=\mathrm{MC}$. Determine the degree measure of angle КМТ.
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Context:
Determine the number of six-letter words where the first and last two letters are the same (e.g., "aabbaa").
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Context:
2014 triangles have non-overlapping interiors contained in a circle of radius 1. What is the largest possible value of the sum of their areas?
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Context:
What is the minimum number of cells in a $3 \times 2016$ board that can be colored such that each cell has at least one adjacent colored cell?
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Context:
21. Let $A B C D E F$ be a hexagon such that the diagonals $A D, B E$ and $C F$ intersect at the point $O$, and the area of the triangle formed by any three adjacent points is 2 (for example, area of $\triangle B C D$ is 2 ). Find the area of the hexagon.
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Context:
Given that $A /_{\ n }^{ 2 }=7A /_{\ n-4 }^{ 2 }$, find the value of $n$.
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Context:
You need to hit a target that is $A=40 \mathrm{~m}$ away and $B=30 \mathrm{~m}$ high with a projectile launched with the smallest possible initial speed. At what angle should the projectile be launched?
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Context:
Find the limit of the following expression as p approaches 0:

$$\lim_{p\to0} \left[p \int_{-\infty}^\infty {{{\ln}^2(1+x)}\over{ (p^2 +x^2)}^2}}\,\mathrm dx\right]$$
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Context:
Given a positive integer $k$, let $f(k)$ be the sum of the $k$-th powers of the primitive roots of $73$. For how many positive integers $k < 2015$ is $f(k)$ divisible by $73?$

[i]Note: A primitive root of $r$ of a prime $p$ is an integer $1 \le r < p$ such that the smallest positive integer $k$ such that $r^k \equiv 1 \pmod{p}$ is $k = p-1$.[/i]
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Context:
Evaluate the limit:
$$\lim_{n\to\infty}\sum_{k=0}^n \binom n k\frac{(-1)^{k}}{k+2} \left(\ln(n)-\frac{1}{k+2}\right)$$
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Context:
When a 10° angle is viewed under a 5x magnifying glass, what is the resulting angle measure in degrees? Express your answer in degrees, including the degree symbol (°).
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Context:
Given a triangle \(A B C\) with \(A B = A C\) and \(\angle A = 110^{\circ}\). Inside the triangle, a point \(M\) is chosen such that \(\angle M B C = 30^{\circ}\) and \(\angle M C B = 25^{\circ}\). Find \(\angle A M C\).
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Context:
A gambler bought $10,000 worth of chips at a casino in denominations of $50 and $200. That evening, the gambler lost 30 chips, and then cashed in the remainder. If the number of $50 chips lost was 3 times the number of $200 chips lost, what is the largest amount of money that the gambler could have received back?
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Context:
In an experiment involving two urns, Urn 1 contains 3 blue balls and 1 red ball, while Urn 2 contains 1 blue ball and 3 red balls. It is equally likely to choose either urn. Players draw a ball at random, replace it, and guess if the urn is majority blue or red based on the ball's color. Each player sees the previous player's guess but not the ball. A cascade occurs if the first two players draw the same color, influencing the third player's guess regardless of their draw. What is the probability that a wrong cascade occurs after the first two players guess?
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Context:
Let $ABC$ be a triangle with $BC = 49$ and circumradius $25$. Suppose that the circle centered on $BC$ that is tangent to $AB$ and $AC$ is also tangent to the circumcircle of $ABC$. Then \[\frac{AB \cdot AC}{-BC + AB + AC} = \frac{m}{n}\] where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.
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Context:
How many pairs \((m, n)\) of positive integers with \(m<n\) fulfill the equation

\[ \frac{3}{2008}=\frac{1}{m}+\frac{1}{n}? \]
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Context:
Let \( A \) be a \( {2n} \times  {2n} \) matrix, with entries chosen independently at random.
