Context:
A pedestrian and a cyclist are moving in one direction along a highway, while a cart and a car are moving in the opposite direction. All participants move at constant speeds (each with their own speed). The cyclist first overtakes the pedestrian, then after some time meets the cart, and then after the same amount of time meets the car. The car first meets the cyclist, then after some time meets the pedestrian, and then after the same amount of time overtakes the cart. The cyclist overtook the pedestrian at 10 o'clock, and the pedestrian met the car at 11 o'clock. When did the pedestrian meet the cart?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( X = \{1, 2, \ldots, 100\} \). How many functions \( f: X \rightarrow X \) satisfy \( f(b) < f(a) + (b - a) \) for all \( 1 \leq a < b \leq 100 \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A number x is mistakenly divided by 10 instead of being processed by a certain operation. The percentage error in the result is 99%. What operation should have been applied to the number x?
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Context:
In a row, there are 99 people - knights and liars (knights always tell the truth, and liars always lie). Each of them said one of two phrases: "To my left, there are twice as many knights as liars" or "To my left, there are as many knights as liars." In reality, there were more knights than liars, and more than 50 people said the first phrase. How many liars said the first phrase?
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Context:
What is the mass percentage of O in Bromic acid?
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Context:
The graph of the function $y=a^{x+2}-2$ ($a>0$, $a\neq1$) always passes through a fixed point A. If point A lies on the line $mx+ny+1=0$, where $m$, $n>0$, then the minimum value of $\frac{1}{m} + \frac{2}{n}$ is \_\_\_\_\_.
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Context:
Let $a > 3$. Determine the value of $a$ given that $f(g(a)) = 16$, where $f(x) = x^2 + 10$ and $g(x) = x^2 - 6$.
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Context:
In triangle  $ABC$ , find the smallest possible value of   $$ |(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)| $$
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Context:
At a certain fruit stand, the price of each apple is 40 cents and the price of each orange is some cents. Mary selects a total of 10 apples and oranges from the fruit stand, and the average (arithmetic mean) price of the 10 pieces of fruit is 48 cents. Mary must put back 2 oranges so that the average price of the pieces of fruit that she keeps is 45 cents. What is the price of each orange?
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Context:
Let \( ABCD \) and \( WXYZ \) be two squares that share the same center such that \( WX \parallel AB \) and \( WX < AB \). Lines \( CX \) and \( AB \) intersect at \( P \), and lines \( CZ \) and \( AD \) intersect at \( Q \). If points \( P, W \), and \( Q \) are collinear, compute the ratio \( AB / WX \).
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Context:
Given the proposition "There exists an $x \in \mathbb{R}$, such that $x^2+1<0$", what is the direct statement of its negation? Express your answer as a complete sentence.
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\boxed{For all $x \in \mathbb{R}$, $x^2+1 \geq 0$.}


