Context:
Let $X_1, X_2, \ldots, X_n, \ldots$ be independent and identically distributed random variables with $P(X_n=0)=P(X_n=1)=P(X_n=2)=P(X_n=3)=\frac{1}{4}$. Define the sequence $\{Y_n\}$ by $Y_0=0$ and for all $n \in \mathbb{N}$, \[ Y_n=\begin{cases} 3 & \text{if } X_n=3, \\ \min\{Y_{n-1},X_n\} & \text{if } X_n<3. \end{cases} \] Compute $\lim_{n \to +\infty} E[Y_n Y_{n-1}]$. \(\text{Express your answer as a single number.}\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient of \(t^{20}\) in the expansion of \((1+t+t^2)^{15}\) using the concept of derivatives.
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Context:
Solve for $c$ in the equation $4 - \cos(-1) = c + 5\ln(c-1)$. You may use the transformation $e^{4 - \cos(1) - c} = (c-1)^5$ to assist in finding the solution.
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Context:
$N$ is a $50-$digit number (in the decimal scale). All digits except the $26^{\text{th}}$ digit (from the left) are $1$. If $N$ is divisible by $13$, what is the $26^{\text{th}}$ digit?
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Context:
平面上有 101 条不同的射线，任两条不共线、没有相同的端点，任三条不共点. 求无序射线对 (a,b) 个数的最大值，使得 a, b 相交，且若交点为 C，则 C 到 a, b 端点的距离相等.
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Context:
In parallelogram \(ABCD\), point \(M\) is located on side \(BC\). Given that \(\angle A = 45^{\circ}\), \(\angle AMD = 90^{\circ}\), and the ratio \(BM : MC = 2 : 3\), determine the ratio of the lengths of the adjacent sides of the parallelogram.
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Context:
Determine the cardinality of the set $S$ of all analytic functions $h: D \rightarrow D$, where $D = \{z \in \mathbb{C}: |z| < 1\}$, such that $h(z)^2 = \overline{h(z)}$ for all $z \in D$. \( \boxed{} \)
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Context:
Determine the dimension of the vector space \(\mathbb{C}\) over the field \(\mathbb{Q}\).
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Context:
Determine the number of field structures that can be defined on the set \(\mathbb{R} \times \mathbb{R}\).
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Context:
Evaluate the limit \( \lim_{c\to\infty} \frac{E\left[ X e^{-\frac{(X-c)^2}{2}} \right]}{E\left[e^{-\frac{(X-c)^2}{2}}\right]} \) where \(X\) is a bounded random variable with \(|X| \le a\).
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Context:
Find the value of $A + B + C$ such that \[ S = \sum_{n=1}^{\infty} \frac{1}{n^2 \binom{2n}{n}} = \frac{A}{B} \zeta(C). \]
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Context:
Find a nonzero solution \((x, y, z) \in \mathbb{F}_7^3\) to the equation:

\[2x^2 + y^2 + 2yz + 2z^2 = 0\]
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Context:
Peter chooses a positive integer \( a \) such that the sum of its digits is 2012. Paul needs to determine this number. On each move, Paul selects a positive integer \( x \), and Peter tells him the sum of the digits of \(|x - a|\). What is the minimal number of moves required for Paul to determine Peter's number with certainty?
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Context:
Determine the least number of sets in a partition of all \(n!\) permutations of \(1, 2, 3, \ldots, n\), such that any two permutations in the same set can be transformed into each other by a sequence of the following moves:
- If \(a\) is the median of \(a, b, c\), replace \(a, b, c\) with \(b, c, a\).
- If \(c\) is the median of \(a, b, c\), replace \(a, b, c\) with \(c, a, b\).
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Context:
Evaluate the integral \( \oint\limits_{|z|=3}\frac{1}{(z^5-1)(z-2)} dz \) using the method of residues, including the residue at infinity.
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Context:
Let \( \{X_n\} \) be a sequence of i.i.d. exponentially distributed random variables with mean 1. Define \( M_n = \max(X_1, X_2, \ldots, X_n) \). Find a sequence \( \{a_n\} \) such that \( P[M_n - a_n \leq x] \) converges to a non-degenerate distribution function \( G(x) \) as \( n \to \infty \), for every \( x \in \mathbb{R} \) where \( G(x) \) is continuous.
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Context:
For $a > 0$, consider the function \( f(x) = \frac{2}{a}\sqrt{1-x}\left(a^2+\frac{1}{1-2x}\right) \) defined for \( 0 \leq x < \frac{1}{2} \). Determine the value of \( a \) for which \( f(x) \) attains its minimum value.
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Context:
Let \( f(n) \) be the sum of the first \( n \) good numbers, where a natural number \( k > 1 \) is called good if there exist natural numbers \( a_1 < a_2 < \cdots < a_k \) such that \( \frac{1}{\sqrt{a_1}} + \frac{1}{\sqrt{a_2}} + \cdots + \frac{1}{\sqrt{a_k}} = 1 \). Find the sum of all values of \( n \) for which \( \frac{f(n+5)}{f(n)} \) is an integer.
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Context:
Find the angle between the tangent line to the logarithmic spiral \(\alpha(t) = e^{-t}(\cos(t),\sin(t))\) at the point where \(t = \phi + 2\pi k\) (for some integer \(k\)) and the ray from the origin at angle \(\phi\).
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Context:
Evaluate the expression $E[E[X|Y]|X]$. What is the result?
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Context:
Calculate the integral part of the number \( T = \frac{2}{1} \cdot \frac{4}{3} \cdot \frac{6}{5} \cdot \frac{8}{7} \cdots \frac{2016}{2015} \). Given that \( T = 2017\int_{0}^{\frac{\pi}{2}} \sin^{2017}(x)dx \), and the identity \( \int_0^{\frac{\pi}{2}} \sin^{2n+1}(x) dx = \frac{2}{3} \cdot \frac{4}{5} \cdot \frac{6}{7} \cdots \frac{2n}{2n+1} \), find the integer part of \( 2017\int_{0}^{\frac{\pi}{2}} \sin^{2017}(x)dx \).
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Context:
Determine the range of values for \( b \) such that the function \( g(x) = \int_{0}^{x} \left|\log_3\left(\log_4\left(\log_2\left(\sin t + b\right)\right)\right)\right| dt \) is decreasing for all \( x \in \mathbb{R} \).
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Context:
Given the context-free grammar with variables $V_n = \{expr, term, factor, number\}$, terminals $V_t = \{(,), -, *, 0...9\}$, productions $P$ as follows:
\[\begin{align} expr &\to expr - expr \mid term \\
term &\to term * factor \mid factor \\
factor &\to number \mid (expr) \\
number &\to 0|1|2|3|4|5|6|7|8|9 \end{align}\]
and start symbol $S = expr$, determine the number of distinct derived trees for the expression $1-2-(3-4)*5*6$. \(\text{Provide your answer as a single number.}\)
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Context:
Given a function \( F: \mathbb{R}^2 \rightarrow \mathbb{R} \) that is \( C^2 \) with \( F_x(1,1)=F_{yy}(1,1)=1 \), \( F_y(1,1)=F_{xx}(1,1)=0 \), and a function \( g: \mathbb{R}^2 \rightarrow \mathbb{R} \) defined by \( g(r,\theta)=F(r\cos\theta,r\sin\theta) \), find the value of \( g_{r\theta}(\sqrt{2},\pi/4) \).
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Context:
Determine the smallest positive integer \( k \) such that \( 2^{w(n)} \leq k \sqrt[4]{n} \) holds for all positive integers \( n \).
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Context:
Find the gradient of the expression \( \mathrm{trace}(A'(X\circ X-1)) \) with respect to the matrix \( X \), where \( \circ \) denotes the Hadamard product, \( A \) is a matrix of positive real values, and \( 1 \) is a matrix of all ones.
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Context:
Given that \( \Im[f'(z)]=6x(2y-1) \) and \( f(0)=3-2i \), \( f(1)=6-5i \), find the value of \( f(1+i) \).
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Context:
For how many consecutive integers \(x\) can the quadratic polynomial \(ax^2 + bx + c\) be a perfect square, given that \(a, b, c \in \mathbb{Z}\), \(a \neq 0\), and \(b^2 - 4ac \neq 0\)?
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Context:
On a $2\times2$ chessboard, a king starts from the upper left corner and moves with equal probability to any adjacent square. Calculate the approximate probability that the king will be in the bottom right corner after exactly 1000 moves.
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Context:
Determine the multiplicity of the root \(x = a\) for the polynomial \(Q(x) = \frac{1}{2}(x-a)(p'(x)+p'(a)) - p(x) + p(a)\), where \(p(x)\) is a polynomial of degree \(n\).
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Context:
Calculate the angle between the line \((6, 8, 5) + t (3, 4, 5)\), where \(t \in \mathbb{R}\), and the sphere \(x^2 + y^2 + z^2 = 25\) at their points of intersection.
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Context:
Determine the minimum number of vertices $m$ such that the complete graph $K_n$ is a vertex minor of the path graph $P_m$. What is the value of $m$?
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Context:
Calculate the flux of the vector field \( \mathbf{F}(x,y,z) = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \) through the cylindrical surface \( S = \{ (x,y,z) \mid x^2 + y^2 \le 9, 0 \le z \le 2 \} \).
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Context:
Let the positive numbers \( x \) and \( y \) satisfy \( x^{3} + y^{3} = x - y \). Find the maximum value of the real number \( \lambda \) such that \( x^{2} + \lambda y^{2} \leq 1 \) always holds.
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Context:
Find the function $f(x)$ that satisfies the integro-differential equation: $$\int_0^x f(u)du - f'(x) = x.$$ Note that the initial condition $f(0)$ is not provided.
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Context:
How many 7-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, 7 such that among the four odd digits, any three are not all adjacent?
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Context:
When two circular rings of diameter $12$ on the Olympic Games Logo intersect, they meet at two points, creating a $60^o$ arc on each circle. If four such intersections exist on the logo, and no region is in $3$ circles, the area of the regions of the logo that exist in exactly two circles is $a\pi - b\sqrt{c}$ where $a$, $b$, $c$ are positive integers and $\sqrt{c}$ is fully simplified find $a + b + c$.
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Context:
A coin with diameter \(d\) and thickness \(t\) is flipped such that it rotates uniformly around a horizontal axis parallel to one of its faces. Determine the ratio \(\frac{d}{t}\) such that the probability of the coin landing on its edge is \(\frac{1}{3}\), assuming that any orientation of the coin is equally likely.
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Context:
If the decimal representation of $n$ is 99…9 (with 20 digits of 9), then the decimal representation of $n^3$ contains     number of digit 9.
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Context:
In triangle \(ABC\), angle \(C\) is a right angle, and \(AC: AB = 4: 5\). A circle with its center on leg \(AC\) is tangent to the hypotenuse \(AB\) and intersects leg \(BC\) at point \(P\), such that \(BP: PC = 2: 3\). Determine the ratio of the radius of the circle to leg \(BC\).
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Context:
Given that \( \sin A + \sin^2 A = 1 \) and \( a \cos^{12} A + b \cos^8 A + c \cos^6 A - 1 = 0 \), find the value of \( b + \frac{c}{a} + b \).
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Context:
Evaluate the integral \( \int_{0}^{\infty} u^{n}e^{-u} \, du \) for \( n \geq 0 \).
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Context:
Find an example of a constructible angle that cannot be quintsected using an unmarked straightedge and compass.
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Context:
The numbers \( 1,2,\ldots ,9 \) are randomly placed into the9squares of a \( 3 \times  3 \)
 grid. Each square gets one number, and each of the numbers is used once.
 What is the probability that the sum of the numbers in each row and each
 column is odd?
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Context:
Given that $A$ is a non-singular square matrix of order $2$ and satisfies the equation $|A + |A|\operatorname{adj}A| = 0$, where $\operatorname{adj}A$ denotes the adjoint of matrix $A$ and $|A|$ denotes the determinant of $A$, evaluate $|A - |A|\operatorname{adj}A|$.
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Context:
Determine the possible rank values for the matrix product $AB$ if $A$ and $B$ are $3 \times 3$ real matrices and $\operatorname{rank}(BA) = 2$. Provide your answer as a set of possible rank values.
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Context:
Determine the values of the parameter \( t \) for which the linear transformation \( \psi: \mathbb{R}^4 \rightarrow \mathbb{R}^3 \) defined by \( \psi([x_1,x_2,x_3,x_4]) = [x_1+x_2+x_3+2x_4,x_1+tx_2+x_3+3x_4,2x_1+x_2+tx_3+3x_4] \) is an epimorphism.
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Context:
The sum of $\lfloor x \rfloor$ for all real numbers $x$ satisfying the equation $16 + 15x + 15x^2 = \lfloor x \rfloor ^3$ is:
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Context:
In triangle $ABC$ with $AC \neq BC$, a triangle $A'B'C$ is formed by rotating $\triangle ABC$ around point $C$. Let $M$, $E$, and $F$ be the midpoints of segments $BA'$, $AC$, and $CB'$ respectively. Given that $EM = FM$, determine the measure of angle $\angle EMF$. \( \text{Provide your answer in degrees.} \)
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Context:
You are given that $17! = 355687ab8096000$ for some digits $a$ and $b$. Find the two-digit number $\overline{ab}$ that is missing above.
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Context:
Calculate the dot product of the bivectors \(\gamma_{23}\) and \(\gamma_{01}\) using the metric \(\eta=(1, -1, -1, -1)\) and the formula for the dot product of two k-blades.
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Context:
Determine the number of non-trivial ring homomorphisms from \( \mathbb{Z}_{12} \) to \( \mathbb{Z}_{28} \).
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Context:
Evaluate the definite integral \(\int_{-1}^{+1} \frac{2 u^{332} + u^{998} + 4 u^{1664} \sin u^{691}}{1 + u^{666}} \, \mathrm{d} \, u\).
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Context:
Let $f (n)$ be a function satisfying the following three conditions for all positive integers $n$:
(a) $f (n)$ is a positive integer,
(b) $f (n + 1) > f (n)$,
(c) $f ( f (n)) = 3n$.
Find $f (2001)$.
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Context:
Maria found a store that sells handmade sweaters for $35, handmade scarves for $25, handmade mittens for $15, and handmade hats for $12. She wants to buy two sweaters, one scarf, one pair of mittens, and one hat for each of her 15 family members. The store offers a 10% discount if she spends more than $800 on her purchase. If Maria has a spending limit of $1500, how much will be left in her savings after buying all these items, considering the discount and a 5% sales tax?
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Context:
Determine the smallest integer $N$ such that the remainder $R_N$ of the series \( \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2} \) after summing the first $N$ terms is less than 0.1. That is, find $N$ such that:
\[
R_N = \sum_{n=N+1}^{\infty} \frac{1}{n(\ln n)^2} < 0.1.
\]
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Context:
A $7 \times 7$ table is filled with non-zero integers. Initially, the border of the table is filled with negative numbers. Each subsequent cell is filled in any order, with the next number being equal to the product of the previously placed numbers closest to it either in the row or column. Determine the maximum number of positive numbers that can be in the table.
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Context:
For $i=1,2,$ let $T_i$ be a triangle with side length $a_i,b_i,c_i,$ and area $A_i.$ Suppose that $a_1\le a_2, b_1\le b_2, c_1\le c_2,$ and that $T_2$ is an acute triangle. Does it follow that $A_1\le A_2$?
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Context:
Find the limit (if it exists) of $$\lim_{x\longrightarrow 0^+}\sum_{n=1}^{\infty}\frac{x}{1+x^2n^2}.$$
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Context:
There are 11 rings arranged in a circle, numbered from 1 to 11. Exactly 9 of these rings are fake, and exactly 2 are real. You can choose any 5 consecutive rings and ask how many real rings are in that selection. What is the minimum number of questions needed to identify at least one real ring?
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Context:
A "fifty percent mirror" is a mirror that reflects half the light shined on it back and passes the other half of the light onward. Two "fifty percent mirrors" are placed side by side in parallel, and a light is shined from the left of the two mirrors. How much of the light is reflected back to the left of the two mirrors?
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Context:
Determine the dimension of the smallest field extension of \( \mathbb{Q} \) that contains a transcendental number, such as \( e \).
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Context:
In triangle $ABC$, an interior point $P$ is defined as "good" if there are exactly 27 rays emanating from $P$ that intersect the sides of the triangle, dividing it into 27 smaller triangles of equal area. Determine the number of good points for a given triangle $ABC$. Provide your answer as a single integer.
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Context:
10. Person A and Person B start from points $A$ and $B$ respectively (Person A starts from $A$), walking towards each other and continuously moving back and forth between the two points. Person A's speed is 4 times that of Person B. It is known that the distance between $A$ and $B$ is $S$ kilometers, where $S$ is a positive integer and has 8 factors. The first time they meet at point $C$ (note: meeting here can refer to meeting face-to-face or catching up from behind), the length of $AC$ is an integer; the second time they meet at point $D$, the length of $AD$ is also an integer; after the second meeting, Person B feels that his speed is too slow, so he borrows a motorcycle from a villager near $D$. By the time he borrows the motorcycle and returns to $D$, Person A has already reached point $E$ (Person A has not yet reached $A$), and the length of $AE$ is also an integer; finally, Person B rides the motorcycle to catch up with Person A, with the motorcycle's speed being 14 times that of Person A, and they both reach point $A$ at the same time. Therefore, the distance between $A$ and $B$ is $\qquad$ kilometers.
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Context:
A boat covers a certain distance downstream in 1 hour, while it comes back in 1.30 hours. The speed of the stream is some value, and the speed of the boat in still water is 15 kmph. What is the speed of the stream?
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Context:
A right rectangular prism $P$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\le b\le c.$ A plane parallel to one of the faces of $P$ cuts $P$ into two prisms, one of which is similar to $P,$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered triples $(a, b, c)$ does such a plane exist?

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


Context:
Given a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), let $M$ and $N$ be two points on the hyperbola that are symmetric about the origin, and let $P$ be any point on the hyperbola. The slopes of lines $PM$ and $PN$ are denoted as $k_{PM}$ and $k_{PN}$, respectively. If $k_{PM} < 0$ and $k_{PN} < 0$, find the maximum value of $k_{PM} + k_{PN}$. Express your answer as a single value.
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Context:
Given a Hidden Markov Model (HMM) with a transition matrix for temperature states (WARM and COLD) and an observation matrix relating tree ring sizes (SMALL, MEDIUM, and LARGE) to temperature, calculate the probability of the sequence of tree ring sizes SSSMMLLL.
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Context:
Let $p$, $q$, $r$, $s$, and $t$ be distinct integers such that $(8-p)(8-q)(8-r)(8-s)(8-t) = -120$. Calculate the sum $p+q+r+s+t$.
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Context:
Determine the number of possible ordered triplets \((a, b, c)\) such that \(\text{LCM}(a, b) = 5^2 \times 3^4\), \(\text{LCM}(b, c) = 5^3 \times 3^4\), and \(\text{LCM}(a, c) = 5^3 \times 3^4\).
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Context:
A point \( D \) is marked on the altitude \( BH \) of triangle \( ABC \). Line \( AD \) intersects side \( BC \) at point \( E \), and line \( CD \) intersects side \( AB \) at point \( F \). It is known that \( BH \) divides segment \( FE \) in the ratio \( 1:3 \), starting from point \( F \). Find the ratio \( FH:HE \).
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Context:
Each of the $5$ sides and the $5$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color?
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Context:
Laura’s House is a 20-mile round trip from her school. The supermarket is 10 miles farther away from the school. If Laura drives to school every morning and two afternoons a week drives to the supermarket. How many miles does Laura driver per week?
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Context:
Iveta sequentially wrote natural numbers consisting of the digits 1, 3, 5, and 7. She used no other digits, proceeded in ascending order from the smallest number, and did not omit any number. She wrote the numbers consecutively to form an exceptionally long number:

$$
1357111315173133 \ldots
$$

What digit is at the 1286th position in this number?