 Every entry is chosen to be 0 or 1,each with probability \( 1/2 \) . Find the
 expected value of \( \det \left( {A - {A}^{t}}\right) \) (as a function of \( n \) ),where \( {A}^{t} \) is the
 transpose of \( A \) .
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Context:
Consider the fiber product \(X \times_Z X\) with two projection arrows \(X \times_Z X \rightrightarrows X\). If we extend this to the fiber product \(X \times_Z X \times_Z X\), how many arrows are associated with this fiber product? Provide the total count of arrows.
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Context:
Find the order of the element $abc$ in the group defined by the presentation $$ G = \langle a,b,c \mid a^2=b^2=c^2=1,\, ac=ca,\, (ab)^3=(bc)^3=1 \rangle.$$
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Context:
Determine the number of conjugacy classes in the dihedral group $D_8$. Use the formula for the number of conjugacy classes given by \( \frac{1}{|G|}\sum|C_{G}(g)| \), where \( C_{G}(g) = \{h \in G ; gh=hg\} \).
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Context:
Evaluate the limit: $$ \lim_{x\to\infty} \left( (x+2017)^{1+\frac{1}{x}} \: -\: x^{1+\frac{1}{x+2017}} \right) $$
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Context:
In an exchange economy with two goods, x and m, and n consumers, half of the consumers have a utility function given by \( U_1(x_1) = 5\ln{x_1} + m_1 \), while the other half have \( U_2(x_2) = 3\ln{x_2} + m_2 \). Initially, each consumer possesses 20 units of good x and 10 units of good m. Determine the maximum amount of good x that the first type of consumers (with utility function \( U_1 \)) can acquire at a Pareto Optimal allocation, given that \( m_1 > 0 \) and \( m_2 > 0 \) for all consumers.
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Context:
What is the average length of a cycle in a permutation of the set \( \{1, 2, 3, \dots, n\} \)?
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Context:
Calculate the expected number of letters in a random string generated from the set \{A, B, C, D\} until the sequence "ABCD" appears consecutively.
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Context:
Evaluate the double integral \( \iint_R xy \, dx \, dy \) over the parallelogram defined by the lines \(2x-y=1\), \(2x-y=3\), \(x+y=-2\), and \(x+y=0\). Use the variable substitution \(u=2x-y\) and \(v=x+y\) to transform the integral.
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Context:
Determine the number of decreasing sequences of positive integers \(b_1 \geq b_2 \geq b_3 \geq \cdots \geq b_7 \leq 1500\) such that \(b_i - i\) is divisible by 3 for \(1 \leq i \le 7\). Express the number of such sequences as \({m \choose n}\) for some integers \(m\) and \(n\), and compute the remainder when \(m\) is divided by 1000.
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Context:
Simplify and write the result as a common fraction: $$\sqrt[4]{\sqrt[3]{\sqrt{\frac{1}{65536}}}}$$
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Context:
Let $ABCD$ be an isosceles trapezoid with $\overline{AD} \parallel \overline{BC}$. The incircle of $\triangle ABC$ has center $I$ and is tangent to $\overline{BC}$ at $P$. The incircle of $\triangle ABD$ has center $J$ and is tangent to $\overline{AD}$ at $Q$. If $PI = 8$, $IJ = 25$, and $JQ = 15$, compute the greatest integer less than or equal to the area of $ABCD$.
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Context:
If $(x-1)^{2011} = a + a_1x + a_2x^2 + \ldots + a_{2010}x^{2010} + a_{2011}x^{2011}$, then find the value of $a_1 + a_2 + a_3 + \ldots + a_{2010} + a_{2011}$.
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Context:
There are 198 positive integers between a certain number and 2000 (both inclusive) that are multiples of 10. What is the lower limit of this range?
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Context:
Jon’s textbooks weigh three times as much as Brandon’s textbooks. Jon has four textbooks with certain weights. Brandon's textbooks weigh 8 pounds. What are the weights of Jon's textbooks if their total weight is three times the weight of Brandon's textbooks?