Context:
There are numbers from 1 to 2013 on the blackboard. Each time, two numbers can be erased and replaced with the sum of their digits. This process continues until there are four numbers left, whose product is 27. What is the sum of these four numbers?
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Context:
In a similar setup, square $PQRS$ is constructed along diameter $PQ$ of a semicircle. The semicircle and square $PQRS$ are coplanar. Line segment $PQ$ has a length of 8 centimeters. If point $N$ is the midpoint of arc $PQ$, what is the length of segment $NS$?
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Context:
Eight distinct points, $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, $P_6$, $P_7$, and $P_8$, are evenly spaced around a circle. If four points are chosen at random from these eight points to form two chords, what is the probability that the chord formed by the first two points chosen intersects the chord formed by the last two points chosen?
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Context:
Given real numbers $x$ and $y$ satisfy $|x-4|+(y-8)^{2}=0$, then the perimeter of an isosceles triangle with sides of length $x$ and $y$ is ____.
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Context:
The Fibonacci sequence is defined as follows: $F_{0}=0, F_{1}=1$, and $F_{n}=F_{n-1}+F_{n-2}$ for all integers $n \geq 2$. Find the smallest positive integer $m$ such that $F_{m} \equiv 0(\bmod 127)$ and $F_{m+1} \equiv 1(\bmod 127)$.
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Context:
The tower clock strikes three times in 12 seconds. For how long will it strike six times?
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Context:
A building has installed 5 colored lights to welcome the 2010 Asian Games in Guangzhou. The sequence in which they shine is not fixed, and each light can only shine in one of the colors: red, orange, yellow, green, or blue, with each of the 5 lights shining in a different color. Let's define the ordered shining of these 5 lights once as one "flash". In each flash, only one light shines per second, and the interval between two consecutive flashes is 5 seconds. Calculate the minimum time required to achieve all different flashes. Express your answer in seconds.
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Context:
Evaluate $2000^3 - 1999 \cdot 2000^2 - 1999^2 \cdot 2000 + 1999^3$
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Context:
Find the total number of cards in a stack where cards are numbered consecutively from 1 through $2n$ and rearranged such that, after a similar process of splitting into two piles and restacking alternately (starting with pile B), card number 252 retains its original position.
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Context:
Find the maximum value of \( x_0 \) for which there exists a sequence \( x_0, x_1, \ldots, x_{1995} \) of positive reals with \( x_0 = x_{1995} \) such that for \( i = 1, \ldots, 1995 \):
\[ x_{i-1} + \frac{2}{x_{i-1}} = 2x_i + \frac{1}{x_i}. \]
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Context:
11 groups perform at a festival. Each day any groups not performing watch the others (but groups performing that day do not watch the others). What is the smallest number of days for which the festival can last if every group watches every other group at least once during the festival?
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Context:
对一切映射 f:\{0,1,2,\cdots,99\}\rightarrow\{0,1,2,\cdots,99\}，求下式的最小可能值：S=\ds{x=0}{99}|f(f(x))-((x+1) \text{mod} 100)|.这里的记号 a \text{mod} 100 表示正整数 a 除以 100 所得的余数.（董子超、吴茁供题）
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Context:
In a redesign of his company's logo, Wei decided to use a larger square and more circles. Each circle is still tangent to two sides of the square and its adjacent circles, but now there are nine circles arranged in a 3x3 grid instead of a 2x2 grid. If each side of the new square measures 36 inches, calculate the total shaded area in square inches.
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Context:
Let $\Delta ABC$ be an equilateral triangle. How many squares in the same plane as $\Delta ABC$ share two vertices with the triangle?
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Context:
A train traveling at an initial speed of 90 kmph and accelerating at a constant rate of 0.5 m/s² overtakes a motorbike traveling at a constant speed of 72 kmph. The train takes 50 seconds to completely pass the motorbike. What is the length of the train in meters, given that the motorbike's length is 2 meters?
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Context:
Teams A and B each have 7 players who will compete in a Go tournament in a predetermined order. The match starts with player 1 from each team competing against each other. The loser is eliminated, and the winner next competes against the loser’s teammate. This process continues until all players of one team are eliminated, and the other team wins. Determine the total number of possible sequences of matches.
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Context:
What is the expected time it takes for a monkey typing random capital letters to produce the sequence "ABBA"? The problem is modeled as a Markov Process. While $26^4$ is a rough estimate, what is the precise expected time?
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Context:
将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+\operatorname{def}$ 是奇数的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
A high school bowling team's 5 members scored a total of 2000 points in their first match. The first bowler scored 1/4 as many points as the second, the second bowler scored 5/3 times as high as the third bowler who didn't exceed 500 points, and the third bowler scored 3/5 as many points as the fourth bowler. The fourth bowler scored 9/10 of what the fifth bowler scored. How many points did the fifth bowler score?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A farmer bought 749 sheep. He sold 700 of them for the price paid for the 749 sheep. The remaining 49 sheep were sold at the same price per head as the other 700. Calculate the percent gain on the entire transaction. Express your answer as a percentage, rounded to two decimal places if necessary.
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Context:
9.5. The cells of a chessboard are painted in 3 colors - white, gray, and black - in such a way that adjacent cells, sharing a side, must differ in color, but a sharp change in color (i.e., a white cell adjacent to a black cell) is prohibited. Find the number of such colorings of the chessboard (colorings that coincide when the board is rotated by 90 and 180 degrees are considered different).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Place the numbers $0, 1, 2, 3, 4, 5, 6, 7$ on the eight vertices of a cube (each vertex having one number, with each number used exactly once) in such a way that the sum of the two numbers at the ends of each edge is always a prime number. What is the maximum sum of the four numbers on one face of the cube?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that a positive integer $N$ can be expressed as
the sum of $k$ consecutive positive integers
$$N=a+(a+1)+(a+2)+\cdots+(a+k-1)$$
for $k=2017$ but for no other values of $k>1.$ Considering all positive integers $N$ with this property, what is the smallest positive integer $a$ that occurs in any of these expressions?
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Context:
An icosahedron is a regular polyhedron with 20 equilateral triangular faces. If the faces are colored either blue or white, what is the maximum number of blue faces that can be colored such that no two blue faces are adjacent?
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Context:
Given that $f \in C^{(1)} [0,1]$, $f(0) = a$, and $\int_{0}^{1} f(x) \, dx = 0$, find the minimum value of $\int_{0}^{1} (f'(x))^2 \, dx$. Provide your answer in terms of $a$ if necessary.
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Context:
Determine the boundary value of the positive integer \(n\) for which the integral \(\int_{a}^{\infty} \frac{1}{x (\ln x)^n} \, dx\) converges, where \(a > 1\).
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Context:
Calculate the conditional expectation \( \mathsf{E}(N_3 \mid N_5 = 5) \), where \( N_t \) is a Poisson process with rate 1, and \( X_t \) and \( Y_t \) are independent Poisson processes that contribute to \( N_t \) with probability \( P = 0.5 \).
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Context:
Evaluate the limit: $$\lim_{x\to\infty}\left(\int_0^xe^{t^2}dt\right)^{\frac{1}{x^2}}$$
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Context:
Evaluate the series \( \sum_{n=1}^\infty \frac{1}{2n-1} \sin((2n-1)x) \) for \( x \in (0, \pi) \).
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Context:
Determine the correct limit for \( r \) when transforming the integral \( \iiint_{E}z\sqrt{x^2+y^2}dV \) to cylindrical coordinates, where \( E \) is the region bounded by the cylinder \( x^2+y^2=2x \) and the planes \( y=0 \), \( z=0 \), and \( z=a \).
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Context:
Given that \( \det(A) = \frac{1}{24} \), solve for the determinant of the matrix expression:
\[
\det\left(\left(\frac{1}{3}A\right)^{-1} - 120 A^*\right)
\] where \( A^* \) is the adjugate of \( A \).
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Context:
Determine the smallest integer n such that there exists a Pythagorean n-tuple \((X_1, X_2, \ldots, X_n)\) where each \(X_i\) is a prime number and satisfies the equation:
\[ X_1^2 + X_2^2 + \cdots + X_{n-1}^2 = X_n^2 \]
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Context:
The roots of the polynomial \( f\left( x\right)  = {x}^{8} + {x}^{7} - {x}^{5} - {x}^{4} - {x}^{3} + x + 1 \) are all roots of unity. We say that a real numbe
 \( r \in  \lbrack 0,1) \) is nice if \( {e}^{2i\pi r} = \cos {2\pi r} + i\sin {2\pi r} \) is a root of the polynomial \( f \) and if \( {e}^{2i\pi r} \) has positive imaginary part
 be the sum of the values of nice real numbers \( r \) . If \( S = \frac{p}{q} \) for relatively prime positive integers \( p,q \) ,find \( p + q \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of subgroups in the dihedral group $D_{16}$ that contain the stabilizer of a vertex in the action of $D_{16}$ on the vertices of an octagon.
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Context:
Calculate the expected value $E[S(t)]$ for the expression $S(t) = \sum_{i=1}^{N(t)} (X_i + T_i)^2$, where $X_i$ are independent and identically distributed random variables with density $f(x) = \frac{1}{2}$ for $0 \leq x \leq 2$, and $T_i$ are the arrival times of a homogeneous Poisson process $N(t)$ with intensity $\lambda = 3$. Assume $N(t)$ is independent of $X_i$.
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Context:
Find the limit of the following expression as \( n \to +\infty \):
\[ \lim_{n \to +\infty} \sqrt[n]{\frac{1}{3^n}+\frac{n^5}{5^n}+\frac{1}{2^{nn^{10^{10}}}}} \]
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Context:
Let \( f\left( z\right)  = \frac{z + a}{z + b} \) and \( g\left( z\right)  = f\left( {f\left( z\right) }\right) \) ,where \( a \) and \( b \) are complex numbers. Suppose that \( \left| a\right|  = 1 \) and \( g\left( {g\left( z\right) }\right)  = z \) for all \( z \)
 for which \( g\left( {g\left( z\right) }\right) \) is defined. What is the difference between the largest and smallest possible values of \( \left| b\right| \) ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an Abelian group, if the orders of elements $a$ and $b$ are both 6, determine the possible values for the order of the element $ab$. Provide your answer as a set of integers.
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Context:
Find the equation of the locus of points where the lengths of the tangents from the circles with equations \( x^2 + y^2 - 6x - 27 = 0 \) and \( x^2 + y^2 + 2x - 8 = 0 \) are equal.
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Context:
Determine the dimension of the coboundary chain-group of a graph with $n$ vertices and $k$ connected components.
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Context:
Evaluate the limit: $$\lim_{x\to2\ \ y\to2}\frac{x^{6}+ \tan (x^{2}-y^{2}) - y^{6}}{\sin(x^{6}-y^{6}) - x^{5}y +xy^{5} + \arctan(x^{2}y -xy^{2})}$$
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Context:
Find the sum of all natural numbers \( n \) such that \((n^2+n+1)^2\) divides the expression \(n^{2195} + n^{2194} + \ldots + n^2 + n + 1\).
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Context:
Calculate the accumulated value of two investments on March 31, 2010, given the force of interest function \( F(t) = \frac{0.02 + 0.01t}{1 + 0.02t + 0.005t^2} \), where \( t \) represents years starting from March 31, 2001. An initial investment of $100 was made on March 31, 2002, and another investment of $200 was made on March 31, 2006. What is the total accumulated value of these investments on March 31, 2010?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The equation of the line joining the complex numbers \(-5 + 4i\) and \(7 + 2i\) can be expressed in the form \(az + b \overline{z} = 38\) for some complex numbers \(a\) and \(b\). Find the product \(ab\).
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Context:
Calculate \( 516489222^{22} \mod 96899 \).
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Context:
让$f（x） = \frac{\sin（x）}{x}$查找$$ \lim_{T\to\infty}\frac{1}{T}\int_0^T\sqrt{1+f'（x）^2}dx.$$
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Context:
Evaluate the integral: \[ \int_{0}^{2\pi} e^{R \cos t} \cos(R \sin t + 3t) \, dt \]
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Context:
Determine the number of non-zero coefficients in the Zhegalkin polynomial representation of the logical expression \(x_1 \lor x_2 \lor x_3 \lor \ldots \lor x_n\).
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Context:
Determine the natural number \( n \) such that for any two sets \( \{ x_1, x_2, ..., x_n \} \) and \( \{ y_1, y_2, ..., y_n \} \) of \( n \) distinct points on the boundary \( \mathbb{S}^1 \) of the unit disk \( \mathbb{D} \), there exists a unique automorphism \( f: \mathbb{D} \rightarrow \mathbb{D} \) with \( f(x_i) = y_i \) for all \( i \).
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Context:
Solve the Euler-Lagrange equation for the functional \( L(y) = \int_{-1}^7 \sqrt{1 + (y')^2} \, dx \) with the initial conditions \( y(0) = 1 \) and \( y(1) = 2 \).
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Context:
Determine the number of zeros of \(\sin \pi x\) within the domain \(D = \{|z - 3 - 4i| < 6\}\).
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Context:
Calculate the mass of fluid flowing through the hemisphere defined by \( x^2 + y^2 + z^2 = 1, z \geq 0 \) in unit time. The fluid flow has a flux density vector \( F(x,y,z) = \langle x, -2x-y, z \rangle \). Let \( n \) denote the unit normal vector pointing out of the sphere. Find the mass of fluid flowing through the hemisphere in the direction of \( n \).
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Context:
Suppose that $a$ and $b$ are positive integers such that $\frac{a}{b-20}$ and $\frac{b+21}{a}$ are positive integers. Find the maximum possible value of $a + b$.
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Context:
Evaluate the limit: \[ \lim_{n\rightarrow \infty}\frac{n!\cdot e^n}{\sqrt{n}\cdot n^n} \] without using Stirling's approximation.
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Context:
Evaluate the limit: $$\lim_{x\rightarrow0}\frac{e^x \sin x - (x+1)\tan x}{x\log\cos(x)}$$
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Context:
Let $Y_1, \ldots, Y_{10}$ be a random sample from a normal distribution with mean $0$ and variance $\sigma^2$. Find the distribution of the statistic $\frac{10 \bar{Y}^2}{S^2}$, where $\bar{Y}$ is the sample mean and $S^2$ is the sample variance.
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Context:
Determine the number of elements of order 15 in the symmetric group \( S_{13} \).
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Context:
Determine all positive integers \(n\) for which it is possible to construct an arithmetic sequence with \(10^n\) terms such that the digit sums of its terms form an arithmetic sequence.
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Context:
In a full binary tree, determine the difference between the sum of the depths of all leaves and the sum of the depths of all internal nodes, expressed in terms of the number of internal nodes.
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Context:
A 3 by 2 rectangle is split into four congruent right-angled triangles. Those four triangles are rearranged to form a rhombus. Calculate the ratio of the perimeter of the rectangle to the perimeter of the rhombus. Express your answer as a simplified ratio (e.g., a:b).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A certain mixture of nuts consists of 5 parts almonds, 2 parts walnuts, 3 parts cashews, and 4 parts pistachios, by weight. You have a budget of $1500 to purchase nuts for making 800 pounds of the mixture. Given that almonds cost $6 per pound, walnuts cost $5 per pound, cashews cost $8 per pound, and pistachios cost $10 per pound, and that almonds should make up at least 30% of the mixture, what is the maximum possible number of pounds of almonds that can be included in the 800-pound mixture while staying within the budget constraints?
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Context:
Given the function $f(x) = e^{1+|x|} - \frac{1}{1+x^2}$, find the range of $x$ for which $f(x) > f(2x-1)$.
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Context:
设$\boldsymbol{u}=\left(u_{1},u_{2},u_{3}\right)$和$\boldsymbol{v}=$$\left(v_{1},v_{2},v_{3}\right)$是空间向量，满足$u_{i}、v_{i}(i=1,2,3)$均为整数，且
$0.9999<\cos\langle\boldsymbol{u},\boldsymbol{v}\rangle<1$。
记$S=\left|u_{1}\right|+\left|u_{2}\right|+\left|u_{3}\right|+\left|v_{1}\right|+\left|v_{2}\right|+\left|v_{3}\right|$。求$[\sqrt{S}]$的最小可能值，其中，$[x]$表示不超过实数$x$的最大整数。
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Context:
A set of six edges of a regular octahedron is called a Hamiltonian cycle if the edges in some order constitute a single continuous loop that visits each vertex exactly once. How many ways are there to partition the twelve edges into two Hamiltonian cycles?
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Context:
The range of the function \( y = \arcsin[\sin x] + \arccos[\cos x] \) (for \( x \in [0, 2\pi) \) and where \([x]\) denotes the greatest integer less than or equal to \( x \)) is ____.
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Context:
Compute the number of ordered quadruples  $(a,b,c,d)$  of distinct positive integers such that  $\displaystyle \binom{\binom{a}{b}}{\binom{c}{d}}=21$ .
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Context:
What is the closed form for the sum $\sum_{i=0}^n {n \choose i} D(i)$, where $D(i)$ represents the number of derangements of $i$ elements? A derangement is a permutation of a set in which no element appears in its original position. For instance, $!3 = 2$ because there are two derangements of the set $\{1,2,3\}$: $\{2,3,1\}$ and $\{3,1,2\}$.
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Context:
Two workers started paving paths in a park simultaneously from point $A$. The first worker paves the section $A-B-C$ and the second worker paves the section $A-D-E-F-C$. They worked at constant speeds and finished at the same time, having spent 9 hours on the job. It is known that the second worker works 1.2 times faster than the first worker. How many minutes did the second worker spend paving the section $DE$?
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Context:
定义在$\left( -8，a \right)$上的奇函数$f\left( x \right)$在区间$\left[ 2，7 \right]$上是增函数，在区间$\left[ 3，6 \right]$上的最大值为$a$，最小值为-1，则$2f\left( -6 \right)+f\left( -3 \right)=$__________．
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Context:
Samantha has 10 green marbles and 5 purple marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 7 marbles. What is the probability that exactly four of the marbles that she removes are green? Express your answer as a decimal.
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Context:
Let $q(x)$ be a quadratic polynomial such that $[q(x)]^2 - x^2$ is divisible by $(x - 2)(x + 2)(x - 5)$. Find $q(10)$.
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Context:
已知存在正整数 $a, b, c$ 满足 $a+b+c=407,10^{n} \mid a b c$ ，则 $n$ 的最大值是多少？
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Context:
In $\triangle ABC$, given $\angle A = 45^{\circ}$ and $\angle B = 60^{\circ}$, find the ratio of the distances from the circumcenter $O$ to each of the three sides of $\triangle ABC$.
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Context:
通项为$a _ { n } = b [ \sqrt { n + c } ] + d$的数列，逐次算得各项是

$1,3,3,3,5,5,5,5,5,··$.

