Context:
设函数 f:\bN_+\rightarrow\bN_+，满足对任意正整数 s,t，均有 f(t^2f(s))=sf(t)^2. 求 f(1998) 的最小可能值.
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Context:
Determine the interval of convergence for the power series:

$$
1-\frac{x}{2 \cdot 2}+\frac{x^{2}}{3 \cdot 2^{2}}-\frac{x^{3}}{4 \cdot 2^{3}}+\ldots+(-1)^{n} \frac{x^{n}}{(n+1) 2^{n}}+\ldots
$$
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Context:
Evaluate the limit: $$\lim_{n\to\infty}\left(\int _{0}^{n} \frac{\mathrm dx}{x^{x^n}}\right)^n$$
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Context:
Let $f(n) = 1 \times 3 \times 5 \times \cdots \times (2n-1)$. Compute the remainder when $f(1) + f(2) + f(3) + \cdots + f(2016)$ is divided by $100$.
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Context:
Among the four faces of a pyramid, the maximum number of right-angled triangles is ____.
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Context:
Solve the equation for real and positive parameters $a$, $b$, and $c$:

$$\sqrt{a+bx}+\sqrt{b+cx}+\sqrt{c+ax}=\sqrt{b-ax}+\sqrt{c-bx}+\sqrt{a-cx}$$
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Context:
Let the odd function $f(x)$ defined on $\mathbb{R}$ satisfy $f(2-x) = f(x)$, and it is monotonically decreasing on the interval $[0, 1)$. If the equation $f(x) = -1$ has real roots in the interval $[0, 1)$, then the sum of all real roots of the equation $f(x) = 1$ in the interval $[-1, 7]$ is $\_\_\_\_\_\_$.
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Context:
The LCM of two numbers is 76176 and HCF is 116. If one of the numbers is 8128, then what is the other number?
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Context:
If the line \( x = \frac{\pi}{4} \) intercepts the curve \( C: (x - \arcsin a)(x - \arccos a) + (y - \arcsin a)(y + \arccos a) = 0 \) at a chord of length \( d \), find the minimum value of \( d \) as \( a \) varies.
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Context:
What is the $\iint_G x^2 z $ where $G$ is the closed cylinder $x^2 + y^2 = 4$, and $0 \le z \le 3$?
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Context:
Arrange $\frac{n(n+1)}{2}$ different numbers randomly into a triangular array as shown in the diagram. Let $M_k$ be the maximum number in the $k$-th row (from top to bottom). Find the probability that $M_1 < M_2 < \cdots < M_n$.
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Context:
Given the functions \(f(x)=\frac{x+2}{1-2x}\) and \(g(x)=\frac{2x+1}{2-x}\), find the minimum number of compositions of \(f\) and \(g\), alternating between the two functions, that result in the identity function.
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Context:
Given a hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) with left and right foci \(F_{1} 、 F_{2}\), a line passing through \(F_{2}\) with an inclination angle of \(\frac{\pi}{4}\) intersects the hyperbola at point \(A\). If the triangle \(	riangle F_{1} F_{2} A\) is an isosceles right triangle, find the eccentricity of the hyperbola. Express your answer as a single value.
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Context:
For \(x \in (-1,1)\), determine the smallest positive integer \(n\) such that the following inequality is satisfied:

\[\left|e^{2x}-\sum_{k=0}^n \frac{(2x)^k}{k!}\right| \leq \frac{|e^{2x}|}{10^{10}}\]
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Context:
Evaluate the expression to 8 decimal places: $$ 16\left(\frac{1}{5}-\frac{1}{3} \times \frac{1}{5^{3}}+\frac{1}{5} \times \frac{1}{5^{5}}-\frac{1}{7} \times \frac{1}{5^{7}}+\frac{1}{9} \times \frac{1}{5^{9}}-\frac{1}{11} \times \frac{1}{5^{11}}\right)-4\left(\frac{1}{239}-\frac{1}{3} \times \frac{1}{239^{3}}\right) $$
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Context:
Consider the function $f:[1,3]\to\mathbf{R}$ given by $f(x)=-x^4+8x^3+ax^2+bx+d$, where $a$, $b$, and $d$ are real constants. Determine the values of $d$ such that $f$ has exactly 3 stationary points in the interval $(1,3)$ and satisfies $f(1)=f(3)=0$. Provide your answer as a set of values for $d$.