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Context:
Determine the smallest positive integer \( n \) such that any coloring of the \( n \) vertices of a regular \( n \)-gon with three colors (red, yellow, and blue) results in four vertices of the same color forming an isosceles trapezoid.
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Context:
Given: Three vertices \( A, B, \) and \( C \) of a square are on the parabola \( y = x^2 \).
Find: The minimum possible area of such a square.
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Context:
Raashan, Sylvia, and Ted play the following game. Each starts with $\textdollar1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $\textdollar1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $\textdollar1$? (For example, Raashan and Ted may each decide to give $\textdollar1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $\textdollar0$, Sylvia will have $\textdollar2$, and Ted will have $\textdollar1$, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their $\textdollar1$ to, and the holdings will be the same at the end of the second round.)
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Context:
There are 3012 positive numbers with both their sum and the sum of their reciprocals equal to 3013. Let $x$ be one of these numbers. Find the maximum value of $x + \frac{1}{x}.$
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Context:
On a straight line $\ell$ there are four points, $A$, $B$, $C$, and $D$ in that order, such that $AB = BC = CD$. A point $E$ is chosen outside the straight line so that when drawing the segments $EB$ and $EC$, an equilateral triangle $EBC$ is formed. Segments $EA$ and $ED$ are drawn, and a point $F$ is chosen so that when drawing the segments $FA$ and $FE$, an equilateral triangle $FAE$ is formed outside the triangle $EAD$. Finally, the lines $EB$ and $FA$ are drawn, which intersect at the point $G$. If the area of triangle $EBD$ is $10$, calculate the area of triangle $EFG$.
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Context:
Given the inequality about $x$, $$\frac {x-a}{x^{2}-3x+2}$$≥0, the solution set is {x|1＜x≤a or x＞2}. Determine the range of values for $a$.
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Context:
Let $S$ be the set of integers of the form $2^x + 2^y + 2^z$, where $x$, $y$, $z$ are pairwise distinct non-negative integers. Determine the 100th smallest element of $S$.
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Context:
The charge for a single room at hotel P for a 5-day stay is 20 percent less than the charge for a single room at hotel R and 10 percent less than the charge for a single room at hotel G for the same duration. Additionally, the price per night at each hotel varies based on the number of rooms booked. If a group books 5 rooms at hotel R, their price per room is 15 percent less than when they book only 1 room. The charge for a single room at hotel R, when 5 rooms are booked, is what percent greater than the charge for a single room at hotel G for a 5-day stay, when no discounts are applied for the number of rooms booked?
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Context:
Inside the triangle $ABC$, a point $D$ is chosen such that $\angle BAD = 60^\circ$ and $\angle ABC = \angle BCD = 30^\circ$. It is given that $AB = 15$ and $CD = 8$. Find the length of segment $AD$. If necessary, round the answer to 0.01 or write the answer as a common fraction.
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Context:
In triangle \( KLM \), with all sides distinct, the bisector of angle \( KLM \) intersects side \( KM \) at point \( N \). A line through point \( N \) intersects side \( LM \) at point \( A \), such that \( MN = AM \). Given that \( LN = a \) and \( KL + KN = b \), find \( AL \).
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Context:
Let $A=\{0,1\}$ and $B=\{0,1,2\}$. Calculate the total number of possible mappings from set A to set B, where each element in A corresponds to a unique element in B. Express your answer as a single integer.
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Context:
The sum of five consecutive numbers is 225 and the smallest of these numbers is a multiple of 7. What is the smallest number?
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Context:
Given that \(p, q, \frac{2q-1}{p}, \frac{2p-1}{q} \in \mathbf{Z}\), and \(p > 1\), \(q > 1\), find the value of \(p + q\).
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Context:
For a sequence of distinct real numbers $b_1, b_2, \ldots, b_k$ in the interval $[1, 2023]$, what is the minimum value of $k$ such that there must exist $b_i$ and $b_j$ satisfying $0 < b_i - b_j < 1 + 3\sqrt[3]{b_i \cdot b_j}$?