其中每一个正奇数$m$恰好连续出现$m$次.上述$b，c，d$是待定的整数.求$b+c+d $的值.
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Context:
The base radius of a cone is $3cm$, and the slant height is $6cm$. Find the central angle of the sector in the unfolded diagram of the cone.
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Context:
Let $f(n) = \sum^n_{i=1}\frac{gcd(i,n)}{n}$. Find the sum of all positive integers $ n$ for which $f(n) = 6$.
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Context:
What is the ratio of the sides of a triangle in which the sum of the lengths of the altitudes taken two at a time corresponds to the ratio 5:7:8?
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Context:
10. Given the sequence $\left\{a_{n}\right\}$ defined by $a_{1}=\frac{2}{3}, a_{n+1}=a_{n}^{2}+a_{n-1}^{2}+\cdots+a_{1}^{2}\left(n \in \mathbf{N}^{*}\right)$. If for any $n$ $\in \mathbf{N}^{*}, \frac{1}{a_{1}+1}+\frac{1}{a_{2}+1}+\cdots+\frac{1}{a_{n}+1}<M$ always holds, find the minimum value of $M$.
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Context:
A moving point P starts from vertex A of the cube ABCD-A1B1C1D1 and moves along the edges to vertex C1 and then back to A. If the path exactly passes through 6 different edges, such a path is called the "optimal path". The number of "optimal paths" is \_\_\_\_\_\_ (answer in digits).
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Context:
已知植圆 $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$ 的左右焦点分别为 $F_{1}$ 与 $F_{2}$, 点 $P$ 在直线 $l: x-\sqrt{3} y+8+2 \sqrt{3}=0$ 上. 当 $\angle F_{1} P F_{2}$取最大值时，比 $\frac{\left|P F_{1}\right|}{\left|P F_{2}\right|}$ 的值为 \sqrt{m}-n 的形式，求 m - n 的值。
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Context:
Let $P_k(x) = (x-k)(x-(k+1))$. Kara picks four distinct polynomials from the set $\{P_1(x), P_2(x), P_3(x), \ldots , P_{12}(x)\}$ and discovers that when she computes the six sums of pairs of chosen polynomials, exactly two of the sums have two (not necessarily distinct) integer roots! How many possible combinations of four polynomials could Kara have picked?
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Context:
9.5. Find the maximum value of the expression $(\sqrt{36-4 \sqrt{5}} \sin x-\sqrt{2(1+\cos 2 x)}-2) \cdot(3+2 \sqrt{10-\sqrt{5}} \cos y-\cos 2 y) \cdot$ If the answer is not an integer, round it to the nearest integer.
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Context:
In quadrilateral \(ABCD\), points \(X, Y, Z\) are the midpoints of segments \(AB, AD, BC\), respectively. It is known that \(XY\) is perpendicular to \(AB\), \(YZ\) is perpendicular to \(BC\), and the measure of angle \(ABC\) is \(100^\circ\). Find the measure of angle \(ACD\).
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Context:
Josephine receives a bill from the hospital for 18,000$. The bill consists of various components with the following percentage allocations: X% of the bill is for medication, Y% is for imaging tests, and Z% is for a surgical procedure. The remaining bill has P% earmarked for overnight stays and Q% for doctor's fees. There is also a fixed fee of 300$ for food, 450$ for consultation, and 600$ for physical therapy sessions. The rest of the bill is for the ambulance ride. Given X=35%, Y=15%, Z=25%, P=10%, and Q=5%, how much did the ambulance ride cost?
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Context:
A rectangular piece of paper measures $4$ units by $5$ units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called basic if
$\mathrm{(i)}$ all four sides of the rectangle are segments of drawn line segments, and
$\mathrm{(ii)}$ no segments of drawn lines lie inside the rectangle.
Given that the total length of all lines drawn is exactly $2007$ units, let $N$ be the maximum possible number of basic rectangles determined. Find the remainder when $N$ is divided by $1000$.
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Context:
The numbers \(1, 2, 3, \ldots, 400\) are written on 400 cards. Two players, \(A\) and \(B\), play the following game:

1. In the first step, \(A\) takes 200 cards for themselves.
2. \(B\) then takes 100 cards from both the remaining 200 cards and the 200 cards that \(A\) has, totaling 200 cards for themselves, and leaves the remaining 200 cards for \(A\).
3. In the next step, \(A\) takes 100 cards from both players' cards, totaling 200 cards for themselves, leaving the remaining 200 cards for \(B\).

This process continues until \(B\) completes the 200th step. At the end, the sums of the numbers on the cards held by \(A\) and \(B\) are calculated as \(C_A\) and \(C_B\) respectively. Then, player \(A\) pays player \(B\) the difference \(C_B - C_A\).

What is the maximum difference that \(B\) can achieve if both players play optimally?
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Context:
Consider a product \(a_{1} a_{2} \ldots a_{100}\) written on a board, where \(a_{1}, \ldots, a_{100}\) are natural numbers. We consider 99 expressions, each of which is obtained by replacing one of the multiplication signs with an addition sign. It is known that exactly 32 of these expressions are even. What is the maximum number of even numbers among \(a_{1}, a_{2}, \ldots, a_{100}\) that could exist?
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Context:
Given that the internal angles of a regular triangle and a dodecagon are 60° and 150° respectively, according to the conditions for plane tiling, the sum of the internal angles at a vertex must be 360°. Determine the number of possible ways to tile a plane using regular triangles and dodecagons. Express your answer as a whole number.
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Context:
Player A and player B are playing a game of Go. If player A plays as black first, the probability of player A winning is $\frac{1}{3}$; if player B plays as black first, the probability of player B winning is $\frac{1}{2}$. Assuming each game is independent and there are no draws, the loser of the first game plays first in the second game. If player A and player B play two games, with the first game having an equal probability of player A or player B playing as black first, then the probability of each player winning one game is ______.
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Context:
If the cost price of 80 articles is equal to the selling price of 60 articles, and an additional 15% tax is levied on the cost price, while a discount of 10% is given on the selling price, then determine the effective gain or loss percentage.
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Context:
In an equilateral triangle $\triangle ABC$ inscribed in the unit circle $C(0,1)$, find the maximum value of $\overline{PA}\cdot\overline{PB}\cdot\overline{PC}$, where $P$ is a variable point in the closed disk $\bar{D}(0,2)$. Provide your answer as a single numerical value.
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Context:
Evaluate the double integral \( \iint_{D} \frac{y^2}{x} \, dx \, dy \), where \( D \) is the region bounded by the curves \( y = x^2 \), \( y = 4x^2 \), \( xy = 1 \), and \( xy = 5 \).
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Context:
Evaluate the integral \( \iint_R \left[5+\frac{1}{4}\left( \frac{x^2}{c^2}-\frac{y^2}{d^2}\right)\right] \, dx \, dy \) where \( R \) is the region defined by \( \frac{x^2}{c^2}-\frac{y^2}{d^2} \ge 1 \). Use a suitable transformation to simplify the integral.
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Context:
Find the number of ordered triples \((a, b, c)\) of natural numbers such that \(a^2b^3c^4 = 2^{10} \times 3^{14}\) and \(\gcd(a, b, c) = 1\).
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Context:
Evaluate the line integral \( \int_0^\pi F \cdot dr \) for the vector field \( F(x,y)=\langle 2y^2+y^2\cos(xy^2),4xy+2xy\cos(xy^2)\rangle \) along the path \( r(t)=\langle e^t\sin(t),e^t\cos(t)\rangle \), where \( t \in [0,\pi] \).
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Context:
Calculate the probability that $X_1 + X_2 < 2$ and $X_1 - X_2 > 0$, given that $(X_1, X_2)$ follows a bivariate normal distribution with mean $\mu = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and covariance matrix $\Sigma = \begin{pmatrix} 3 & -1\\ -1 & 3 \end{pmatrix}$. Note that the events are not independent.
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Context:
Find the largest real $m$, such that for all positive real $a, b, c$ with sum $1$, the inequality $\sqrt{\frac{ab} {ab+c}}+\sqrt{\frac{bc} {bc+a}}+\sqrt{\frac{ca} {ca+b}} \geq m$ is satisfied.
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Context:
Using Stokes' Theorem, evaluate the line integral of the vector field \(\vec{F} = -y^3 \hat{i} + x^3 \hat{j} - z^3 \hat{k}\) along the curve \(C\), which is the intersection of the cylinder \(x^2 + y^2 = 1\) and the plane \(x + y + z = 1\).
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Context:
Determine the values of \( p \) for which the series \( \sum\limits_{n=1}^\infty n(1+n^2)^p \) converges.
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Context:
On a chessboard, \( n \) white rooks and \( n \) black rooks are placed such that rooks of different colors do not threaten each other. Find the maximum possible value of \( n \).
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Context:
What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are $(\cos 40^\circ,\sin 40^\circ)$, $(\cos 60^\circ,\sin 60^\circ)$, and $(\cos t^\circ,\sin t^\circ)$ is isosceles?
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Context:
In the circular sector \( OAB \), the central angle of which is \( 45^{\circ} \), a rectangle \( KMRT \) is inscribed. Side \( KM \) of the rectangle lies along the radius \( OA \), vertex \( P \) is on the arc \( AB \), and vertex \( T \) is on the radius \( OB \). The side \( KT \) is 3 units longer than the side \( KM \). The area of the rectangle \( KMRT \) is 18. Find the radius of the circle.
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Context:
There are 6 positive integers, from a certain number to 100, inclusive, that are not divisible by even integers greater than 1. What is the starting number of this range?
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Context:
Two distinct squares of the $8 \times 8$ chessboard $C$ are said to be adjacent if they have a vertex or side in common. Also, $g$ is called a $C$-gap if for every numbering of the squares of $C$ with all the integers $1, 2, \ldots, 64$, there exist two adjacent squares whose numbers differ by at least $g$. Determine the largest $C$-gap $g$.
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Context:
Given a linear operator $T$ on an $n$-dimensional vector space $V$ (over $\mathbb R$), I want to find an orthonormal basis for $V$ in which the matrix of $T$ is sparse (has many zeros). What is the largest number $k=k(n)$ such that every real $n\times n$ matrix is orthogonally equivalent to a matrix with at least $k$ zero entries?
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Context:
Let the side lengths of a triangle be integers \( l \), \( m \), and \( n \), with \( l > m > n \). It is given that \( \left\{\frac{3^{l}}{10^{4}}\right\} = \left\{\frac{3^{m}}{10^{4}}\right\} = \left\{\frac{3^{n}}{10^{4}}\right\} \), where \(\{x\} = x - [x]\) and \([x]\) denotes the greatest integer less than or equal to \( x \). Find the minimum perimeter of such a triangle.
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Context:
Misha noticed that the tram passed by him in 2 seconds, and it went through a tunnel 96 meters long in 10 seconds. Find the speed of the tram (in meters per second), assuming it remains constant throughout the observation period.
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Context:
Given a $2018 \times 4$ grid, tint the cells with red and blue such that each row and each column has an equal number of red and blue cells. Determine the number of ways, $M$, to tint the grid with the mentioned requirement and find $M \pmod{2018}$.
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Context:
In triangle $ABC$, the sides opposite angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and the area of $ABC$ is $S$. It is given that $(a^2 + b^2)\tan C = 8S$, and $\sin A \cos B = 2\cos A \sin B$. Find the value of $\cos A$.
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Context:
$ 500 will become $ 1000 in 5 years find the rate of interest ?
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Context:
"Drawing a pie to satisfy hunger" is a ______ event.
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Context:
Julia's garden has a 3:7 ratio of tulips to daisies. She currently has 35 daisies. She plans to add 30 more daisies and wants to plant additional tulips to maintain the original ratio. How many tulips will she have after this addition?
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Context:
In an alphabet with $n>1$ letters, a word is defined as any finite sequence of letters where any two consecutive letters are different. A word is called "good" if it is not possible to delete all but four letters from it to obtain a sequence of the form $a a b b$, where $a$ and $b$ are different letters. Find the maximum possible number of letters in a "good" word.
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Context:
The sum of the coefficients of the quadratic equation $5x(1-x) = 3$ is ____.
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Context:
The math scores of a high school's college entrance examination, denoted as $\xi$, approximately follow a normal distribution $N(100, 5^2)$. Given that $P(\xi < 110) = 0.98$, find the value of $P(90 < \xi < 100)$.
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Context:
Compute $\lim_{x \to -1} f(x)$ for a function $f: \mathbb R \to \mathbb R$ given by the following limit equation:

$$4 = \lim_{x \to -1} \left( \frac{f(x)+2}{x+1} - \frac{x}{x^2-1} \right) \tag{1}$$
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Context:
求不定方程 $x^{2}+615=2^{y}$ 的正整数解的组数。
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Context:
Given the digits 1, 2, 3, 4, and 5, create a five-digit number without repetition, with 5 not in the hundred's place, and neither 2 nor 4 in the unit's or ten-thousand's place, and calculate the total number of such five-digit numbers.
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Context:
Place five different files, labeled A, B, C, D, and E, into seven drawers numbered consecutively from 1 to 7. Each drawer can contain at most one type of file. If files A and B must be placed in adjacent drawers, and files C and D must also be placed in adjacent drawers, then find the total number of different ways to place the files into the drawers that meet these conditions. Express your answer as a single integer.
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Context:
Given the sets:
\[ M=\left\{(x, y) \mid \tan(\pi y) + \sin^2(\pi x) = 0 \right\}, \]
\[ N=\left\{(x, y) \mid x^{2}+y^{2} \leq 2\right\}, \]

find the number of elements in the intersection \( M \cap N \). Express your answer as a whole number.
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Context:
Five people participate in a relay game, each randomly choosing between two options: left path and right path. If more than 2 people choose the same path, each of them gets 1 point; if fewer than 3 people choose the same path, each of them gets 0 points. Let $\xi$ denote the score of a player named Xiaoqiang. Calculate the expected value of $\xi$, $E\xi$. Express your answer as a simplified fraction.
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Context:
Given the expression
$$
\left(\left(\cdots\left(\left((x-2)^{2}-2\right)^{2}-2\right)^{2}-\cdots-2\right)^{2}-2\right)^{2},
$$
which contains $k$ nested sets of parentheses, remove the parentheses and combine like terms to obtain a polynomial. Find the coefficient of $x^{2}$ in this polynomial.
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Context:
Find the unique pair of positive integers $(a, b)$ with $a<b$ for which $$\frac{2020-a}{a} \cdot \frac{2020-b}{b}=2$$
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Context:
An alien from the planet OMO Centauri writes the first ten prime numbers in arbitrary order as $U$, $W$, $XW$, $ZZ$, $V$, $Y$, $ZV$, $ZW$, $ZY$, and $X$. Each letter represents a nonzero digit. Each letter represents the same digit everywhere it appears, and different letters represent different digits. Also, the alien is using a base other than base ten. The alien writes another number as $UZWX$. Compute this number (expressed in base ten, with the usual, human digits).
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Context:
Given that $f(x)$ is an even function defined on $\mathbb{R}$, $g(x) = f(x) + x^2$, and $g(x)$ is monotonically increasing when $x \in (-\infty, 0]$, find the solution set of the inequality $f(x+1) - f(x-1) + 4x > 0$.
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Context:
The integer $n$ has exactly six positive divisors, which are $1 < a < b < c < d < n$. Let $k = a - 1$. If the $k$-th divisor (according to the above ordering) of $n$ is equal to $(1 + a + b)b$, find the highest possible value of $n$.
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Context:
已知$f(x)$为定义在$\mathbf{R}$上的奇函数，当$x\leqslant0$时，
$$f(x)=\left|x+\frac{1}{2}\sin\alpha\right|+\left|x+\frac{3}{2}\sin\alpha\right|-2\sin\alpha$$
其中，$\alpha\in(-\pi,\pi)$。若对任意实数$x$，均有$f(x)\geqslant f(x+2)$，则$\alpha$的最小值
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Context:
Senya has three straight sticks, each 24 centimeters long. Senya broke one of them into two parts such that with the two pieces of this stick and the two whole sticks, he could form the contour of a right triangle. How many square centimeters is the area of this triangle?
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Context:
A coin with probability of heads p is being tossed repeatedly. Consider the 4 state Markov chain given by the results of the previous toss and the toss before that. Using Markov chain hitting time arguments, find the expected time we need to wait until we see two consecutive heads.
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Context:
分解因式：$\left( xy+1 \right)\left( x+1 \right)\left( y+1 \right)+xy=$_______.
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Context:
Given $a \in \mathbb{R}$, let the set $S = \{x \mid 2ax^2 - x \leq 0\}$ and $T = \{x \mid 4ax^2 - 4a(1-2a)x + 1 \geq 0\}$. If $S \cup T = \mathbb{R}$ (where $\mathbb{R}$ is the set of all real numbers), then the range of values for the real number $a$ is ______.
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Context:
Eighteen children solved a problem from the April issue of KöMaL. They want to discuss their solutions via email. What is the minimum number of emails they need to send to ensure that each of them learns about every solution? (Each email counts as many times as the number of people it is sent to.)
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Context:
在平面直角坐标系 $x O y$ 中, 设 $A B$ 是抛物线 $y^{2}=4 x$ 的过点 $F(1,0)$ 的弦, $\triangle A O B$ 的外接圆交抛物线于点 $P$ (不同于点 $O, A, B)$. 若 $P F$ 平分 $\angle A P B$, 求 $|P F|$ 的所有可能值。原始答案为\sqrt{m}-n的格式，请给出m - n的值。
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Context:
Weiming Real Estate Company sold a house to Mr. Qian at a 5% discount off the list price. Three years later, Mr. Qian sold the house to Mr. Jin at a price 60% higher than the original list price. Considering the total inflation rate of 40% over three years, Mr. Qian actually made a profit at a rate of     % (rounded to one decimal place).
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Context:
Given positive numbers \( a \) and \( b \) satisfying \( a + b = 1 \), find the minimum value of \( M = \sqrt{1 + 2a^2} + 2 \sqrt{\left(\frac{5}{12}\right)^2 + b^2} \).
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Context:
Three sportsmen called Primus, Secundus, and Tertius take part in a race every day. Primus wears the number '1' on his shirt, Secundus wears '2', and Tertius wears '3'.

On Saturday, Primus wins, Secundus is second, and Tertius is third. Using their shirt numbers this result is recorded as '123'.

On Sunday, Primus starts the race in the lead with Secundus in second. During Sunday's race:
- Primus and Secundus change places exactly 9 times,
- Secundus and Tertius change places exactly 10 times,
- Primus and Tertius change places exactly 11 times.