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Context:
Find the derivative of the function \( f(z) = z \text{Re}(z) + \bar{z} \text{Im}(z) + \bar{z} \) at the point \( z_0 \).
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Context:
For every $a \in \mathbb N$ denote by $M(a)$ the number of elements of the set
\[ \{ b \in \mathbb N | a + b \text{  is a divisor of } ab \}.\]
Find $\max_{a\leq 1983} M(a).$
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Context:
Calculate the sum \( \sum_{k=1}^{240} f(k) \), where the function \( f(n) \) is defined as follows for each positive integer \( n \):
\[
f(n) = \begin{cases}
0, & \text{if } n \text{ is a perfect square,} \\
\left\lfloor \frac{1}{\{\sqrt{n}\}} \right\rfloor, & \text{if } n \text{ is not a perfect square,}
\end{cases}
\]
with \( \lfloor x \rfloor \) denoting the greatest integer less than or equal to \( x \), and \( \{x\} = x - \lfloor x \rfloor \).
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Context:
Let $p$ be an odd prime, and let $(p^p)!=mp^k$ for some positive integers $m$ and $k$. Find in terms of $p$ the number of ordered pairs $(m,k)$ satisfying $m+k\equiv0\pmod p$.
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Context:
If the function $y=-x^{3}+ax$ is strictly decreasing on $\left[1,+\infty \right)$, then the range of values for $a$ is ______.
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Context:
Determine the number of differentiable functions on $(0, \infty)$ that satisfy the equation $f^{-1} = f'$. Provide your answer as a single integer.
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Context:
Find a subgroup $H$ of the multiplicative group $\Bbb{R}\setminus\{0\}$ such that for any $x, y \in \Bbb{R}\setminus\{0\}$, $xy^{-1} \in H$ if and only if $x$ and $y$ have the same sign.
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Context:
Let \( f(z) = \sin\left(z + \mathrm{e}^{3z}\right) \). Find \( \frac{\partial f}{\partial \bar{z}}(z) \).
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Context:
Given that \( \det(X) = 1 \) for the matrix \( X = \begin{bmatrix} a & 1 & d \\ b & 1 & e \\ c & 1 & f \end{bmatrix} \) and \( \det(Y) = 4 \) for the matrix \( Y = \begin{bmatrix} a & 1 & d \\ b & 2 & e \\ c & 3 & f \end{bmatrix} \), find the determinant of the matrix \( Z = \begin{bmatrix} a & 5 & d \\ b & 7 & e \\ c & 9 & f \end{bmatrix} \).
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Context:
Given the vector field \( \vec{F}(x,y) = (y + xg(x), y^2) \) and the condition \( \vec{F}(1,1) = (3,1) \), find the function \( g(x) \) such that the partial derivatives \( \vec{F}_x \) and \( \vec{F}_y \) are orthogonal. Determine \( g(x) \).
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Context:
An urn contains 4 red balls and 3 blue balls. Picking a red ball earns you $1, while picking a blue ball costs you $1. You may stop picking balls at any time. What is the expected value of this game if you play optimally?
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Context:
Compute the fundamental group of the space obtained by taking two copies of the 2-sphere $S^2$, denoted as $X_1$ and $X_2$, and a circle $S^1$, denoted as $X_3$. Identify the north pole $N_i$ of each $X_i$ with a point $n$ on $X_3$ and the south pole $S_i$ with a point $s$ on $X_3$, for $i = 1, 2$. What is the fundamental group of the resulting space?
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Context:
Determine the radius of convergence for the complex series \(\sum_{n=0}^\infty c_nz^n\), where the coefficients \(c_n\) are given by \(c_n=(1/n!)\int_1^\infty t^{-1}e^{-t}(\log t)^ndt\).