(E. Novotná)
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Context:
Given that a > 0, b < 0, and |a| < |b|, determine the correct ordering of the variables a, b, -a, and -b based on their magnitude and sign. Express your answer using the greater than (>) symbol to denote the ordering.
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Context:
若复数 $z_{1}, z_{2}$ 分别满足 $\left|z_{1}-3 \mathrm{i}\right|=2,\left|z_{2}-8\right|=1$ ，则在复平面上所有由复数 $z_{1}-z_{2}$ 对应的点围成的面积是 $x\pi$，求 $x$ 的值。
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Context:
The surface of a 3 x 3 x 3 Rubik's Cube consists of 54 cells. What is the maximum number of cells you can mark such that the marked cells do not share any vertices?
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Context:
What is the largest divisor by which $29 \cdot 14$ leaves the same remainder when divided by $13511, 13903,$ and $14589$?
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Context:
设正实数 $c_{1}, c_{2}, c_{3}$ 使得对任意定义在 $[-\pi, \pi]$ 上的三次多项式 $f(x)$ 均有 $\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(x) {d} x=$ $c_{1} f(-\pi)+c_{2} f(0)+c_{3} f(\pi)$, 则 $\left[\frac{c_{2}+c_{3}}{c_{1}}\right]$ 的最大可能值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest \( n \) such that the sequence of positive integers \( a_1, a_2, \ldots, a_n \) has each term \( \leq 15 \) and \( a_1! + a_2! + \ldots + a_n! \) has last four digits 2001.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知双曲线$\frac{x^2}9-\frac{y^2}{16}=1$ 的左、右焦点分别为$F_1$、$F_2.$ 过$F_1$ 的直线与左支交于 $A$、$B$ 两点，且 $\frac{|AF_1|}{|AF_2|}=\frac12$ .则 $\triangle AF_1F_2$ 与$\triangle ABF_2$ 的内切圆面积之比为 $\qquad.$ 请给出比值两数之和。
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Context:
Given the parabola $C: y^2 = 4x$ with focus $F$, and a line $l$ passing through point $P(-1, 0)$ with slope $k$ ($k > 0$) intersects the parabola $C$ at points $A$ and $B$. The lines $AF$ and $BF$ intersect the parabola $C$ again at points $M$ and $N$ respectively. If $\frac {|AF|}{|FM|} + \frac {|BF|}{|FN|} = 18$, then $k = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\frac{3}{4}$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\frac{1}{12}$. What is the least possible number of faces on the two dice combined?
Please reason step by step, and put your final answer within \boxed{}.


Context:
An airline company operates passenger transportation between four cities A, B, C, and D. The prices for some of the direct flights between these cities are as follows: AB is 2000 yuan; AC is 1600 yuan; AD is 2500 yuan; CD is 900 yuan; BC is 1200 yuan. If the company's pricing for flights is directly proportional to the straight-line distance between the cities (assuming all four cities are on the same plane), calculate the price for the straight-line distance between B and D. Express your answer in yuan as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Marvin had a birthday on Tuesday, May 27 in the leap year $2008$. In what year will his birthday next fall on a Saturday?
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Context:
For each integer $n\geqslant2$, determine the largest real constant $C_n$ such that for all positive real numbers $a_1, \ldots, a_n$ we have
\[\frac{a_1^2+\ldots+a_n^2}{n}\geqslant\left(\frac{a_1+\ldots+a_n}{n}\right)^2+C_n\cdot(a_1-a_n)^2\mbox{.}\]
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Context:
Ben initially had 18 marbles and John had 17 marbles. Ben gave half of his marbles to John, and another quarter of his marbles to his friend Lisa. Additionally, John gave 1/3 of the marbles he received from Ben to Lisa as well. After these transactions, how many more marbles did John have than Ben?
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Context:
Two circles touch in  $M$ , and lie inside a rectangle  $ABCD$ . One of them touches the sides  $AB$  and  $AD$ , and the other one touches  $AD,BC,CD$ . The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in  $M$  divides  $AB$  and  $CD$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given eight small squares, in how many different ways can six of these squares be shaded so that the shaded squares form the net of a cube? Express your answer as a single integer.
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Context:
请写出一个开口向上，与y轴交点纵坐标为-2，且经过点(1,2)的抛物线关系式。
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Context:
Let $X_1, X_2, \ldots, X_n$ be i.i.d. random variables with a uniform distribution on $[0,1]$. Define $Z = \max\{X_1, X_2, \ldots, X_n\}$. Calculate the probability $P(X_2 < a \mid Z = z, X_2 < z)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Galois group of the polynomial \( f(x) = x^5 - x + 1 \) over the finite field \( \mathbb{F}_7 \).
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Context:
How many cyclic subgroups are there in the symmetric group $S_5$?
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Context:
A circle is circumscribed around quadrilateral $EFGH$ as follows: $EF$ subtends a central angle of $100^\circ$, and $GH$ subtends a central angle of $120^\circ$. Find the sum of the angles $\angle EFG + \angle EHG$.
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Context:
Given the line $l: (2\lambda+1)x + (\lambda+2)y + 2\lambda + 2 = 0$ ($\lambda \in \mathbb{R}$) has equal intercepts on the x-axis and y-axis, then $\lambda = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a drawer Sandy has 5 pairs of socks, each pair a different
 color. On Monday Sandy selects two individual socks at
 random from the 10 socks in the drawer. On Tuesday Sandy
 selects 2 of the remaining 8 socks at random and on
 Wednesday two of the reaining 6 socks at random. The
 probability that Wednesday is the first day Sandy selects
 matching socks is \( \frac{m}{n} \) ,where \( m \) and \( n \) are relatively prime
 positive integers. Find \( m + n \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the distribution that is the limit of the sequence \( \langle f_n, \varphi \rangle = n(\varphi(\frac{1}{n})-\varphi(\frac{-1}{n})) \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Among the 8 vertices, the midpoints of the 12 edges, the centers of the 6 faces, and the center of a cube (totaling 27 points), how many groups of three collinear points are there?
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Context:
Determine the values of $p \in \mathbb{R}$ for which the series \( \sum^{\infty}_{n=4}{\frac{1}{n\log (n)\log( \log(n))^p}} \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a rectangle \(ABCD\) with an area of 1 (including its boundary), there are 5 points such that no three points are collinear. Determine the minimum number of triangles, formed by these 5 points as vertices, that have an area of at most \(\frac{1}{4}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the equation of the hyperplane in \(\mathbb{R}^4\) that passes through the points \(P_1 = (1, 2, 1, 4)\), \(P_2 = (0, 1, 2, 1)\), \(P_3 = (-2, 1, 1, 0)\), and \(P_4 = (1, 1, 1, 0)\).
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Context:
Compute the distance from the constant function $g(x) = 0$ to the set $A = \{f \in \mathcal{C}([0,1]) \mid f(0) = 1 \text{ and } \int_0^1 f^2 = 2\}$ in the normed space $(\mathcal{C}([0,1]), \|\cdot\|_\infty)$ with the supremum norm.
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Context:
There are 35 even integers between a certain number and 95. What is the starting number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $X$ that satisfies the equation involving recursive $n$-th roots:
\[ \sqrt[n]{X+\sqrt[n]{X+\sqrt[n]{X+ \dots}}}  =\sqrt[n]{X\sqrt[n]{X\sqrt[n]{X \dots}}} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the set of values for $x$ that makes the fraction $\frac{{x-1}}{{x+1}}$ meaningful. Express your answer using mathematical notation to describe the condition for $x$.
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Context:
Determine the order of the best constant \(\lambda = \lambda(n)\) such that for any \(n \times n\) complex matrix \(A\) with trace zero, there exist \(n \times n\) matrices \(B\) and \(C\) satisfying \(A = BC - CB\) and \(\|B\| \cdot \|C\| \le \lambda \|A\|\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
When the two diagonals of an $8 \times 10$ grid are drawn, calculate the number of the $1 \times 1$ squares that are not intersected by either diagonal.
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Context:
Let \( B = (B_t)_{t \geq 0} \) be a standard Brownian motion and define \( M_t = 4B_t^2 + e^{4B_t -8t} -4t \). Find \( \mathbb{E}(M_\sigma) \) for \( \sigma = \inf\{t \geq 0 : |B_t| = 1\} \).
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Context:
Find the minimum value of \( P = \dfrac{1}{(a-b)^2} + \dfrac{1}{(b-c)^2} + \dfrac{1}{(c-a)^2} \) given that \( a, b, c \in \mathbb{R}^+ \) and \( a^2 + b^2 + c^2 = 3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \[ I = \frac{1}{2\pi i}\oint_{|z|=1}\frac{2(1-\cos(z))e^z}{z^4}dz \] using the identity of Laurent's expansion \[ a_{n} = \frac{1}{2\pi i}\oint_{C}\frac{f(z)}{(z-z_{0})^{n+1}}dz. \]
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Context:
Compute the differential of the i-th coordinate map \(x_i: \mathbb{R}^n \to \mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
6. Given that the base of the tetrahedron $S-ABC$ is an equilateral triangle, the projection of point $A$ on the side face $SBC$ is $H$, which is the orthocenter of $\triangle SBC$, the dihedral angle $H-AB-C$ is $30^{\circ}$, and $SA=2\sqrt{3}$. Then the volume of the tetrahedron $S-ABC$ is
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of distinct subsets that can be formed from a set of 10 subsets using the operations of union, intersection, complement, and symmetric difference, including the possibility of empty subsets.
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Context:
Determine the domain of the function \( f(x) = \int_0^1 \frac{t^x - 1}{\ln(t)} \, dt \), where \( x \) is a real number.
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Context:
The rules for a race require that all runners start at $A$, touch any part of the 1500-meter wall, touch any part of the opposite 1500-meter wall, and stop at $B$. What is the minimum distance a participant must run? Assume that $A$ is 400 meters directly south of the first wall, and that $B$ is 600 meters directly north of the second wall. The two walls are parallel and are 1500 meters apart. Express your answer to the nearest meter.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the system of quadratic equations in $n$ variables $x_1, x_2, \ldots, x_n$:

$$ax_1^2+bx_1+c=x_2$$
$$ax_2^2+bx_2+c=x_3$$
$$\vdots$$
$$ax_n^2+bx_n+c=x_1$$

where $a,b,c \in \mathbb{R}$ and $a \neq 0$. If $(b-1)^2 > 4ac$, determine the number of distinct solutions to this system.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Minimize the expression \( \frac{a_1^TXa_1+a_4^TXa_4}{a_1^TXa_1 a^T_4Xa_4-a_2^TXa_2 a_3^T X a_3} \) subject to the constraints \( \text{trace}(X) = 1 \) and \( X \succeq 0 \), where \( X \) is a matrix and \( a_k \in \mathbb{R}^n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all the triples of integers $ (a, b,c)$ such that:
\[ \begin{array}{ccc}a+b+c &=& 24\\ a^{2}+b^{2}+c^{2}&=& 210\\ abc &=& 440\end{array}\]
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Context:
In the $xyz$ space, consider a right circular cylinder with radius of base 2, altitude 4 such that
\[\left\{
\begin{array}{ll}
x^2+y^2\leq 4 &\quad  \\
0\leq z\leq 4 &\quad
\end{array}
\right.\]
Let $V$ be the solid formed by the points $(x,\ y,\ z)$ in the circular cylinder satisfying
\[\left\{
\begin{array}{ll}
z\leq (x-2)^2 &\quad  \\
z\leq y^2 &\quad
\end{array}
\right.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three fair twenty-sided dice are rolled and arranged in decreasing order. Find the sum of the numerator and denominator of the expected value of the sum of the two largest dice, expressed in simplest form as \( \frac{p}{q} \), where \( p \) and \( q \) are relatively prime positive integers.
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Context:
In a table $4\times 4$ we put $k$ blocks such that
i) Each block covers exactly 2 cells
ii) Each cell is covered by, at least, one block
iii) If we delete a block; there is, at least, one cell that is not covered.
Find the maximum value of $k$.
Note: The blocks can overlap.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sum of the first fifty positive odd integers subtracted from the sum of the first fifty positive even integers, each decreased by 3, calculate the result.
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Context:
What is the minimum number of queries required to determine the number in each cell of a $7\times7$ board with $49$ distinct numbers, given that you can query any $3$ cells at a time to find out their set of numbers?
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Context:
Calculate the determinant of the matrix \(A=\begin{pmatrix} \sin\alpha & \cos\alpha & a\sin\alpha & b\cos\alpha & ab \\ -\cos\alpha & \sin\alpha & -a^2\sin\alpha & b^2\cos\alpha & a^2b^2 \\ 0 & 0 & 1 & a^2 & b^2 \\ 0 & 0 & 0 & a & b \\ 0 & 0 & 0 & -b & a \end{pmatrix}\) where \(\alpha,a,b \in \mathbb{R}\).
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Context:
A train 1500 m long can cross an electric pole in 120 sec. Find the speed of the train.
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Context:
Each segment with endpoints at the vertices of a regular 100-sided polygon is colored red if there is an even number of vertices between the endpoints, and blue otherwise (in particular, all sides of the 100-sided polygon are red). Numbers are placed at the vertices such that the sum of their squares equals 1, and the product of the numbers at the endpoints is allocated to each segment. Then, the sum of the numbers on the red segments is subtracted by the sum of the numbers on the blue segments. What is the maximum possible result?
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Context:
Calculate the value of the polynomial $f(x) = 8x^4 + 5x^3 + 3x^2 + 2x + 1$ at $x = 2$ using the Horner's method. What is the value of $v_2$?
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Context:
For a Brownian Motion $X_t$ starting at $0$, calculate the probability $\mathbb{P}(X_{2T}>0, X_{T}>0)$, given a time $T>0$. Provide your answer as a single probability value.
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Context:
Find the value of \( k \) such that the length of the parabola \( p(x) = kx^2 + d \) is equal to the length of the catenary, both passing through the points \((-2, \cosh(2))\) and \((2, \cosh(2))\). The equation for the equality of lengths is given by:
\[ 2\sinh(2) = \frac{1}{2} \cdot (4k\cdot \sqrt{1+(4k)^2} + \arsinh(4k)) \]
Solve this equation for \( k \).
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Context:
There are $N$  permutations $（a_1，a_2，\dots，a_{30}）$ of $1,2，\dots，30$ such that for $m\in\{2,3,5\}$, $m$  divides $a_{n+m}-a_n$ for all integers $n$ with $1\leq n <n+m\leq 30$. Find the remainder when $N$  is divided by 1000.
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Context:
For an $n \times n$ square matrix $A$, express the determinant of its cofactor matrix in terms of $\det(A)$ and $n$. What is the determinant of the cofactor matrix of $A$?
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Context:
Determine all integer values of $n$ such that $m = \sqrt{n - 5} + \sqrt{n + \sqrt{n - 5}}$ is an integer.
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Context:
In a group of 15 people, there will be five murders. Each murder involves one killer and one victim, except for the third murder, which involves one killer and two victims. After each murder, the killer and victim(s) are removed from the group. In the third murder, one of the victims can also be the killer, but the other victim must be different. The order of victims in the third murder matters. Calculate the total number of possible combinations of killers and victims for all five murders.
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Context:
Given point \( A(4,0) \) and \( B(2,2) \), while \( M \) is a moving point on the ellipse \(\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1\), the maximum value of \( |MA| + |MB| \) is ______.
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Context:
Given complex numbers $z_1, z_2, z_3$ such that $z_1 + z_2 + z_3 = 2$, $z_1^2 + z_2^2 + z_3^2 = 3$, and $z_1z_2z_3 = 4$, evaluate the expression:

$$\frac{1}{z_1z_2 + z_3 - 1} + \frac{1}{z_2z_3 + z_1 - 1} + \frac{1}{z_3z_1 + z_2 - 1}$$
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Context:
a circular road runs round a circular ground . if the difference between the circumferences of the outer circle and inner circle is 66 metres , the width of the road is :
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Context:
Triangles $ABC$ and $ADF$ have areas $4014$ and $14007,$ respectively, with $B=(0,0), C=(447,0), D=(1360,760),$ and $F=(1378,778).$ What is the sum of all possible $x$-coordinates of $A$?
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Context:
Calculate the limit: $$ \lim_{y\to\infty}\int_{0}^{\infty}  \frac{y\cos^2(x/y)}{y+x^4} \, dx $$
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Context:
Let \( W_n \) be the region in the \(xy\)-plane that is bounded by the \(y\)-axis and the parabola \(x=y^2-n^2\). Determine the value of \(n\) such that the volume obtained by rotating \(W_n\) about the \(y\)-axis is 16 times the volume obtained by rotating \(W_n\) about the \(x\)-axis.
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Context:
Find the value of \(\sum_{k=1}^{60} \sum_{n=1}^{k} \frac{n^{2}}{61-2 n}\).
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Context:
Let $x, y, z$ be real numbers such that $xy + yz + zx = 3$, $x + y + z = 5$, and $xyz = 2$. Calculate the value of:

$$\frac{1}{xy + z - 2} + \frac{1}{yz + x - 2} + \frac{1}{zx + y - 2}$$
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Context:
In the ring $B = \mathbb{Z}_2[x]/(x^3 + x^2 + x + 1)$, find an element that, when multiplied by $[x^2 + 1]$, results in zero.
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Context:
Fifteen balls, consisting of 3 balls each of 5 different colors, are to be arranged in a triangular formation. Determine the number of distinct arrangements possible if rotations of the triangle are considered equivalent.
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Context:
Find the directional derivative of the function implicitly defined by \( z + x\ln(z) + xe^{xy} - 1 = 0 \) at the point \( P = (0, 1) \) in the direction of the vector \( \mathbf{v} = \langle 4 \sqrt{3}, 3 \sqrt{3} \rangle \).
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Context:
Find the codimension of the subspace \( A = \overline{\operatorname{span}}\{S^n(1,4,-1,0,0,\ldots):n=0,1,2,\ldots\} \) in \( \ell_2 \), where \( S \) is the right shift operator defined by \( Se_i = e_{i+1} \).
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Context:
Determine the smallest integer \(\alpha\) such that the error of the truncated Taylor series of \(\sqrt{e}\) at \(x=0\) with \(\alpha\) terms is less than 0.001.
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Context:
Given the moment generating function of a random variable X:
\[ M_X(t) = \frac{1}{(1-2500t)^4} \]
calculate the standard deviation of the claim size for this class of accidents.
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Context:
Find the limit of the sum \( \frac{1}{n} \sum_{k=1}^n n^{1/k} \) as \( n \) approaches infinity.
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Context:
Two boys and three girls stand in a row for a photo. If boy A does not stand at either end, and exactly two of the three girls are adjacent, determine the number of different arrangements.
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Context:
A retailer purchases shirts from a wholesaler and then sells the shirts in her store at a retail price that is 80 percent greater than the wholesale price. If the retailer decreases the retail price by a certain percentage, this will have the same effect as increasing the wholesale price by 44.000000000000014 percent. What is the percentage decrease of the retail price?
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Context:
Given the sequence $\sqrt{2}$, $\sqrt{5}$, $2\sqrt{2},\ldots$, find the term number of $2\sqrt{5}$ in this sequence. Express your answer as a whole number.
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Context:
Given the function \( f(x) = x + \frac{2}{1 \cdot 3}x^3 + \frac{2 \cdot 4}{1 \cdot 3 \cdot 5}x^5 + \frac{2 \cdot 4 \cdot 6}{1 \cdot 3 \cdot 5 \cdot 7}x^7 + \cdots \) for all \( x \in (0,1) \), and knowing that \( f\left(\frac{1}{2}\right) = \frac{\pi}{a\sqrt{b}} \) where \( a, b \in \mathbb{R} \), find the value of \( |a+b| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A bridge can tolerate a mass without structural damage that follows a normal distribution with mean 200,000 kg and standard deviation 20,000 kg. Car weights follow a normal distribution with mean 1,300 kg and standard deviation 130 kg. Determine the minimum number of cars needed on the bridge to cause structural damage with a probability greater than 0.1.
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Context:
Mixture A is a certain percentage oil and the rest is material B by weight. If 2 more kilograms of oil are added to the 8 kilograms mixture A, 6 kilograms of mixture A must be added to make a 70% material B in the new mixture. What is the percentage of oil in the original mixture A?
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Context:
Determine the smallest number of lines needed such that any ray starting from a point $P$ in the plane intersects at least $k$ of these lines, where $k$ is a positive integer.
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Context:
Find the locus of a point $P$ such that the sum of the slopes of the normals from $P$ to the hyperbola $xy = c^2$ is a constant $k$ where $k > 0$. What is the equation of this locus?
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Context:
A wall 8 feet high is located 1 foot away from a house. Determine the length \( L \) of the shortest ladder that can be placed over the wall to reach the house. Assume the ladder forms a right triangle with the wall and the ground, where the height is \( y \), the base is \( 1 + x \), and the hypotenuse is \( L \).
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Context:
Given that the average score for six students is 92.5, the highest score is 99, and the lowest score is 76, with all scores being unique integers, calculate the minimum possible score for the student who ranks 3rd from the highest.
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Context:
Calculate the number of different deal combinations possible in the standard version of the Klondike card game, considering the order of the cards in the deck and the red-on-black stacking rule.
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Context:
A man sold 20 articles for a certain amount and gained 20%. He should sell 29.99999625000047 articles to incur a loss of 20%. What was the initial selling price?
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Context:
For a given prime number $p$, consider the sequence defined by:

\[ a_1 = p, \]

\[ a_2 = 2p + 1, \]

\[ a_3 = 2(2p + 1) + 1 = 4p + 3, \]

\[ \ldots \]

\[ a_n = 2^{n-1}p + 2^{n-1} - 1. \]