How will Sunday's result be recorded?
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Context:
Given that $\cos(\pi+\theta) = -\frac{1}{2}$, find the value of $\tan(\theta - 9\pi)$.
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Context:
Given the sequence $\{a_n\}$: $\frac{1}{2}$, $\frac{1}{3} + \frac{2}{3}$, $\frac{1}{4} + \frac{2}{4} + \frac{3}{4}$, $\frac{1}{5} + \frac{2}{5} + \frac{3}{5} + \frac{4}{5}$, ..., then the sum of the first $n$ terms of the sequence $b_n = \frac{1}{a_n a_{n+1}}$ is \_\_\_\_\_\_.
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Context:
Martha writes down a random mathematical expression consisting of 3 single-digit positive integers with an addition sign "$+$" or a multiplication sign "$\times$" between each pair of adjacent digits. For example, her expression could be $4 + 3 \times 3$, with a value of 13. Each positive digit is equally likely, each arithmetic sign ("$+$" or "$\times$") is equally likely, and all choices are independent. What is the expected value (average value) of her expression?
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Context:
Given that quadrilateral ABCD is a cyclic quadrilateral inscribed in circle O, with AC = sqrt(3)BD and angle ADC = 2*angle BAD. If AB*CD + BC*AD = 4*sqrt(3), calculate the radius of circle O. Express your answer as a single number.
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Context:
A sequence $(c_n)$ is defined as follows: $c_1 = 1$, $c_2 = \frac{1}{3}$, and
\[c_n = \frac{2 - c_{n-1}}{3c_{n-2}}\] for all $n \ge 3$. Find $c_{100}$.
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Context:
What is the number of ways in which one can choose  $60$  unit squares from a  $11 \times 11$  chessboard such that no two chosen squares have a side in common?
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Context:
12 men are equivalent to as many women as are equal to 20 boys. All together, they earn a combined Rs. 300. What are the men's wages?
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Context:
There are $n$ students standing one behind the other in a circle, with heights $h_{1}<h_{2}<\ldots<h_{n}$. At any moment, if a student of height $h_{k}$ is directly behind a student of height less than or equal to $h_{k-2}$, they can choose to switch places with that student. Determine the maximum number of exchanges that can occur among all initial configurations and possible exchange choices.
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Context:
A pet store has 17 bird cages. Each cage contains a different number of parrots and parakeets. The first cage has 2 parrots and 7 parakeets. Each subsequent cage has one additional parrot and two additional parakeets more than the preceding cage. If 3 cages are temporarily empty due to cleaning, how many birds are there in total presently hosted in the cages of this pet store?
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Context:
Find the positive integer $k$ such that the roots of $x^3 - 15x^2 + kx -1105$ are three distinct collinear points in the complex plane.
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Context:
Let $p$ and $q$ be any two prime numbers greater than 100. The minimum value of the greatest common divisor of $p^2-1$ and $q^2-1$ is.
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Context:
The coefficient of the $x^2$ term in the expansion of $((a+x)(1 + \sqrt{x}))^5$ is $15$. Find the sum of all the coefficients in the expansion.
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Context:
A rectangle with a length of 6 cm and a width of 4 cm is rotated around one of its sides. Find the volume of the resulting geometric solid (express the answer in terms of $\pi$).
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Context:
Find the minimum value of $\int_{0}^{1} (f'(x))^2 \, dx$ given that $f \in C^{(1)} [0,1]$, $f(1) = b$, and $\int_{0}^{1} f(x) \, dx = 0$.
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Context:
Determine $x^2+y^2+z^2+w^2$ if

$\frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1$$\frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1$$\frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1$$\frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1$
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Context:
3. A student wrote a program for recoloring a pixel into one of 128 different colors. These colors he numbered with natural numbers from 1 to 128, and the primary colors received the following numbers: white color - number 1, red - 5, orange - 13, yellow - 21, green - 45, blue - 75, indigo - 87, violet - 91, black - 128. If the initial color of the pixel has a number \( n \leq 19 \), then the student's program recolors it to the color with the number \( n+4 \), and if the initial color of the pixel has a number \( n \geq 20 \), then the pixel is recolored to the color with the number \( |129-2n| \). Initially, the pixel was red. The student applied his program 2019 times sequentially. What color did the pixel end up as a result?
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Context:
In an investment, the amount increases by a certain percentage of itself each year, with interest compounding annually. In the first year, the investment increases by 1/9th of its value, and in the second year, it increases by 1/6th of its value. If the present value is Rs. 64,000, how much will the investment be worth after two years?
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Context:
The diagonal \( AC \) of the inscribed quadrilateral \( ABCD \) is the diameter of the circumscribed circle \( \omega \). From point \( D \), a line is drawn perpendicular to the segment \( BC \), and it intersects the circle \( \omega \) again at point \( E \). Find the ratio of the areas of triangle \( BCD \) and quadrilateral \( ABEC \).
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Context:
Two concentric circles have radii of 1 and 2 units, respectively. How small can the area of a cyclic quadrilateral be, which is inscribed in the larger circle and contains the smaller circle?
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Context:
The leading coefficient $a$ in the quadratic polynomial $P(x) = ax^2 + bx + c$ is greater than $100$. Determine the maximum number of integer values for $x$ such that $|P(x)| < 50$.
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Context:
当 $\frac{\pi}{4} \leq x \leq \frac{\pi}{2}$ 时, $y=\sin ^{2} x+\sqrt{3} \sin x \cos x$ 的取值范围是 $[a, \frac{c}{b}]$。请给出$a + b+c$的值。
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Context:
Given $60 < x < 84$ and $28 < y < 33$, find the range of $x - y$.
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Context:
Right triangle DEF has leg lengths DE = 18 and EF = 24. If the foot of the altitude from vertex E to hypotenuse DF is F', then find the number of line segments with integer length that can be drawn from vertex E to a point on hypotenuse DF.
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Context:
A point $Q$ is chosen inside $\triangle DEF$ such that lines drawn through $Q$ parallel to $\triangle DEF$'s sides decompose it into three smaller triangles $u_1$, $u_2$, and $u_3$, which have areas $3$, $12$, and $15$ respectively. Determine the area of $\triangle DEF$.
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Context:
给定正整数 $n(n \geqslant 3)$ 。对于 $1,2, \cdots, n$的任意一个排列 $P=\left(x_{1}, x_{2}, \cdots, x_{n}\right)$, 若 $i<j<k$, 则称 $x_{j}$ 介于 $x_{i}$ 和 $x_{k}$ 之间（如在排列（ $1,3,2,4$ ）中，3 介于 1 和 4 之间， 4 不介于 1 和 2 之间）。设集合 $S=\left\{P_{1}, P_{2}, \cdots, P_{m}\right\}$的每个元素 $P_{i}$ 是 $1,2, \cdots, n$ 的排列. 已知 \{1,2, \cdots, n\} 的任意三个不同数中都有一个数，它在每个 $P_{i}(1 \leqslant i \leqslant m)$ 中都不介于另外两个数之间。求 $m$ 的最大值。请给出当 $n = 5$ 时，$m$ 的最大值。
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Context:
A couple agreed to split their expenses equally, including the salary of their house help. While doing her job, the house help had an accident; thus, she underwent a medical procedure which cost her $128. The couple decided to pay half of the medical expenses and the other half will be deducted from her $160 salary. If the husband paid for the entire half of the hospital bill, how much does he still need to pay so that they split the expenses for the house help equally?
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Context:
In triangle \(ABC\), the radii of the circumscribed and inscribed circles are given as \(R\) and \(r\) respectively. Let \(A_1, B_1, C_1\) be the points where the angle bisectors of triangle \(ABC\) intersect the circumscribed circle. Find the ratio of the areas of triangles \(ABC\) and \(A_1B_1C_1\).
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Context:
Daniel is 85 years old, Elise is 55 years old, and Ishaan is 15 years old. How many years will it take until Daniel is 3.5 times as old as Ishaan, and Elise is twice as old as Ishaan?
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Context:
There are 100 chips numbered from 1 to 100 placed in the vertices of a regular 100-gon in such a way that they follow a clockwise order. In each move, it is allowed to swap two chips located at neighboring vertices if their numbers differ by at most $k$. What is the smallest value of $k$ for which, by a series of such moves, it is possible to achieve a configuration where each chip is shifted one position clockwise relative to its initial position?
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Context:
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\angle BAC = 120^{\circ}$ and the angle bisector of $\angle BAC$ intersects $BC$ at $D$, if $AD = 1$, then the minimum value of $b + 4c$ is ____.
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Context:
On the reverse sides of 2005 cards, different numbers are written (one number on each card). In one query, you are allowed to point to any three cards and learn the set of numbers written on them. What is the minimum number of queries needed to determine the numbers written on each card?
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Context:
Joe's quiz scores were 88, 92, 95, 81, and 90, and then he took one more quiz and scored 87. What was his mean score after all six quizzes?
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Context:
9 friends decided to divide the hotel bill evenly. The bill was a certain amount, and they paid $124.11 since 1 cent is the smallest unit. What was the original bill amount?
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Context:
Given that \( n \) is a positive integer, \( P \) is a prime number, and \( pn \) has exactly 8 positive divisors, arrange them in ascending order as \( 1=d_{1}<d_{2}< \cdots <d_{8}=pn \). Additionally, let \( d_{17p-d_{3}}=\left(d_{1}+d_{2}+d_{3}\right)\left(d_{3}+d_{4}+13p\right) \). Find \( n \).
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Context:
Evaluate the integral \( \int_0^\infty xI_0(2x)e^{-x^2}\,dx \), where \( I_0(x) \) is the modified Bessel function of the first kind, defined by \( I_0(x) = \frac{1}{\pi} \int_0^\pi e^{x\cos\theta}\,d\theta \).
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Context:
Allan brought 6 balloons and Jake brought 3 balloons to the park. Jake then bought 4 more balloons at the park. Meanwhile, Emily brought 5 balloons to the park. How many more balloons did Jake have than the combined total of Allan and Emily in the park?
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Context:
Evaluate the integral \( \int\limits_{0}^{\infty} {\frac{2}{3x^{4/3}} \sqrt{\frac{\lambda}{2\pi}} \exp \left( \frac{-\lambda (x^{2/3}-\mu)^2}{2\mu^2 x^{2/3}}  \right) dx} \) for \(\mu>0, \lambda>0\).
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Context:
Let $x, y, z$ be positive real numbers satisfying $2 x y z=3 x^2+4 y^2+5 z^2$.
Find the minimum of the expression $P=3 x+2 y+z$.
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Context:
Evaluate the sum of the infinite series: $$\sum_{k=5}^{\infty}\binom{k-1}{k-5}\frac{k^3}{2^{k}}.$$
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Context:
Adam has a box with $15$ pool balls numbered from $1$ to $15$. He picks out $5$ of them and sorts them in increasing order. He then calculates the four differences between each pair of adjacent balls and finds that exactly two of these differences are equal to $1$. How many different selections of $5$ balls could he have drawn from the box?
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Context:
Sara has an ice cream cone with every meal. The cone has a height of $2\sqrt2$ inches and the base of the cone has a diameter of $2$ inches. Ice cream protrudes from the top of the cone in a perfect hempisphere. Find the surface area of the ice cream cone, ice cream included, in square inches.
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Context:
In the context of the ratio test for determining the radius of convergence of a power series \( \sum c_{n}z^n \), if \( \limsup \frac{c_{n}}{c_{n+1}} \neq \liminf \frac{c_{n}}{c_{n+1}} \), which should be considered: the lim sup or the lim inf?
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Context:
Let $a$ and $b$ be two positive reals such that the following inequality
    \[ ax^3 + by^2 \geq xy - 1 \] is satisfied for any positive reals $x, y \geq 1$. Determine the smallest possible value of $a^2 + b$.