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Context:
Brian goes fishing a certain number of times as often as Chris, but catches 2/5 times fewer fish than Chris per trip. If Brian caught 400 fish every time he went fishing, they caught 13600 fish altogether when Chris went fishing 10 times. What is the ratio of the number of times Brian goes fishing to the number of times Chris goes fishing?
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Context:
James buys 4 packs of sodas with 10 sodas each, 3 packs of juice with 8 juice boxes each, and 2 packs of water bottles with 15 bottles each. He also buys 7 additional energy drinks. He already has 12 sodas and 5 juice boxes at home. James drinks 3 sodas, 2 juice boxes, and 1 water bottle from Monday to Wednesday. From Thursday to Sunday, he drinks 2 sodas, 4 juice boxes, 1 water bottle, and 1 energy drink per day. Assuming that he drinks at the same rate throughout the entire week, how many total beverages does he drink by the end of the week?
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Context:
Find the limit of the sequence \(a_n = \frac{\sqrt[n]{\binom{2n+3}{n+2}}}{\sqrt[n]{(n+2)!}}.\)
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Context:
Calculate the spectral radius of the Volterra operator $A$ acting on $L^2[0,1]$, defined by $$A(f)(x)=\int_0^x f(t) dt,$$ using the spectral radius formula for bounded linear operators: $$\rho(A)=\lim_{n\rightarrow \infty} \|A^n\|^{1/n}.$$
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Context:
At an international conference, “red” world countries, “blue” world countries, and "green" world countries are the participants. The ratio of “red” world participants to “blue” world participants is 3:2, and the ratio of “blue” world participants to "green" world participants is 5:4. If one-third of “red” world participants are left-handed, two-thirds of “blue” world participants are left-handed, and one-fourth of "green" world participants are ambidextrous (and not considered left-handed), then what is the fraction of the participants who are left-handed, considering participants from all three worlds?
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Context:
Find the continuous least squares trigonometric polynomial \( S_{2}(x) \) for \( f(x) = 2x - 1 \) on the interval \([-\pi, \pi]\). Express \( S_{2}(x) \) in the form \( \frac{a_{0}}{2} + a_{1}\cos(x) + a_{2}\cos(2x) + b_{1}\sin(x) \).
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Context:
Determine the number of positive integers \( x \) less than 10000 such that the difference \( 2^x - x^2 \) is not divisible by 7.
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Context:
Let \( S \) be the set of all nondegenerate triangles formed from the vertices of a regular octagon with side length 1. Find the ratio of the largest area of any triangle in \( S \) to the smallest area of any triangle in \( S \).
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Context:
Evaluate the integral \( \int_C \frac{e^{1/z^2}}{z-1} \, dz \), where \( C \) is the positively oriented circle \( |z| = \frac{1}{2} \).
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Context:
Determine the dimension of the vector space \( \mathbb{C}[[x,y]]/(2xy-y^2,x^2-2xy) \).
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Context:
Let $S(n)$ be the sum of digits for any positive integer $n$ (in decimal notation). Let $N=\sum_{k=10^{2003}}^{10^{2004}-1} S(k)$. Determine $S(N)$.
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Context:
Given a Poisson distribution with a double mode at $X=1$ and $X=2$, calculate the probability $P(X=0)$. Express your answer as a decimal or a fraction.
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Context:
Given $a > 0$, let proposition $p$: the function $y = a^{x}$ is monotonically increasing on $\mathbb{R}$; proposition $q$: the inequality $ax^{2} - ax + 1 > 0$ holds for $\forall x \in \mathbb{R}$. If $p$ and $q$ are false, and $p$ or $q$ is true, find the range of the real number $a$.
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Context:
In a triangular prism $ABC-A_{1}B_{1}C_{1}$, a line is drawn through the midpoint of any two edges. Among these lines, the number of lines that are parallel to the plane $ABB_{1}A_{1}$ is $\_\_\_\_\_\_\_\_$.