Determine the longest sequence of consecutive terms that are all prime numbers. What is the length of this sequence?
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Context:
Evaluate the limit: $$\lim_{n \to \infty} \int_{0}^{1} \sin^2\left(\dfrac{1}{ny^2}\right) \, dy$$
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Context:
Among the cross-sections of a cube, the one with the most sides is a ____-sided polygon.
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Context:
Let \( \{X(t), t \geq 0\} \) be a Brownian motion with drift parameter \( \mu = 3 \) and variance parameter \( \sigma^2 = 9 \). If \( X(0) = 10 \), find \( P(X(0.5) > 10) \).
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Context:
Let $a,$ $b,$ $c$ be real numbers such that $9a^2 + 4b^2 + 25c^2 = 1.$ Find the maximum value of
\[8a + 3b + 5c.\]
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Context:
Determine the radius of convergence for the power series \(\sum_{n=1}^\infty \left(\frac{z-i}{z+2i}\right)^n\) using the Cauchy criterion.
Please reason step by step, and put your final answer within \boxed{}.


Context:
1614. 600 corn seeds are planted with a probability of 0.9 for each seed to germinate. Find the boundary of the absolute value of the deviation of the frequency of germinated seeds from the probability $p=0.9$, if this boundary is to be guaranteed with a probability $P=0.995$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For the integer numbers $i, j, k$ satisfying the condition $i^2 + j^2 + k^2 = 2011$, find the largest value of $i + j + k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest number of natives, \( N \), such that after crossing a river using a single boat, each native knows at least \( k \) additional jokes apart from their own. Each crossing involves two natives rowing to the right and one returning alone, and jokes are only shared between the two natives in the boat during each crossing.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A vessel with a capacity of 8 liters is filled with a mixture of oxygen and nitrogen, with oxygen comprising $16 \%$ of the vessel's capacity. Some amount of this mixture is released from the vessel and the same amount of nitrogen is added. Then, the same quantity of this mixture is released again and the same quantity of nitrogen is added again. In the new mixture, oxygen makes up $9 \%$. What quantity of the mixture is released from the vessel each time?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \( \triangle ABC \), \( \angle BAC = 80^\circ \), \( \angle ABC = 60^\circ \). Point \( D \) is an interior point such that \( \angle DAB = 10^\circ \) and \( \angle DBA = 20^\circ \). Find the measure of \( \angle ACD \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of elements in the ring \( \mathbb{F}_2[x]/(x^5 + x^3 + 1) \) that are divisible by \( x^2 + 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest value of \( y \) in the interval \([0, 0.1]\) such that \( P(P > y \mid X = 0) \leq 0.01 \), where \( n = 528174 \), \( X \sim \text{Binomial}(n, P) \), and \( P \sim U[0, 0.1] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
An export company plans to open four direct sales stores in five candidate cities, with no more than two stores in the same city. Calculate the total number of different plans the company can have for establishing its direct sales stores. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( I = \int_0^{2\pi}\sum_{n=1}^{N}a_n \sin (nt)\cos(n\xi) \sum_{n=1}^{N} a_n\sin(n\xi)\cos(n\tau) \, d\xi \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In Yang's number theory class, Michael K, Michael M, and Michael R take a series of tests. Afterwards, Yang makes the following observations about the test scores:

(a) Michael K had an average test score of  $90$ , Michael M had an average test score of  $91$ , and Michael R had an average test score of  $92$ .

(b) Michael K took more tests than Michael M, who in turn took more tests than Michael R.

(c) Michael M got a higher total test score than Michael R, who in turn got a higher total test score than Michael K. (The total test score is the sum of the test scores over all tests)

What is the least number of tests that Michael K, Michael M, and Michael R could have taken combined?

*Proposed by James Lin*
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Context:
The graphs of the quadratic polynomials \( f(x) \) and \( g(x) \) intersect at the point \( (3 ; 8) \). The polynomial \( f(x) + g(x) \) has a single root at 5. Find the leading coefficient of the polynomial \( f(x) + g(x) \).
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Context:
For a positive integer \( N \), we color the positive divisors of \( N \) (including 1 and \( N \) ) with four colors. A coloring is called multichromatic if whenever \( a, b \) and \( \operatorname{gcd}(a, b) \) are pairwise distinct divisors of \( N \), then they have pairwise distinct colors. What is the maximum possible number of multichromatic colorings a positive integer can have if it is not the power of any prime?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the quadratic variation of the stochastic process $Y(t)$ over the interval $[-1, 2]$, where $Y(t)$ is defined as follows:

$$Y(t) = \begin{cases} 0 & -1 \le t \le 0 \\\ Z(t) - tZ(1) & 0 < t < 1 \\\ 1 - t & 1 \le t \le 2 \end{cases}$$

and $Z(t)$ is a standard Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of paths from the origin to the point $(a, b)$ (where $a$ and $b$ are positive integers and $a \geq b$) that consist only of rightward and upward movements and do not cross the line $y = x - \frac{1}{2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the joint density function \( f(x, y) = 2 \) for \( 0 < y < x < 1 \), find the probability \( P(X - Y > z) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A whale goes on a feeding frenzy that lasts for 12 hours. For the first hour, it catches and eats x kilos of plankton. In every hour after the first, it consumes an incrementally increasing amount of additional plankton based on the sequence '2n' (2, 4, 6, ...). By the end of the frenzy, the whale will have consumed a whopping accumulated total of 960 kilos of plankton. How many kilos did the whale consume in the eighth hour?
Please reason step by step, and put your final answer within \boxed{}.


Context:
$A(-1,1)$, $B$, and $C$ are points on the hyperbola $x^2 - y^2 = 1$. If $\triangle ABC$ is a regular triangle, find the area of $\triangle ABC$. The original answer is in the form of k\sqrt{m}, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{x \to 0} \int_0^1 \frac{1}{x} \left(e^{-x^2(1+t)}(1+t)^{-1}-(1+t)^{-1}\right) dt. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ S(n) $ be the sum of the squares of the positive integers less than and coprime to $ n $. For example, $ S(5) = 1^2 + 2^2 + 3^2 + 4^2 $, but $ S(4) = 1^2 + 3^2 $. Let $ p = 2^7 - 1 = 127 $ and $ q = 2^5 - 1 = 31 $ be primes. The quantity $ S(pq) $ can be written in the form $$ \frac{p^2q^2}{6}\left(a - \frac{b}{c} \right) $$ where $ a $, $ b $, and $ c $ are positive integers, with $ b $ and $ c $ coprime and $ b < c $. Find $ a $.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the following equation in positive numbers:
$$ (2a+1)(2a^2+2a+1)(2a^4+4a^3+6a^2+4a+1) = 828567056280801 $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two complete subgraphs $G_a$ and $G_b$ with $a \geq 3$ and $b \geq 3$ vertices each, where they share exactly two vertices, what is the number of Hamiltonian cycles in the graph $G$ formed by these subgraphs?
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Context:
A deck of $2n$ cards numbered from $1$ to $2n$ is shuffled, and $n$ cards are dealt to each player $A$ and $B$. The players, $A$ and $B$, alternately discard a card face up, starting with $A$. The game ends when the sum of the discarded cards is first divisible by $2n + 1$, and the last person to discard wins. What is the probability that $A$ wins if neither player makes a mistake?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n \to \infty}\left(\frac{1^{1/x}+2^{1/x}+\ldots+n^{1/x}}{n}\right)^{nx} \]
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Context:
In $\triangle ABC$, if $\sin A:\sin B:\sin C=7:8:13$, then $C=$______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let

$$
2^{x}=\left(1+\tan 0.01^{\circ}\right)\left(1+\tan 0.02^{\circ}\right)\left(1+\tan 0.03^{\circ}\right) \ldots\left(1+\tan 44.99^{\circ}\right)
$$

Find \( x \). If necessary, round the answer to the nearest 0.01.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that point $P(1,-1)$ is on the terminal side of angle $φ(-π < φ < 0)$, determine the interval where the function $y=3\cos (x+φ)$, $x∈[0,π]$ is strictly decreasing.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a subset of \( \mathbb{R} \) that is homeomorphic to its complement, where the complement is endowed with the subspace topology, and the subset is not open.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\bigcup_{n \in \mathbb{Z}} [2n, 2n + 1)}