[i]Proposed by Fajar Yuliawan[/i]
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Context:
If $\displaystyle p=\sum^{r}_{k=0}\binom{n}{2k}\binom{n-2k}{r-k}$ and $\displaystyle q=\sum^{n}_{k=r}\binom{n}{k}\binom{2k}{2r}\bigg(\frac{3}{4}\bigg)^{n-k}\bigg(\frac{1}{2}\bigg)^{2k-2r}$ , where $n\geq 2r$. Then value of $\displaystyle \frac{p}{q}$
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Context:
若 $\triangle A B C$ 的内角 $A, B, C$ 满足 $\sin A=\cos B=\tan C$,  $\cos ^{3} A+\cos ^{2} A-\cos A$ 的值为$\frac{n}{m}$，求$m+n$的值。
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Context:
Let \( A_{11} \) denote the answer to problem 11. Determine the smallest prime \( p \) such that the arithmetic sequence \( p, p + A_{11}, p + 2A_{11}, \ldots \) begins with the largest possible number of primes.
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Context:
What is the probability that the greatest common divisor of two natural numbers, $a$ and $b$, is twice a prime number? Denote the set of prime numbers by $P$. Express your answer as a numerical probability.
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Context:
A rectangular array of chairs is an arrangement of the chairs in rows and columns such that each row contains the same number of chairs as every other row, and each column contains the same number of chairs as every other column. With the condition that there must be at least two chairs in every row and column, and all the chairs in the room must be included, how many arrays are possible in a classroom containing $48$ chairs?
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Context:
The area of the closed shape formed by the graph of the function $y=2\sin x$, where $x \in \left[\frac{\pi}{2}, \frac{5\pi}{2}\right]$, and the lines $y=\pm2$ is what?
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Context:
In a convex quadrilateral \( {ABCD} \) it is given that \( \angle {CAB} = {40}^{ \circ  },\angle {CAD} = {30}^{ \circ  },\angle {DBA} = {75}^{ \circ  } \) , and \( \angle {DBC} = {25}^{ \circ  } \) . Find
 \( \angle {BDC} \) .
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Context:
Calculate the limit: \( \lim_{x\to0} \sum_{n=1}^\infty \frac{\sin x}{4+n^4x^4} \)
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Context:
Determine the cardinality of the set of equivalence classes of well-orders on \( \mathbb{N} \) under the equivalence relation \( R \equiv R' \leftrightarrow (\mathbb{N},R) \cong (\mathbb{N},R') \), where \( R \) and \( R' \) are well-orders on \( \mathbb{N} \).
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Context:
What is the minimum number of vertices that must be selected from a hexagonal grid so that there are always two vertices among them such that the midpoint of the segment connecting them is also a vertex of the grid?
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Context:
Two students are preparing to register for the independent admission tests of Zhejiang University, Fudan University, and Shanghai Jiao Tong University, with the requirement that each person can choose at most two schools. Calculate the number of different registration results.
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Context:
Consider two circles $k_1$ and $k_2$ centered at points $A$ and $B$ with radii $r_1$ and $r_2$, respectively. A third circle $k$ is tangent to both $k_1$ and $k_2$ from the inside, with radius $r_3$. Determine the limit of the ratio $r_3 / r_1$ as the radius $r_1$ approaches 0, while keeping the centers $A$ and $B$ fixed.
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Context:
Calculate \( \lim_{\lambda \rightarrow \infty} T_{\frac{\sin(\lambda x)}{x}} \) in the sense of distributions \( \mathcal{D}'(\mathbb{R}) \).
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Context:
A fancy bathroom scale is calibrated in Newtons. This scale is put on a ramp, which is at a $40^\circ$ angle to the horizontal. A box is then put on the scale and the box-scale system is then pushed up the ramp by a horizontal force $F$. The system slides up the ramp at a constant speed. If the bathroom scale reads $R$ and the coefficient of static friction between the system and the ramp is $0.40$, what is $\frac{F}{R}$? Round to the nearest thousandth.
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Context:
Ostrovsky $M$.

A number from 1 to 144 is guessed. You are allowed to select one subset of the set of numbers from 1 to 144 and ask whether the guessed number belongs to it. You have to pay 2 rubles for a "yes" answer and 1 ruble for a "no" answer. What is the minimum amount of money needed to surely guess the number?
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Context:
On the lateral sides \( A B \) and \( C D \) of trapezoid \( A B C D \), points \( P \) and \( Q \) are marked such that line \( P Q \) is parallel to \( A D \), and segment \( P Q \) is divided by the diagonals of the trapezoid into three equal parts. Find the length of the base \( B C \), given that \( A D = a \), \( P Q = m \), and the intersection point of the diagonals of the trapezoid lies inside the quadrilateral \( B P C Q \).
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Context:
记 I=120\displaystyle\int_0^1\sqrt{\df{1-x}{1+x}}\ \dd x，则 [I]=__________.
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Context:
Evaluate the improper integral \( \int_{-\infty}^\infty{\delta(t^2)} \, dt \) using the properties of the Dirac delta function.
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Context:
In a bag, there are $n$ red balls and $5$ white balls, where $n \in \mathbb{N}^*$. Two balls are randomly drawn from the bag each time. If the probability of drawing two balls of different colors is denoted as $p(n)$, then the maximum value of $p(n)$ is ____.
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Context:
Determine the signature of the quadratic form associated with the matrix:

$$ \begin{pmatrix}
    1 & 2 & 0 \\
    2 & 0 & 1 \\
    0 & 1 & 1 \\
    \end{pmatrix} $$
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Context:
Compute the limit:
$$ \lim_{n\to +\infty}\frac{1}{n^2}\sum_{i=1}^{n}\log\binom{n}{i}.$$
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Context:
From the set $M = \{(x,y) \mid (|x|-1)^{2} + (|y|-1)^{2} < 4\}$, where $x$ and $y$ are integers, randomly select a point $P(x,y)$. If the probability that $xy \geqslant k$ (where $k > 0$) is $\frac{6}{25}$, then the maximum value of $k$ is ______.
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Context:
A cross-pentomino is a shape that consists of a unit square and four other unit squares, each sharing a different edge with the first square. If a cross-pentomino is inscribed in a circle of radius $R$, what is $100R^2$?
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Context:
What is the minimum number of operations required to transform 2014 into 1, where an operation is defined as adding, subtracting, multiplying, or dividing by a single-digit number (1-9)?
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Context:
In triangle $\triangle ABC$, the sides opposite to the angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b=50\sqrt{6}$, $c=150$, and $C=60°$, find the angle $B$.
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Context:
Determine the Big-O complexity of the recurrence relation: \[ T(n) = \sqrt{n} \cdot T(\sqrt{n}) + \sqrt{n} \]
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Context:
An amount of money will become $1000 in 5 years with a rate of interest of 20%. What is the initial amount of money?
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Context:
Provide an example of a real sequence \( \{a_n\} \) such that \( \lim\limits_{n\to\infty}(a_{n+1}-a_n)=0 \), but \( \lim\limits_{n\to\infty}a_n \) does not exist finitely.
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Context:
Let \(a, b \in \mathbb{N}\) be two coprime numbers. Determine the number of ordered pairs \((a, b)\) such that \(\frac{b}{a} + \frac{101b}{5050a} \in \mathbb{N}\), given that \(5050=2\cdot5\cdot505\).
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Context:
Solve the differential equation \( \frac{dy}{dx} = \frac{x^2-1}{y^2-1} \) with the initial condition \( y(0) = 0 \).
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Context:
The triangle $ABC$ is isosceles with $AB = BC$. The point $F$ on the side $[BC]$ and the point $D$ on the side $AC$ are the feet of the internal bisector drawn from $A$ and the altitude drawn from $B$, respectively, such that $AF = 2BD$. Find the measure of the angle $\angle ABC$.
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Context:
For how many ordered triples $(x,y,z)$ of nonnegative integers less than $20$ are there exactly two distinct elements in the set $\{i^x, (1+i)^y, z\}$, where $i^2 = -1$?
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Context:
Solve in prime numbers:
$  \{\begin{array}{c}\\ \ 2a - b + 7c = 1826  \\3a + 5b + 7c = 2007 \end{array}$
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Context:
Let $a, b$ and $c$ be the distinct solutions to the equation $x^3-2x^2+3x-4=0$. Find the value of
$$\frac{1}{a(b^2+c^2-a^2)}+\frac{1}{b(c^2+a^2-b^2)}+\frac{1}{c(a^2+b^2-c^2)}.$$
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Context:
The five tallest buildings in Los Angeles in 1985 had a mean height of 733 feet. The tallest of the five buildings has a height of 858 feet, the shortest of the five 625 feet. If a new building were constructed with a height of 885 feet, by how many feet would it increase the mean height of the five tallest buildings of the city?
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Context:
Find the coefficient of \(x^{18}\) in the expansion of \((x+1)(x+2)\cdots(x+10)(2x+1)(2x+3)\cdots(2x+19)\).
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Context:
Determine the number of 3-Sylow subgroups in a simple group with 168 elements. The possible values for the number of 3-Sylow subgroups are 1, 4, 7, or 28. Given that the group is simple, find the correct value for the number of 3-Sylow subgroups.
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Context:
Today, Ivan the Confessor prefers continuous functions $f:[0,1] \rightarrow \mathbb{R}$ satisfying $f(x)+f(y) \geq|x-y|$ for all pairs $x, y \in[0,1]$. Find the minimum of $\int_{0}^{1} f$ over all preferred functions.
The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
What is the minimal number of cuts necessary to find the center of a cylindrical cake using a knife, given that you can only score the surface of the cake? Is it possible to determine the center with fewer than 7 cuts?
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Context:
Determine whether the statement "A given non-terminal A in a given context-free grammar (CFG) is ever used in the generation of a word" is decidable or undecidable.
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Context:
Let $A$ be a $55 \times 55$ diagonal matrix with characteristic polynomial $(x-c_1)(x-c_2)^2(x-c_3)^3 \ldots (x-c_{10})^{10}$, where $c_1, c_2, \ldots, c_{10}$ are distinct. Let $V$ be the vector space of all $55 \times 55$ matrices $B$ such that $AB = BA$. Determine the dimension of $V$.
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Context:
Determine the values of $a$ for which the matrix product $[R]_{\alpha}[S]_{\alpha}$ is skew-symmetric, given the matrices $[S]_{\alpha}$ and $[R]_{\alpha}$.
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Context:
设$a_{1},a_{2},\cdots,a_{k}\left(k\in\mathbf{Z}_{+}\right)$均为大于1的整数，且满足$$\left(a_{1}!\right)\left(a_{2}!\right)\cdots\left(a_{k}!\right)\mid2017!$$。当$k$变化时，求$\sum\limits_{i=1}^{k}a_{i}$的最大值。
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Context:
Find all complex numbers \( z \) such that \( \sin(z) = 0 \).
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Context:
Given eight distinguishable rings, let  $n$  be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand.  The order of rings on each finger is significant, but it is not required that each finger have a ring.  Find the leftmost three nonzero digits of  $n.$
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Context:
Vertices $A, B, C$ of a equilateral triangle of side $1$ are in the surface of a sphere with radius $1$ and center $O$. Let $D$ be the orthogonal projection of $A$ on the plane $\alpha$ determined by points $B, C, O$. Let $N$ be one of the intersections of the line perpendicular to $\alpha$ passing through $O$ with the sphere. Find the angle $\angle DNO$.
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Context:
Find the least positive integer such that the product of its digits is $8! = 8  \cdot 7  \cdot6  \cdot 5  \cdot 4  \cdot 3  \cdot 2 \cdot 1$.
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Context:
7. A row of 100 coins is laid out. All the coins look the same, but somewhere among them are 50 counterfeit ones (and the rest are genuine). All genuine coins weigh the same, and the counterfeit ones may weigh differently, but each counterfeit is lighter than a genuine one. Can at least 34 genuine coins be found with a single weighing on a balance scale without weights? (O. Dmitriev, R. Zhenodarov)
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Context:
Determine whether the double integral \( I = \iint_{\mathbb{R}^2} \frac{x^2}{(1+x^2)(x^2 + y^2)^{3/2}}\,dx\,dy \) converges.
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Context:
Given an $L^1(\mathbb{R})$ function $f$ such that $f(x) = -f(-x)$, does it follow that $\lim_{\xi \to \infty} \hat{f}(\xi) = \hat{f}(0)$, given that $\hat{f}(\xi) = -\hat{f}(-\xi)$ and $\lim_{\xi \to \infty} \hat{f}(\xi) = 0$?
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Context:
Suppose $G$ is a Lie group acting smoothly and freely, but not necessarily properly, on a smooth manifold $M$. If $M/G$ is known to be a topological manifold, determine whether the dimension of $M/G$ is equal to $\dim M - \dim G$.
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Context:
In a \(\sigma\)-finite measure space \((X, \mathcal{M}, \mu)\) and the Lebesgue measurable space \((\mathbb{R},\mathcal{L}, m)\), consider a function \(f:X \to \mathbb{R}\) that is integrable (\(f \in L^1(X)\)). Let \(\Gamma = \{(x, f(x)): x\in X\} \subset X \times \mathbb{R}\). Determine if \(\Gamma\) is measurable with respect to the product \(\sigma\)-algebra \(\mathcal{M} \times \mathcal{L}\), and if it is, compute the measure \((\mu \times m)(\Gamma)\).
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Context:
Given integers $f(i)$ for all integers $i=1,2,\ldots,2^{n-1}$ such that $1\le f(i)\le i$, determine if there exist integers $a_{1},a_{2},\ldots,a_{n}$ such that $1=a_{1}<a_{2}<\cdots<a_{n}\le 2^{n-1}$, and $f(a_{1})\le f(a_{2})\le \cdots\le f(a_{n})$. Answer "yes" or "no."
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Context:
Determine the values of $p$ for which the closure of $L^1(\mathbb{R}) \cap L^p(\mathbb{R})$ is equal to $L^p(\mathbb{R})$. Provide your answer as a set of values for $p$.
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Context:
Find the value of $k \in \mathbb{R}$ such that \( \lim_{n\to\infty} \frac{\tan n}{n^k} = 0 \).
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Context:
Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions

(a) $0\leq A<B<C\leq99$,
(b) there exist integers $a$, $b$, and $c$, and prime $p$ where $0\leq b < a < c < p$,
(c) $p$ divides $A-a$, $B-b$, and $C-c$, and
(d) each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithmetic sequences.

Find $N$.
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Context:
Determine the points $p$ in the set $L = \{ (x,y,z) \mid f(x,y,z) = c \}$, where $f(x,y,z) = \langle x^2 + y^2 + z^2, (x - \frac{1}{2})^2 + y^2 \rangle$ and $c = (1, a^2)$ for $0 < a < \frac{3}{2}$, such that the rank of the Jacobian matrix $Df(p)$ is strictly less than 2. The Jacobian matrix is given by:
\[Df(p) = \begin{bmatrix} 2x & 2y & 2z \\ 2x-1 & 2y & 0 \end{bmatrix}\]
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Context:
Suppose $P(x)$ is a degree $n$ monic polynomial with integer coefficients such that $2013$ divides $P(r)$ for exactly $1000$ values of $r$ between $1$ and $2013$ inclusive. Find the minimum value of $n$. \( \boxed{n} \)
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Context:
Consider a $2 \times n$ grid where each cell is either black or white, which we attempt to tile with $2 \times 1$ black or white tiles such that tiles have to match the colors of the cells they cover. We first randomly select a random positive integer $N$ where $N$ takes the value $n$ with probability $\frac{1}{2^n}$. We then take a $2 \times N$ grid and randomly color each cell black or white independently with equal probability. Compute the probability the resulting grid has a valid tiling.
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Context:
Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement?

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Context:
Determine whether the extension \( \mathbb{K}(X) \supset \mathbb{K}(u) \) is algebraic, where \( u = \frac{X^3}{X+1} \), and find the degree \([\mathbb{K}(X):\mathbb{K}(u)]\).
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Context:
Find the maximum number of elements which can be chosen from the set $ \{1,2,3,\ldots,2003\}$ such that the sum of any two chosen elements is not divisible by 3.
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Context:
Let $X$ be an uncountable set with the cocountable topology. Determine whether $(X, T)$ is a $T_1$ space.
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Context:
Is the Ramsey number $R(0,a)$ defined, and if so, what is its value?
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Context:
Determine whether the series \( \sum_{n=1}^{\infty}\left(\cos\frac{1}{n}\right)^{k(n)} \) converges, where \( k(n)=\frac{1}{\sin^a\left(\frac{1}{n}\right)} \) and \( a \in \mathbb{R} \). Use the approximations \( \cos\frac{1}{n} \approx 1-\frac{1}{2n^2} \) and \( \sin^a\left(\frac{1}{n}\right) \approx \frac{1}{n^a} \) to analyze the behavior of the series. Consider applying the comparison or limit comparison test.
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Context:
A bag contains three tiles marked $A$, $L$, and $I$. Muhammad Ali wants to pick the letters $A$, $L$, and $I$ in that order. He randomly pulls one tile from the bag. If the letter $A$ is drawn, he keeps it. If the letter pulled is not $A$, he puts it back into the bag. He repeats this process for the next tile, keeping it if it is $L$ and putting it back otherwise. Determine the probability that Muhammad draws from the bag at most 10 times. Express this probability as a fraction $\frac{x_1}{x_2}$, where $x_1$ and $x_2$ are relatively prime numbers. Compute the remainder when $x_1 + x_2$ is divided by 1000.
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Context:
12. If $\sec \alpha \sqrt{1+\tan ^{2} \alpha}+\tan \alpha \sqrt{\csc ^{2} \alpha-1}=\tan ^{2} \alpha$, determine the sign of the product $\sin (\cos \alpha) \cdot \cos (\sin \alpha)$.
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Context:
Determine whether the dimension of the symplectic group $Sp(2n, F)$ is $2n^2 + n$ when $F$ is the algebraic closure of a finite field.
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Context:
Determine the maximum number of elements in the set \( \mathcal{S} \) of all linear maps \( \mathbf{L} (\mathbb{C}^3) \) where no two maps are similar and satisfy the equation \( \mathbf{A^{10} - 3A^9 + 2A^8 = 0} \).
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Context:
Points $E$ and $F$ are located on square $ABCD$ so that $\triangle BEF$ is equilateral. What is the ratio of the area of $\triangle DEF$ to that of $\triangle ABE$?

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Context:
Given the in-degree and out-degree sequences of a directed graph without loops or multiple edges, determine how many of the following sequences correspond to a valid graph:

(a) \(d_{in}=(0,1,2,3), d_{out}=(2,2,1,1)\)

(b) \(d_{in}=(2,2,1), d_{out}=(2,2,1)\)

(c) \(d_{in}=(1,1,2,3,3), d_{out}=(2,2,3,1,2)\)

Find the number of sequences that can form a valid directed graph.
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Context:
Three members of the Euclid Middle School girls' softball team had the following conversation.
Ashley: I just realized that our uniform numbers are all 2-digit primes.
Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month.
Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month.
Ashley: And the sum of you two uniform numbers is today's date.