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Context:
Determine the value of \( r \) such that for a simple random walk defined by independent random variables \( Z_1, Z_2, \ldots, Z_n \), where each \( Z_j \) is either 1 or -1 with equal probability, the probability that the sum \( S_n = \sum_{j=1}^n Z_j \) satisfies \(|S_n| \leq r\) is 0.5.
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Context:
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$. If $a_2a_4a_6a_8=120$ and $\frac{1}{a_4a_6a_8} + \frac{1}{a_2a_6a_8} + \frac{1}{a_2a_4a_8} + \frac{1}{a_2a_4a_6} = \frac{7}{60}$, then the value of $S_9$ is ______.
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Context:
Determine the number of homomorphisms from the dihedral group $D_6$ to the dihedral group $D_5$. A homomorphism is a function $\varphi: D_6 \rightarrow D_5$ such that for all $x, y \in D_6$, $\varphi(xy) = \varphi(x)\varphi(y)$. Provide the total number of such homomorphisms.
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Context:
3. There are 195 different cards with numbers $1, 5, 7, 5^{2}, 7^{2}, \ldots, 5^{97}, 7^{97}$ (each card has exactly one number, and each number appears exactly once). In how many ways can 3 cards be chosen so that the product of the numbers on the chosen cards is a square of an integer divisible by 35?
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Context:
For a list of n elements where n >= 3, determine the maximum number of inversions required to order the list using the inversion technique, where inverting the initial segments of the list is allowed.
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Context:
Determine the minimum value of \( n \) for which the function \( y(x) = x\sin x \) is a solution to an \( n^{th} \) order linear non-homogeneous differential equation with constant coefficients.
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Context:
Find the real value(s) of $a (a \ne -1)$ for which the limit
$$ \lim_{ n \to \infty} \frac{ 1^a+2^a\cdots+n^a}{(n+1)^{a-1}[ (na+1)+(na+2) \cdots+(na+n)]} = \frac{1}{60}$$
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Context:
Cagney can frost a cupcake every 15 seconds, while Lacey can frost every 40 seconds. They take a 10-second break after every 10 cupcakes. Calculate the number of cupcakes that they can frost together in 10 minutes.
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Context:
Evaluate the limit as \( x \to 0 \):
\[
\lim_{x \to 0} \frac{(\sin 2x - 2x\cos x)(\tan 6x + \tan\left(\frac{\pi}{3} - 2x\right) - \tan\left(\frac{\pi}{3} + 4x\right))}{x\sin x \tan x\sin 2x}.
\]
Use the facts that \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \) and \( \lim_{x \to 0} \frac{\tan x}{x} = 1 \) to evaluate the limit without using L'Hôpital's rule or Taylor series.
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Context:
Find a function \( f:[0,\infty) \rightarrow [0,\infty) \) that is convex and increasing, such that \( (1/z)f(z) \) is non-convex in \((0,\infty)\).
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Context:
Calculate the limit of the function involving the Gamma function for a given integer value of \( n \):

\[ f(n) = \lim_{c \rightarrow 0} \frac{\Gamma(-n + c) + \Gamma(-n - c)}{2} \]

Find \( f(n) \) for \( n = 0 \).
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Context:
There are 20 piles of stones, each with 2006 stones. An operation involves taking one stone from each of any 19 piles and placing them into the remaining pile. After fewer than 20 operations, one pile has 1990 stones, and another pile has between 2080 and 2100 stones. How many stones are in this pile?
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Context:
A metal bar at a temperature of $20^{\circ} \mathrm{C}$ is placed in water at a temperature of $100^{\circ} \mathrm{C}$. After thermal equilibrium is established, the temperature becomes $80^{\circ} \mathrm{C}$. Then, without removing the first bar, another identical metal bar also at $20^{\circ} \mathrm{C}$ is placed in the water. What will be the temperature of the water after thermal equilibrium is established?
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Context:
Using the numbers $2, 3, \cdots, 2019$ to form 1009 fractions, with each number appearing only once in either the numerator or the denominator, select the largest fraction among these 1009 fractions. Find the minimum value of this largest fraction.