Context:
$4\cos {50}^{0}-\tan {40}^{0}=$_______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
After eating a hearty meal they went to see the Buckingham palace. There, Rachel learned that 705 visitors came to the Buckingham palace that day. If there were some visitors the previous day and 457 visitors the day before that. How many visitors were there the previous day if there were 57 more visitors on that day than on the previous 245 days?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine how many roots of the equation \( z^4 + z^3 + 1 = 0 \) are located in the first quadrant of the complex plane using Rouche's Theorem.
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Context:
In the plane of square ABCD, there is a point P such that △PAB, △PBC, △PCD, and △PDA are all isosceles triangles. How many such points P are there? Express your answer as a single integer.
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Context:
For a real-valued random variable $X \geq 0$, we have the inequality from Jensen's inequality:

$$1 - \frac{1}{1+E \left[x\right]} \geq E \left[ 1 - \frac{1}{1+x} \right].$$

Find the tightest possible constant $\epsilon_1$ such that:

$$ \left| 1 - \frac{1}{1+E \left[x\right]} - E \left[ 1 - \frac{1}{1+x} \right] \right| \leq \epsilon_1 \cdot \text{Var}(x).$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the kernel of the ring homomorphism $\psi: k[p, q, r] \rightarrow k[u^2, v^2, uv]$ defined by $\psi(p) = u^2$, $\psi(q) = uv$, and $\psi(r) = v^2$, where $k$ is a field.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Twenty percent off 60 is one-third more than what number?
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Context:
Call an integer interesting if the integer is divisible by the sum of its digits. For example, $27$ is divisible by $2 + 7 = 9$, so $27$ is interesting. How many $2$-digit interesting integers are there?
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Context:
A box contains 5 balls marked with the number 1, 3 balls marked with the number 3, and 2 balls marked with the number 9. We randomly select a ball 100 times with replacement. What is the probability that the product of the numbers on the selected 100 balls is between \(3^{60}\) and \(3^{100}\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive real number $d,$ such that for all nonnegative real numbers $x, y,$ and $z,$
\[
\sqrt{xyz} + d |x^2 - y^2 + z^2| \ge \frac{x + y + z}{3}.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
C2) Let $n$ be a strictly positive integer. Roger has a rectangular garden of dimensions $(2 n+1) \times(2 n+1)$, and he places barriers to divide his garden into several rectangular cells. Once completed, he will have thus formed exactly two horizontal rectangles $k \times 1$ and two vertical rectangles $1 \times k$ for each even integer $k$ between 1 and $2 n+1$, as well as a single $1 \times 1$ square. How many ways can Roger do this?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A coin is flipped multiple times until we get an odd number of heads followed by a tail. Given $n \in \mathbb{N}^{*}$, find the number of sequences of $n$ flips.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the exact value of the series
\[\frac{1}{5 + 1} + \frac{2}{5^2 + 1} + \frac{4}{5^4 + 1} + \frac{8}{5^8 + 1} + \frac{16}{5^{16} + 1} + \dotsb.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
A circle of radius 2 rolls along a flat surface without slipping. A fixed point on the circle, located 3 units away from its center, moves along a curve. Let $A$ be the midpoint of the arc where the point is at its highest point, and let $B$ be a point where the path intersects the surface. Calculate the ratio $\frac{v_A}{v_B}$, where $v_A$ and $v_B$ are the speeds of the point at $A$ and $B$, respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The number 890 has the following property: by changing any of its digits by 1 (either increasing or decreasing), one can obtain a number that is a multiple of 11. Find the smallest three-digit number with the same property.
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Context:
# 8. Problem 8

Given trapezoid $A B C D$. A line parallel to the bases intersects the lateral sides $A B$ and $C D$ at points $M$ and $N$ respectively. A point $E$ is taken on the base $B C$. Segments $A E$ and $E D$ intersect $M N$ at points $S$ and $T$. The areas of triangles $A M S, S E T, T N D$ are 12, 8, and 15, respectively. What is the minimum possible area of the trapezoid under these conditions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{R\rightarrow \infty} \int_{C_R} \frac{6z^6 + 5z^5}{z^6 + z^5 + 10}dz, \] where \( C_R=\{z \in \mathbb{C} : |z|=R  \} \) for \( R>0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the remainder when $$\sum_{k=1}^{30303} k^{k}$$ is divided by 101.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The income of a person is composed of three different sources (A, B, and C) with the following ratios to the total income: 3:2:1. The income from source A is taxed at 10%, source B at 15%, and source C is tax-free. If the person's total income is Rs. 18000, and their expenditures are in the ratio of 5:4 with their income, determine their total savings after paying taxes.
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Context:
Find the value of $x$ such that the following equation holds: $$\frac{2}{\sqrt{\pi}}\int\limits_{0}^{x}e^{-t^2}dt = 0.7$$
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Context:
Let $Y_1 < ... < Y_8$ be the order statistics of $n$ independent observations from a continuous distribution with a 70th percentile of $\pi_{0.7} = 27.3$. Calculate the probability $P(Y_7 < 27.3)$. Provide your answer as a probability value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the number of positive integers that divide at least two of the integers in the set $\{1^1,2^2,3^3,4^4,5^5,6^6,7^7,8^8,9^9,10^{10}\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be a $2n \times 2n$ matrix, with entries chosen independently at random. Every entry is chosen to be 0 or 1, each with probability $1/2$. Find the expected value of $\det(A-A^t)$ (as a function of $n$), where $A^t$ is the transpose of $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABC$ be a triangle with $BC=49$ and circumradius $25$. Suppose that the circle centered on $BC$ that is tangent to $AB$ and $AC$ is also tangent to the circumcircle of $ABC$. Then
\[\dfrac{AB \cdot AC}{-BC+AB+AC} = \frac{m}{n}\]
where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the origin is inside the ellipse \( k^{2} x^{2}+y^{2}-4 k x+2 k y+k^{2}-1=0 \), what is the range of the parameter \( k \)?
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Context:
(5) Let two regular tetrahedra $P-ABC$ and $Q-ABC$ be inscribed in the same sphere. If the dihedral angle between a lateral face and the base of the regular tetrahedron $P-ABC$ is $45^{\circ}$, then the tangent value of the dihedral angle between a lateral face and the base of the regular tetrahedron $Q-ABC$ is $\qquad$.
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Context:
Find the leading order behavior of \( \left(\frac{d^n}{dx^n}\right) \left.\Gamma(x)\right|_{x=1} \) as \( n \rightarrow \infty \).
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Context:
In a company, some pairs of people are friends (if $A$ is friends with $B$, then $B$ is friends with $A$). It turns out that among every set of 100 people in the company, the number of pairs of friends is odd. Find the largest possible number of people in such a company.
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Context:
Evaluate the surface integral \( \iint_S f(x,y,z) \, dS \) where \( S \) is the part of the cone \( z = \sqrt{x^2+y^2} \) above the circle \( x^2+y^2 \leq 2x \) and \( f(x,y,z) = x^2+y^2 + z^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) with left and right foci as \(F_{1} 、 F_{2}\), a line passing through \(F_{2}\) with an inclination angle of \(\frac{\pi}{4}\) intersects the hyperbola at a point \(A\). If the triangle \(	riangle F_{1} F_{2} A\) is an isosceles right triangle, find the eccentricity of the hyperbola. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \frac{1}{2\pi i}\int_\mathcal{C} |1+z+z^2+\cdots+z^{2m-1}|^2~dz \) where \(\mathcal{C}\) is the unit circle.
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Context:
Find the maximum possible value of $8 \cdot 27^{\log_6 x} + 27 \cdot 8^{\log_6 x} - x^3$ as $x$ varies over the positive real numbers.
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Context:
Calculate the upper bound for the integral: $$\int_{0}^{+\infty}e^{-A\sqrt{x^2+B}}\cos(Cx)\frac{x'}{\sqrt{x'^2-B}}dx'$$ where A, B, and C are real numbers, and A and B are positive.
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Context:
A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of 50 meters per second,while
 each of the spiders has a speed of \( r \) meters per second. The spiders choose their starting positions,and choose the fly’s
 starting position, with the requirement that the fly must begin at a vertex. Each bug knows the position of each other bug at all
 times, and the goal of the spiders is for at least one of them to catch the fly. What is the maximum \( c \) so that for any \( r < c \) , the
 fly can always avoid being caught?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a Markov chain $X$ with state space $E = \{1, 2, 3, 4\}$ and transition matrix
$$P = \begin{bmatrix}
0 & 0 & 1 & 0 \\
0 & 0.4 & 0.6 & 0\\
0.8 & 0 & 0.2 & 0\\
0.2 & 0.3 & 0 & 0.5\\
\end{bmatrix}$$
compute the expected value $E[f(X_5)f(X_6)|X_4 = 4]$, where the function $f$ has values $2, 4, 7,$ and $3$ at states $1, 2, 3,$ and $4$ respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
若方格 P 的左右两侧分别有 a,b 个方格，上下两侧分别有 c,d 个方格，其中 a,b,c,d 是正整数，且满足 (a-b)(c-d)=0，则称由这些方格所组成的图形为一个"十字星".现有一张由 2014 个方格所组成的 38\times53 方格表，求该方格表中十字星的个数.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \iint_D |x-y| \, dx \, dy \), where the region \( D \) is defined by \( D= \{(x, y) \in \mathbb{R}^2 : x \ge 0, 0 \le y \le 3-2x \} \). Determine the appropriate limits of integration and handle the modulus in the integrand.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the value of the infinite product \( \prod_{n=0}^\infty a_n \), where the sequence \( \{a_n\} \) is defined recursively by \( a_0 = \frac{1}{2} \) and \( a_n = 1 + (a_{n-1} - 1)^2 \) for \( n \geq 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the degree of the field extension $[F_p(X,Y) : F_p(X^p,Y^p)]$, where $F_p$ is a finite field of characteristic $p$. Provide your answer as a single integer.
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Context:
A list of integers has mode 32 and mean 22. The smallest number in the list is 10. The median  m of the list is a member of the list. If the list member  m were replaced by  m+10, the mean and median of the new list would be 24 and  m+10, respectively. If  m were instead replaced by  m-8, the median of the new list would be  m-4. What is  m?
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Context:
Find the closure of the open interval $(0, 1)$ in the lower limit topology $\tau_l$ on $\mathbb{R}$, which is generated by the subbasis consisting of all half-open intervals $[a, b)$. Recall that the closure of a set is the smallest closed set containing that set.
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Context:
Given a mapping f: {1, 2, 3} → {1, 2, 3} that satisfies f(1) < f(2) < f(3) and f(f(x)) = 3x, find the value of f(1). Express your answer as a single integer.
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Context:
Tom's algebra notebook consists of 50 pages, with pages 1 and 2 on the front and back of the first sheet of paper, pages 3 and 4 on the second sheet, and so on. One day, Tom left the notebook on the table while he went out, and his roommate took away several consecutive pages. The average of the remaining page numbers is 19. Calculate the number of pages that the roommate took away.
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Context:
Determine the minimum number of questions needed to find the positions of all numbers in a 64-cell grid, where each cell contains a unique number from 1 to 64. Each question allows you to find out the list of numbers situated in the cells you choose. What is the minimum number of questions required?
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Context:
Evaluate the expression: $$\frac{29\int_0^1 (1-x^4)^7\,dx}{4\int_0^1 (1-x^4)^6\,dx}$$ without expanding.
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Context:
Evaluate the improper integral: $$\int_{-\infty}^{\infty} \frac{\cos^2 x}{(x^2-\pi ^2)} dx$$
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Context:
Suppose $A B C D$ is a convex quadrilateral with $\angle A B D=105^{\circ}, \angle A D B=15^{\circ}, A C=7$, and $B C=C D=5$. Compute the sum of all possible values of $B D$.
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Context:
In a plane, there are 2018 points such that all pairwise distances between them are distinct. For each point, mark the closest point among the remaining points. What is the minimal number of points that are marked?
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Context:
Determine the order of the pole of the product of two holomorphic functions \( h(z) \) and \( j(z) \), where \( h(z) \) has a zero of order \( m > 0 \) and \( j(z) \) has a zero of order \( n > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider \(A \in \mathcal{M}_{2020}(\mathbb{C})\) such that
\[ A + A^{\times} = I_{2020} \]
\[ A \cdot A^{\times} = I_{2020} \]
where \(A^{\times}\) is the adjugate matrix of \(A\), i.e., the matrix whose elements are \(a_{ij} = (-1)^{i+j} d_{ji}\), where \(d_{ji}\) is the determinant obtained from \(A\), eliminating the row \(j\) and the column \(i\).