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Context:
The Razorback shop makes $210 off each jersey and a certain amount off each t-shirt. During the Arkansas and Texas Tech game, they sold 177 t-shirts and 23 jerseys. A t-shirt costs $30 more than a jersey. How much does the shop make off each t-shirt?
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Context:
A 6-liter solution is 40% alcohol. How many liters of pure alcohol must be added to produce a solution that is 50% alcohol?
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Context:
Sasha places rooks on an empty chessboard: the first one anywhere he wants, and each subsequent rook in such a way that it attacks an odd number of previously placed rooks. What is the maximum number of rooks he can place in this manner?
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Context:
设集合 $A$ 中元素全为正整数, 且对任意 $x, y \in A(x \neq y)$, 有 $|x-y| \geqslant \frac{1}{25} x y$, 问 $A$ 中最多有几个数?
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Context:
I have four distinct rings that I want to wear on my right hand (five distinct fingers). One of these rings is a Canadian ring that must be worn on a finger by itself, and the rest can be arranged however I want. If I have two or more rings on the same finger, then I consider different orders of rings along the same finger to be different arrangements. How many different ways can I wear the rings on my fingers?
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Context:
Hugo can fold a small box in 3 seconds and a medium one in twice that time. Tom can fold both the small and medium boxes in 4 seconds. They want to leave as early as possible, and it takes them 7200 seconds to fold a certain number of small boxes and 1800 medium boxes together. How many small boxes do they fold?
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Context:
An eight-sided die, with faces numbered from 1 to 8, is tossed three times. Given that the sum of the first two tosses equals the third, calculate the probability that at least one "2" is tossed.
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Context:
5% of people in a village in Sri Lanka died by bombardment, 15% of the remainder left the village on account of fear. The population is now reduced to 3294. What was the initial population of the village?
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Context:
In a ten-mile race $\textit{First}$ beats $\textit{Second}$ by $2$ miles and $\textit{First}$ beats $\textit{Third}$ by $4$ miles. If the runners maintain constant speeds throughout the race, by how many miles does $\textit{Second}$ beat $\textit{Third}$?
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Context:
A TV station once continuously played 5 different commercial advertisements during a certain time period. Now, it plans to only keep 2 of these commercial advertisements, add one new commercial advertisement, and two different public service announcements (PSAs). The two PSAs cannot be played consecutively nor can they be placed at the beginning or end of the sequence. Calculate the total number of different sequences that can be arranged.
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Context:
Our third and final item comes to us from Germany, I mean Geometry. It is known that a regular $n$-gon can be constructed with straightedge and compass if $n$ is a prime that is 1 plus a power of 2. It is also possible to construct a $2 n$-gon whenever an $n$-gon is constructible, or a $p_{1} p_{2} \cdots p_{m}$-gon where the $p_{i}$ 's are distinct primes of the above form. What is really interesting is that these conditions, together with the fact that we can construct a square, is that they give us all constructible regular $n$-gons. What is the largest $n$ less than $4,300,000,000$ such that a regular $n$-gon is constructible?
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Context:
The eight vertices of a cube are randomly labelled with the integers from 1 to 8 inclusive. Judith looks at the labels of the four vertices of one of the faces of the cube. She lists these four labels in increasing order. After doing this for all six faces, she gets the following six lists: $(1,2,5,8),(3,4,6,7),(2,4,5,7),(1,3,6,8),(2,3,7,8)$, and $(1,4,5,6)$. Identify the label of the vertex of the cube that is farthest away from the vertex labelled 2. Express your answer as a single integer representing the vertex label.
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Context:
It is given that $x = -2272$, $y = 10^3 + 10^2c + 10b + a$, and $z = 1$ satisfy the equation $ax + by + cz = 1$, where $a, b, c$ are positive integers with $a < b < c$. Find $y$.