What number does Caitlin wear?
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Context:
An error of 7% in excess is made while measuring the side of a parallelogram, and an error of 3% in deficiency is made while measuring the angle. What will be the percentage error in the calculated area?
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Context:
Suppose $X$ and $Y$ are independent and identically distributed random variables with probability density function $f(x)$, $E[X]=E[Y]=m$, and $Z=(X-Y)\mathrm{sign}(X-m)$. Determine if $E[Z]>0$ is true.
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Context:
Determine the Euler-Lagrange equation for the functional \( \int_{0.5}^2 (y')^2(1 + y'^2) \,dx \) subject to the boundary conditions \( y(0.5) = 20 \) and \( y(2) = 2 \).
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Context:
Suppose $\epsilon$ and $X$ are random variables, and $\epsilon_i$ and $x_i$ are the $i$-th realizations of these random variables. Determine if the condition $E[\epsilon_i|x_i] = 0$ for all $i$ implies $E[\epsilon | X] = 0$. Answer 'yes' or 'no'.
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Context:
Given real numbers $b_0,b_1,\ldots, b_{2019}$ with $b_{2019}\neq 0$, let $z_1,z_2,\ldots, z_{2019}$ be the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019}b_kz^k.
\]
 Let $\mu = (|z_1|+ \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\ldots, z_{2019}$ to the origin.  Determine the largest constant $M$ such that $\mu\geq M$ for all choices of $b_0,b_1,\ldots, b_{2019}$ that satisfy
\[
1\leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019.
\]
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Context:
Chords $ AB$ and $ CD$ of a circle intersect at a point $ E$ inside the circle. Let $ M$ be an interior point of the segment $ EB$. The tangent line at $ E$ to the circle through $ D$, $ E$, and $ M$ intersects the lines $ BC$ and $ AC$ at $ F$ and $ G$, respectively. If
\[ \frac {AM}{AB} = t,
\]
find $\frac {EG}{EF}$ in terms of $ t$.
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Context:
某建筑物的底部是周长约是75.36米的圆，它的占地面积大约是多少平方米？
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Context:
Given that the largest real root of the equation \(x^{3} - 7x^{2} + 1 = 0\) is \(t\), what is the remainder when \(\left[t^{20100}\right]\) is divided by 7?
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Context:
Jack and Jenny start hiking from the trailhead to the fire tower at the top of the mountain. Jenny, carrying a very heavy backpack, walks very slowly. Jack starts at a speed of 4 mph and, upon reaching halfway, where the trail becomes steep, his speed decreases to 2 mph. After reaching the fire tower, Jack immediately returns and descends the steep trail at a speed of 3 mph, meeting Jenny halfway down. What was Jenny's average speed (in mph) before meeting Jack? Express your answer as a fraction or decimal value.
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Context:
(2009•宝山区一模)某抛物线形拱桥的跨度为20米，拱高是4米，在建桥时，每隔4米需用一根柱支撑，其中最高支柱的高度是多少？
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Context:
Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
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Context:
Suppose matrices A and B satisfy the equation \( A^{-1} + B^{-1} = (A+B)^{-1} \). Given that \(|A| = -3\), determine the possible values of \(|B|\).
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Context:
Evaluate \( \lim_{n\to\infty} n^2 \int_0^1 x^n f(x) \, dx \) given that \( f(x) = (x-1) e^{-\frac{1}{x+5}} \) and \( f(1) = 0 \).
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Context:
Let $\{W_t\}$ be a standard Brownian motion and define the process $\{X_t\}$ by $X_t= {W_t}^{2}-t ,\ \ t\geq0.$ Calculate $P(X_5>0\mid W_3=1)$. Provide your answer as a probability value.
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Context:
Determine the minimum dimension n such that the system of differential equations \( x' = Ax \) is satisfied, where \( A \) is an \( n \times n \) matrix and the first coordinate of the solution \( x(t) \) is given by \( x_1(t) = 5t^2\cos(3t) + 2\sin(3t) - e^{2t}\sin(3t) + 4t \). What is the minimum value of n?
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Context:
In a simple linear regression model $Y_i = \beta_0 + \beta_1x_i + \epsilon_i$ where $\epsilon_i \sim^{indep} N(0, \sigma^2)$ for $i = 1,...,n$, let $\hat{\beta_{0}}$ and $\hat{\beta_{1}}$ be the maximum likelihood estimators of $\beta_0$ and $\beta_1$. The $i$th residual is $\hat{\epsilon_{i}} = Y_i - \hat{Y_{i}}$, where $\hat{Y_i} = \hat{\beta_{0}} + \hat{\beta_{1}}x_i$. Derive the covariance $Cov(\hat{\epsilon_{i}},\hat{Y_i})$. \( \text{Express your answer in terms of known quantities.} \)
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Context:
Let $(a_n)_n\geq 0$ and $a_{m+n}+a_{m-n}=\frac{1}{2}(a_{2m}+a_{2n})$ for every $m\geq n\geq0.$ If $a_1=1,$ then find the value of $a_{2007}.$
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Context:
Given the function $F = e^a b(c(t))$, where $a \in \mathbb{R}$ and $b(c(t))$ is a function with $c(t)$ as its argument, determine the derivative of $F$ with respect to $\dot{c}(t)$. Is this derivative equal to zero?
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Context:
Who is credited with the conjecture that all odd squares are solitary numbers?
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Context:
Calculate the expected number of days it will take for two independent seeds to sprout, given that the probability of a seed not sprouting on day \( n \) is \( \frac{1}{2^n} \).
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Context:
4. Let $z=1-2 i$. Find $\frac{1}{z}+\frac{2}{z^{2}}+\frac{3}{z^{3}}+\cdots$.
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Context:
Let $A,B,C$ be angles of an obtuse triangle with
$$
\begin{align*} \cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\ \cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9} \end{align*}
$$
There are positive integers $p$, $q$, $r$, and $s$ for which $$\cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B = \frac{p-q\sqrt{r}}{s},$$ where $p+q$ and $s$ are relatively prime and $r$ is not divisible by the square of any prime. Find $p+q+r+s$.

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Context:
A trapezoid has area $2\, m^2$ and the sum of its diagonals is $4\,m$. Determine the height of this trapezoid.
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Context:
A $4 \times 4 \times h$ rectangular prism can fit 8 small spheres with a radius of 1 and 1 large sphere with a radius of 2. Determine the minimum value of $h$.
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Context:
How many pairs of positive integers \((x, y)\) satisfy the equation \(x^2 - 10! = y^2\)?
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Context:
Let \( V \) be a finite-dimensional vector space, and let \( \mathcal{L}(V) \) denote the space of all endomorphisms of \( V \). For any \( \phi \in \mathcal{L}(\mathcal{L}(V)) \), there is a unique bilinear map \( \mathcal{L}(V) \otimes \mathcal{L}(V) \to \mathcal{L}(V) \) (denoted by \( \Phi(\phi) \)) satisfying \( \Phi(\phi) (a \otimes b) = \phi(ab) \) for any \( a, b \in \mathcal{L}(V) \). Determine the rank of the linear map \( \Phi \).
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Context:
Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$?
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Context:
Determine the additional variable \( \eta \) required to make the transformation \( \Phi : (x,t) \rightarrow (\xi,\eta) \) a diffeomorphism when using Boltzmann's transformation with \( \xi = \frac{x}{2\sqrt{t}} \) in the context of converting Fick's second law to an ordinary differential equation.
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Context:
Determine whether the sequence \(\sqrt[n]{(n+1)!}\) converges or diverges.
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Context:
Consider a Poisson process $N_t$ and a continuous process of finite variation $X_t$. Is the quadratic covariation $\langle X, N \rangle_t$ equal to zero?
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Context:
In trapezoid $ABCD$, sides $AB$ and $CD$ are parallel, and diagonal $BD$ and side $AD$ have equal length. If $m\angle DBC=110^\circ$ and $m\angle CBD =30^\circ$, then $m \angle ADB=$

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Context:
Determine whether \( f(n!) \to +\infty \) as \( n \to +\infty \), where \( f(x) = x (1 + [x] - x) \).
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Context:
Chris and Paul each rent a different room of a hotel from rooms $1 - 60$. However, the hotel manager mistakes them for one person and gives ”Chris Paul” a room with Chris’s and Paul’s room concatenated. For example, if Chris had $15$ and Paul had $9$, ”Chris Paul” has $159$. If there are $360$ rooms in the hotel, what is the probability that ”Chris Paul” has a valid room?

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Context:
Find the sum of all integers $n$ with $2 \le n \le 999$ and the following property: if $x$ and $y$ are randomly selected without replacement from the set $\left\{ 1,2,\dots,n \right\}$, then $x+y$ is even with probability $p$, where $p$ is the square of a rational number.

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Context:
Determine the dimension of the vector space \(V = \mathbb{R}^+ \times \mathbb{R}^+ \times \mathbb{R}^+\), where \(\mathbb{R}^+\) is the set of all positive real numbers. The operations in \(V\) are defined as \((x_1, y_1, z_1) \oplus (x_2, y_2, z_2) = (x_1x_2, y_1y_2, z_1z_2)\) and \(\alpha(x, y, z) = (x^\alpha, y^\alpha, z^\alpha)\) for \(\alpha \in \mathbb{R}\). Provide a basis for \(V\) that includes the vector \((e, e, 1)\).
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Context:
A cylinder is filled with gas at atmospheric pressure (103.3 kPa). Assuming the gas is ideal, calculate the work done (in joules) during the isothermal compression of the gas by a piston that moves inward by 1.0 meter. The initial height of the gas column is 2.0 meters, and the radius of the cylinder is 0.4 meters. Use the equation of state \( \rho V = \text{const} \), where \( \rho \) is the pressure and \( V \) is the volume.
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Context:
Evaluate the Riemann-Stieltjes integral \( \int_0^3 x^3 \, \text{d}\left[\frac{x}{2}\right] \) using the properties that \([x/2] = 0\) for \(0 \leq x < 2\) and \([x/2] = 1\) for \(2 \leq x < 4\).
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Context:
Find the largest positive integer $n$ such that the following is true:
There exists $n$ distinct positive integers $x_1,~x_2,\dots,x_n$ such that whatever the numbers $a_1,~a_2,\dots,a_n\in\left\{-1,0,1\right\}$ are, not all null, the number $n^3$ do not divide $\sum_{k=1}^n a_kx_k$.
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Context:
Let \( u_n \) be a sequence defined by \( u_0 = 1 \) and \( u_{n+1} = 1 + \frac{n}{u_n} \). Determine whether the sequence \( u_n \) diverges to \( +\infty \).
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Context:
Evaluate the contour integral \( \int_\gamma \frac{\mathrm{Re} (z)}{z - \beta} dz \), where \( \beta \) is a complex number such that \( 0 < |\beta| < 4 \) and \( \gamma \) is the circle oriented in the negative sense, centered at the origin with radius 5.
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Context:
Let $\pi_1$ and $\pi_2$ be two distinct irreducible representations of a finite group $G$, both having the same degree. Determine whether the following sum equals the zero matrix:
\[\sum_{g \in G}  \pi_1(g)   \pi_2(g^{-1})^\top = 0.\]
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Context:
(fidgetboss_4000) In the diagram below, $ABC$ is an isosceles right triangle with a right angle at $B$ and with a hypotenuse of $40\sqrt2$ units. Find the greatest integer less than or equal to the value of the radius of the quarter circle inscribed inside $\triangle ABC$.

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Context:
In a plane covered with equilateral triangles, each vertex is shared by 6 triangles. Initially, one triangle is colored red. In each subsequent step, color all triangles that share at least one vertex with a red triangle and are not yet colored. How many triangles are colored in the 100th step?
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Context:
Suppose that the series \(\sum_{j=1}^{\infty}a_j\) is convergent, with \(a_j\geq 0\). Determine whether the following limit is true: \[\lim_{n\to \infty}S_n=0,\qquad S_n:=\sum_{j=1}^{n-1} \frac{a_j}{n-j}.\]
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Context:
Find the value of $a$ such that:

$$\lim_{x\to\frac{\pi}{6}}\bigg(\sqrt3 \tan x\bigg)^\frac{(a+1)\tan (3x)}{\sin x} = e^\frac{-8\sqrt3}{3}$$
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