Find the maximum number of matrices verifying these conditions such that any two of them are not similar.
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Context:
If the circle $x^2+y^2=4$ is externally tangent to the circle $(x-t)^2+y^2=1$, then the value of the real number $t$ is \_\_\_\_\_\_.
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Context:
It is now 3:15:20 PM, as read on a 12-hour digital clock. In 305 hours, 45 minutes, and 56 seconds, the time will be $X:Y:Z$. What is the value of $X + Y + Z$?
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Context:
Compute the integral \( \int_0^\infty \mathrm{si}(x)\mathrm{d}x \), where \( \mathrm{si}(x) = \int_x^\infty \frac{\sin(t)}{t}\mathrm{d}t \) for all \( x > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of ways to make change for a dollar using generating functions. Specifically, determine the coefficient of the \(x^{100}\) term in the expansion of the generating function \(\frac{1}{(x-1)(x^5-1)\cdots(x^{50}-1)}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $P(x)$ be a polynomial of degree at most 6 that satisfies $P(k) = k!$ for $k = 0, 1, 2, 3, 4, 5, 6$. Compute the value of $P(7)$.
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Context:
Find the closure of the set \( M = \{(a_1, a_2, \ldots) : \text{all but finitely many } a_i = 0\} \) in the space \( l^{\infty} \).
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Context:
How many ways can one color the squares of a $6 \times 6$ grid red and blue such that the number of red squares in each row and column is exactly 2?
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Context:
Find an integer $n$ such that there exist natural numbers $k \geq 2$ and positive rational numbers $a_1, a_2, \ldots, a_k$ satisfying the equation $a_1 + a_2 + \cdots + a_k = a_1 a_2 \cdots a_k = n$. What is the value of $n$?
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Context:
Along the diagonals of the bases \( AC \) and \( B_1D_1 \) of the cube \( ABCD A_1B_1C_1D_1 \) with side length \( a \), two ants, Gosha and Lesha, are crawling. They started simultaneously from points \( A \) and \( B_1 \) respectively, with constant speeds. Lesha's speed was twice the speed of Gosha, and they finished when Lesha reached point \( D_1 \). What was the minimum distance between Gosha and Lesha during their movement?
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Context:
For any finite non empty set $X$ of integers, let $max(X)$ denote the largest element of $X$ and $|X|$ denote the number of elements in $X$. If $N$ is the number of ordered pairs $(A, B)$ of finite non-empty sets of positive integers, such that
$$\max(A) \times |B| = 12;$$
$$|A| \times \max(B) = 11$$
and $N$ can be written as $100a + b$ where $a, b$ are positive integers less than 100, find $a + b$.
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Context:
Determine the Girsanov kernel for the process $X_t = W_t + \cos t$, where $W_t$ is a standard $\mathbb{P}$-Brownian motion, such that $X_t$ becomes a standard $\mathbb{Q}$-Brownian motion.
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Context:
Compute the intersection multiplicity of the elliptic curve given by \(x^3 + xz^2 + z^3 + y^2z + yz^2 = 0\) and the curve \(x = 0\) at the point \((0, 1, 1)\). Note: \(f(x, y, z) = x^3 + xz^2 + z^3 + y^2z + yz^2 \in \mathbb{F}_2[x, y, z]\).
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Context:
Evaluate the limit: $$\lim_{n \to \infty} n \int_0^\pi \{ -x \}^n \, dx$$ where $\{ x \}$ denotes the fractional part of the real number $x$. Determine the value of this limit.
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Context:
Evaluate the integral \( \int_0^{2\pi} \frac{e^{|\sin x|}\cos(x)}{1+e^{\tan x}} \, dx \).
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Context:
Given the function $f(x)= \sqrt{3}\sin 2\omega x-\cos 2\omega x$ where $\omega\in(0,1)$, if the graph of $f(x)$ passes through the point $\left( \dfrac{\pi}{6},0\right)$, then the interval of monotonic increase of $f(x)$ on the interval $[0,\pi]$ is \_\_.
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Context:
$\mathrm{Compute}\sum_{i=0}^{\frac{q-1}{2}}\left\lfloor\frac{ip}{q}\right\rfloor+\sum_{j=0}^{\frac{p-1}{2}}\left\lfloor\frac{jq}{p}\right\rfloor\mathrm{if}p=51\;\mathrm{and}\;q=81.$
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Context:
For a set $S \subseteq \mathbb{N}$, define $f(S) = \{\left\lceil \sqrt{s} \right\rceil \mid s \in S\}$. Find the number of sets $T$ such that $\vert f(T) \vert = 2$ and $f(f(T)) = \{2\}$.
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Context:
Determine the winding number of the function \( f(z) = z^3 + \frac{1}{(z-1)^2} \) around the origin as \( z \) moves counterclockwise along the circle \( |z| = 2 \).
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Context:
Find the limit of the solution to the initial value problem \(x'(t)=Ax(t)+g(t)\) with \(x(0)=x_0\), given that all eigenvalues of the matrix \(A\) have negative real parts (\(\mathrm{Re}(\alpha_j)<0\)), and \(\lim_{t\to\infty} |g(t)|=g_0\).
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Context:
In cube \( ABCDA_1B_1C_1D_1 \), point \( M \) is located on edge \( AB \). A rectangle \( MNLK \) is inscribed in square \( ABCD \) such that vertex \( M \) is on \( AB \) and the other vertices \( N, L, K \) are on different sides of the square. The rectangle \( M_1N_1L_1K_1 \) is the orthogonal projection of \( MNLK \) onto the top face \( A_1B_1C_1D_1 \). Given that the diagonals of quadrilateral \( MK_1L_1N \) are perpendicular, determine the ratio \( AM:MB \).
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Context:
Let  $r_1, r_2, ..., r_{2021}$  be the not necessarily real and not necessarily distinct roots of  $x^{2022} + 2021x = 2022$ . Let  $S_i = r_i^{2021}+2022r_i$  for all  $1 \le i \le 2021$ . Find  $\left|\sum^{2021}_{i=1} S_i \right| = |S_1 +S_2 +...+S_{2021}|$ .
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Context:
On a strip of bus tickets numbered from 000000 to 999999, blue paint marks those tickets where the sum of the digits in even positions equals the sum of the digits in odd positions. Determine the largest difference between the numbers of two consecutive blue tickets.
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Context:
Evaluate the limit: \[ \lim _{N\rightarrow \infty } \left( \,{\frac {{N}^{3}}{8\sum _{k=0}^{N-1}({k}^{2}+\left\lfloor k/2\right\rfloor)}} \right) \]
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Context:
Evaluate the integral \( \displaystyle\int_{|z|=2}\frac{e^{1/z^2}}{1-z}dz \).
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Context:
All the $7$ digit numbers containing each of the digits $1, 2, 3, 4, 5, 6, 7$ exactly once, and not divisible by $5$ are arranged in increasing order. Find the $200^{th}$ number in the list.
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Context:
Find the largest 9 digit integer in which no two digits are the same and the sum of each pair of adjacent digits is prime. That is, the sum of the first two digits is prime, the sum of the second and third digits is prime, the sum of the third and fourth digits is prime, and so on.
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Context:
Factorize the polynomial $x^{27} + 1$ in the field $\mathbb{F}_3[x]$ using Fermat's Little Theorem and the Freshman's Dream.
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Context:
Consider a machine with two parts, X and Y. The time until part X fails follows an Exponential distribution with rate \( \lambda = 1 \), and the time until part Y fails follows an Exponential distribution with rate \( \lambda = 0.05 \). When part X fails, it is replaced in 1 time unit, during which the machine is not operational and part Y does not age. When part Y fails, the machine is considered dead. Calculate the expected time until the machine is dead.
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Context:
Identify the quotient group $\mathbb{C}^*/P$, where $\mathbb{C}^*$ is the multiplicative group of non-zero complex numbers and $P$ is the subgroup of positive real numbers.
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Context:
Evaluate the limit: \[ \lim_{n \to \infty} \frac{5^{\frac{n!}{(2n)!}} - 4^{\frac{1}{n!}}}{3^{\frac{n!}{(2n)!}} - 2^{\frac{1}{n!}}} \]
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Context:
Determine the values of $x$ for which the series $$\sum_{n=1}^\infty \frac{\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)(\sin(n x))}{n}$$ converges.
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Context:
Use the Laurent series of \( e^{1/z} \) on the punctured plane \( \mathbb{C} - \{0\} \) to compute \( \frac{1}{\pi} \int_0^\pi e^{\cos\theta}\cos(\sin\theta-n\theta) \, d\theta \) for \( n = 0, 1, 2, \ldots \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the limit \( \lim_{\varepsilon \searrow 0} \varepsilon \mathbf{E}(|X|^{1-\varepsilon}) \), where \( X \) is a Cauchy random variable with density \( f(x) = \frac{1}{\pi} \frac{1}{1+x^2} \).
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Context:
How many non-isomorphic planar graphs (V, E) exist with 6 vertices (|V| = 6) and 10 edges (|E| = 10), given that each vertex has a degree of 3 or greater?
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Context:
What is the Lebesgue measure of the set $A$ in [0,1], defined as the set of numbers with continued fraction expansions that admit an infinite subsequence where each term is divisible by its predecessor?
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Context:
In rectangle ABCD, a circle is inscribed such that it touches the longer sides AD and BC. The diagonal BD intersects the circle at points E and F, creating the chord EF. If BE = 3, EF = 9, and FD = 16, determine the length of side BC.
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Context:
Consider the polynomial \( \prod_{n = 1}^{1996}\left(1 + nx^{3^n}\right) = \sum_{n=0}^{m}a_{n}\,x^{k_{n}} \), where the exponents \( k_{n} \) are in ascending order and the coefficients \( a_{n} \) are nonzero. Determine the coefficient \( a_{1996} \).
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Context:
[ [Ratio of areas of triangles with a common base or common height] Rearrangement of areas Pentagons

Given a convex pentagon $A B C D E$. The area of each of the triangles $A B C, B C D, C D E, D E A, E A B$ is $S$. Find the area of the given pentagon.