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Context:
A retired worker receives an annual pension that is proportional to the square root of the number of years they served. If they serve $a$ additional years, their pension increases by $p$ dollars. If they serve $b$ additional years $(b \neq a)$, their pension increases by $q$ dollars. What is their annual pension (expressed in terms of $a$, $b$, $p$, and $q$)?
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Context:
In January, an air-conditioning unit costs $470. In February, there was a 12% discount on the unit due to a clearance sale. Following that, in March, after restocking, the price increased by 8% due to production cost adjustments. In April, a Spring promotion offered a 10% discount on all air-conditioning units. In June, another 5% price increase was applied due to increased demand during the summer months. In August, the company reduced the price by 7% during the mid-year sales event. In October, prices increased again by 6% in preparation for the holiday season. In November, the company offers a seasonal promotion with a 15% discount on the final price. How much will an air-conditioning unit cost in November considering all these changes throughout the year?
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Context:
What is the minimum width that an infinite strip of paper must have in order to cut out any triangle with an area of 1?
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Context:
In the triangle $A_{1} A_{2} A_{3}$, the point $B_{i}$ is the quarter point closest to $A_{i}$ on the side $A_{i} A_{i+1}$, and the point $C_{i}$ is the quarter point closest to $A_{i+1}$ on the same side $\left(i=1,2,3, A_{4} \equiv A_{1}\right)$. What fraction of the area of triangle $A_{1} A_{2} A_{3}$ is the area of the common part of triangles $B_{1} B_{2} B_{3}$ and $C_{1} C_{2} C_{3}$?
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Context:
In a candy store, a salesperson placed 91 candies of several kinds in a row on the counter. It turned out that between any two candies of the same kind, there is an even number of candies. What is the minimum number of kinds of candies that could be?
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Context:
In a right-angled triangle $LMN$, suppose $\sin N = \frac{5}{13}$ with $LM = 10$. Calculate the length of $LN$.
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Context:
After giving a discount of 15% on the initial selling price and then offering a further discount of rs. 55, the shopkeeper still gets a profit of 25%. If the cost price is rs. 220, find the markup percentage?
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Context:
Find the Taylor series expansion of the function $x(t) = \frac{1}{1+\frac{1}{2}\Omega^2t^2}$ up to the term $t^4$, given the differential equation $\ddot x(t)=3\Omega^2 x^2(t) -4\Omega^2 x^3(t)$.
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Context:
21. (1999 National Team Selection Test) For non-negative real numbers $x_{1}, x_{2}$, $\cdots, x_{n}$ satisfying the condition $x_{1}+x_{2}+\cdots+x_{n}=1$, find the maximum value of $\sum_{j=1}^{n}\left(x_{j}^{4}-x_{j}^{5}\right)$.
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Context:
\frac{3}{2} + \frac{5}{4} + \frac{9}{8} + \frac{17}{16} + \frac{33}{32} + \frac{65}{64} - 7 =
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Context:
I'm stuck with this expression while solving. Can you help me solve it?

$$\lim_{n \to \infty}n(\frac{\cos(\frac{2n+1}{n^2+n})\sin(\frac{1}{n^2+n})}{\cos(\frac{2n+3}{(n+1)^2+n+1})\sin(\frac{1}{(n+1)^2+n+1})}-1)$$
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Context:
Consider a large square divided into a grid of \(5 \times 5\) smaller squares, each with side length \(1\) unit. A shaded region within the large square is formed by connecting the centers of four smaller squares, creating a smaller square inside. Calculate the ratio of the area of the shaded smaller square to the area of the large square.
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Context:
A game show offers a contestant three prizes $A, B$ and $C$, each of which is worth a whole number of dollars from $＄$ $1$ to $＄$ $9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order $A, B, C$. As a hint, the digits of the three prices are given. On a particular day, the digits given were $1, 1, 1, 1, 3, 3, 3$ . Find the total number of possible guesses for all three prizes consistent with the hint.
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