#
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Context:
Let the roots of $ax^2+bx+c=0$ be $r$ and $s$. The equation with roots $ar+b$ and $as+b$ is:
$
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Context:
The inclination angle of the line $x+ \sqrt {3}y+c=0$ is \_\_\_\_\_\_.
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Context:
Determine the values of $b \in \mathbb{R}$ for which the symmetric part of the matrix \( \begin{pmatrix} 4 & -1 & b \\ -1 & 2 & -2 \\ 0 & -2 & 1 \end{pmatrix} \) is positive definite.
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Context:
A point is randomly selected on the segment \([6, 11]\). Let \( k \) be the selected value. Determine the probability that the roots of the quadratic equation \( (k^{2}-2k-24)x^{2}+(3k-8)x+2=0 \) satisfy the condition \( x_{1} \leq 2x_{2} \).
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Context:
Determine the integer value of \( a \) such that the equation \( x^3 - 3x + a = 0 \) has three distinct roots between 0 and 1.
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Context:
A particle performs a random walk on the vertices of a cube. At each step, it remains at its current vertex with probability \( \frac{1}{4} \), or moves to one of its neighboring vertices, each with probability \( \frac{1}{4} \). Let vertices \( A \) and \( D \) be diametrically opposite. If the walk starts at vertex \( A \), determine the mean number of steps until the particle visits vertex \( B \), which is one step away from \( A \), for the first time.
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Context:
$已知函数f(x) = \frac{1}{3}x^{3}-x^{2}.$
$若f(x)在区间(-1, m]上的取值范围是 [-\frac{4}{3},0]，求m的取值范围.$
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Context:
Evaluate the limit $\lim\limits_{n \to \infty} nx_n$ for the sequence defined by $x_2 \in \left(0,\frac{\pi}{2}\right)$ and $x_{n+1} = \left(1-\frac{1}{n}\right)\sin x_n$ for $n \ge 2$. Note that $x_{n+1} \le \sin x_n \le x_n$ and $|x_{n+1}| \le 1$, which implies the convergence of $\{x_n\}$ to $0$.
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Context:
Calculate the integral \( \int_D{f^*\omega } \) where \( f:D \rightarrow \mathbb{R}^3 \) is given by \( f(u,v)=(u,v,u^2+v^2+1) \) with \( D=[0,1]\times [0,1] \), and \( \omega=y\,\mathrm{d}y\wedge \mathrm{d}z+xz\,\mathrm{d}x\wedge \mathrm{d}z \).
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Context:
Find the limiting distribution of \( \frac{Y_n - n}{\sqrt{2n}} \) as \( n \to \infty \), where \( Y_n \sim \chi^2(n) \), using moment generating functions.
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Context:
Define a new function $\$N$ such that $\$N = 0.75N + 2$. Calculate $\$(\$(\$30))$.
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Context:
Evaluate \( {\int }_{0}^{\frac{\pi }{4}}\frac{1}{\cos \theta }\sqrt{\frac{1 + \sin \theta }{\cos \theta }}{d\theta } \) .
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Context:
Given a Brownian motion $X_t$ with $X_0 = 0$ and $X_1 > 0$, find the probability that $X_2 < 0$. Express your answer as a single value.
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Context:
Determine the least integer $k$ for which the following scenario could hold true: In a chess tournament with $24$ players, every pair of players plays at least $2$ and at most $k$ games against each other. At the end of the tournament, it turns out that every player has played a different number of games.
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Context:
点集 $\left\{(x, y) \left\lvert\, \lg \left(x^{3}+\frac{1}{3} y^{3}+\frac{1}{9}\right)=\lg x+\lg y\right.\right\}$ 中元素个数为多少？
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Context:
Let \( f = \sum_{i \ge m} a^i t^t \) (for \( m \in \mathbb{Z} \)) be a formal Laurent power series. Consider its formal differential \( df \). What is the coefficient of degree \(-1\) of the differential form \( \frac{df}{f} \)? Provide an explicit expression for this coefficient.
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Context:
How many ways are there to distribute 6 distinguishable balls into 2 indistinguishable boxes if no box can hold more than 4 balls?
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Context:
Find the number of valid strings of length \( n \) formed from the set \( \{A, B, C, D, E, F, G\} \) that contain at least one of the letters A, B, C, or D using the Principle of Inclusion-Exclusion.
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Context:
Compute the winding number of the loop $\alpha(z)=3z^3+5z^2+2: S^1 \rightarrow \mathbb{C} -$ {$0$} about the origin $0$.
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Context:
Calculate the remainder when \(27^{123}\) is divided by \(123\). Given that the Carmichael function of \(123\) is \(40\) and \(27^{40} \equiv 27 \pmod{123}\), use this information to find the remainder.
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Context:
Find a natural number $a$ such that for all natural numbers $n$, the expression $n^4 + a$ is not a prime number.
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Context:
Let X follow a normal distribution N(-1,6^2), and P(-3 ≤ X ≤ -1) = 0.4. Find the probability P(X ≥ 1). Express your answer as a numerical value.
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Context:
In the triangular pyramid $SABC$, the height $SO$ passes through point $O$ - the center of the circle inscribed in the base $ABC$ of the pyramid. It is known that $\angle SAC = 60^\circ$, $\angle SCA = 45^\circ$, and the ratio of the area of triangle $AOB$ to the area of triangle $ABC$ is $\frac{1}{2 + \sqrt{3}}$. Find the angle $\angle BSC$.
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Context:
One day in October, Nikita looked at the wall clock and saw that four flies had fallen asleep exactly on the numbers 12, 3, 6, and 9.

Find the probability that exactly 20 minutes after Nikita looked at the clock, two of the flies had already been swept away by the minute hand.
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Context:
Let $S(x)$ denote the sum of the digits of a positive integer $x$. Find the maximum possible value of $S(x+2019)-S(x)$.
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Context:
Given a positive harmonic function $u$ on the unit disc $U$ with $u(0) = 1$, determine the maximum possible value of $u(1/2)$. Provide your answer as a bound or specific value.
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Context:
Let \( \triangle ABC \) be a triangle with \( AB = 8 \), \( BC = 15 \), and \( AC = 17 \). Point \( X \) is chosen at random on line segment \( AB \). Point \( Y \) is chosen at random on line segment \( BC \). Point \( Z \) is chosen at random on line segment \( CA \). What is the expected area of triangle \( XYZ \)?
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Context:
In triangle \( \triangle ABC \), the sides opposite to angles \( A \), \( B \), and \( C \) are \( a \), \( b \), and \( c \) respectively. If the angles \( A \), \( B \), and \( C \) form a geometric progression, and \( b^{2} - a^{2} = ac \), then the radian measure of angle \( B \) is equal to ________.
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Context:
There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm. Then, starting from the right end, a line is drawn every 6 cm. The stick is cut at each marked line. How many of the resulting smaller sticks are 3 cm long?
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Context:
Find the last five digits of the sum:
$$1^{100} + 2^{100} + 3^{100} + \ldots + 999999^{100}$$
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Context:
The maximum value of the function $y = \sin 2x + \cos 2x$ where $x \in \mathbb{R}$ is __________.
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Context:
如果一个10位正整数的各数码互不相同，且它是11111的倍数，则称之为“好数”.那么“好数”共有＿个.
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Context:
设 $a 、 b 、 c \in(0,1], \lambda$ 为实数,使得
$$
\frac{\sqrt{3}}{\sqrt{a+b+c}} \geqslant 1+\lambda(1-a)(1-b)(1-c).
$$
求入的最大值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
If ‘ + ’ means ‘ × ’ , ‘ - ‘ means ‘ + ’ , ‘ × ’ means ‘ ÷ ’ and ‘ ÷ ’ means ‘ - ‘ then find the value of ; 6 – some number + 8 × 3 ÷ 25 = 5. What is that number?
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Context:
A dishonest shopkeeper professes to sell three types of pulses (A, B, and C) at the cost price, but he uses different false weights for each. For product A, he uses a false weight of 950 gm for a kg; for product B, he uses a false weight of 900 gm for a kg; and for product C, he uses a false weight of 920 gm for a kg. Additionally, the cost prices of the pulses are as follows: product A costs $20 per kg, product B costs $25 per kg, and product C costs $18 per kg. If the shopkeeper sells 10 kg of each product, calculate his overall percentage gain from selling these products.
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Context:
1. Use the numbers $2,3, \cdots, 2019$ to form 1009 fractions, with each number appearing only once in either the numerator or the denominator. From these 1009 fractions, select the largest one. Find the minimum value of this number.
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Context:
For a positive integer  $n$ , there is a school with  $2n$  people. For a set  $X$  of students in this school, if any two students in  $X$  know each other, we call  $X$  *well-formed*. If the maximum number of students in a well-formed set is no more than  $n$ , find the maximum number of well-formed set.

Here, an empty set and a set with one student is regarded as well-formed as well.
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Context:
Given the parabola $E: x^{2}=4y$ and the circle $F: x^{2}+\left(y-1\right)^{2}=1$, a line $l$ passing through point $F$ intersects the above two curves from left to right at points $A$, $C$, $D$, $B$ respectively. Find the minimum value of $|AC|+2|BD|$.
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Context:
Express the number 1988 as the sum of some positive integers in such a way that the product of these positive integers is maximal.
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Context:
It is known that a regular \( n \)-gon can be constructed with straightedge and compass if \( n \) is a prime that is 1 plus a power of 2. It is also possible to construct a \( 2n \)-gon whenever an \( n \)-gon is constructible, or a \( p_{1}p_{2} \cdots p_{m} \)-gon where the \( p_{i} \)'s are distinct primes of the above form. What is the largest \( n \) less than 4,300,000,000 such that a regular \( n \)-gon is constructible?
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Context:
The sides of triangle ABC measure 6, 8, and 10. A circle with center P and radius 1 rolls inside triangle ABC, always staying tangent to at least one side. When P first returns to its initial position, what is the path length traversed by P?
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