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Context:
Let $X$ and $Y$ be independent random variables, each uniformly distributed on $[0,1]$. Define $Z = \max\{X, Y\}$. Calculate the conditional expectation $\mathbb{E}[X|Z]$. \( \boxed{} \)
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Context:
Evaluate the sum of the shifted sinc function defined as follows: $$g(y) = \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)}.$$
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Context:
What is the area of the portion of the circle defined by \(x^2 - 10x + y^2 = 9\) that lies above the \(x\)-axis and to the left of the line \(y = x-5\)?
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Context:
The average age of a certain number of members of a committee is the same as it was 4 years ago, because an old member has been replaced by a young member. The new member is 40 years younger. How many members are in the committee?
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Context:
Evaluate the triple integral \( \iiint_\Omega{\sqrt{x^2+y^2+z^2}dxdydz} \) where \( \Omega=\{x^2+y^2+z^2\le y\} \).
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Context:
Determine the condition under which the $2\times 2$ stochastic matrix \(A\) given by \(A=\left( \begin{array}{cc} 1-a & a \\ b & 1-b\\ \end{array}\right)\) can be expressed as the square of a transition probability matrix \(P\) for a Discrete-Time Markov Chain, i.e., find the condition such that \(P^{2}=A\).
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Context:
In a 4x4 magic square where all rows, columns, and the two main diagonals must sum to the same given value, what is the minimum number of squares that need to be filled to ensure a unique solution for the entire square?
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Context:
Paco had 39 sweet cookies and some salty cookies. He ate 23 salty cookies and 32 sweet cookies. He ate 9 more sweet cookies than salty cookies. How many salty cookies did Paco have initially?
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Context:
1. Given 50 quadratic trinomials. The graphs of these trinomials have their vertices on the $x$-axis. No three graphs intersect at the same point. What is the minimum number of intersection points they can form?
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Context:
Let $X_i$, $i = 1,\ldots,n$ be i.i.d. Poisson random variables with parameter $\lambda = 2$. Determine the minimum value of $n$ such that $P(1.9 \leq \overline{X} \leq 2.1) \geq 0.90$, where $\overline{X}$ is the sample mean.
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Context:
Evaluate the limit: $$\lim_{x \rightarrow 0^{+}} \frac{1}{9} \left(\frac{1}{\arctan x} - \frac{1}{x}\right).$$
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Context:
Determine the number of zero divisors in the ring \( \mathbb{Z}_5[x]/(x^3-2) \).
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Context:
Find the splitting field of the polynomial \( f(x) = x^3 + x^2 + 1 \) over the field \( \mathbb{Z}_2 \).
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Context:
It is known that the occurrence of a certain symbol in the text is between \(10.5\%\) and \(11\%\) of the length of the text (where the length of the text is understood as the total number of characters in the text). Find the minimum possible length of the text.
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Context:
设无穷等比数列 $\left\{a_n\right\}$ 的公比 $q$ 满足 $0<|q|<$ 1 。若 $\left\{a_n\right\}$ 的各项和等于 $\left\{a_n\right\}$ 各项的平方和，将公比 $q$ 表示为 $\frac{m}{n}$ 的形式（其中 $m, n$ 为整数且 $\operatorname{gcd}(m, n)=1$ ), 设 $a_2$ 的取值范围是两个区间，将这两个区间的端点分别表示为 $\frac{a_1}{b_1}$ （左区间左端点）、 $\frac{a_2}{b_2}$ （左区间右端点）、 $\frac{a_3}{b_3}$ （右区间左端点）、 $\frac{a_4}{b_4}$ （右区间右端点），且所有分数都是既约分数（分子分母只有公因数1）。请提供 $a_1+b_1+a_2+b_2+a_3+b_3+a_4+b_4$ 的值。
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Context:
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. The area of $\triangle ABC$ is $S$, and $(a^{2}+b^{2})\tan C=8S$, also $\sin A\cos B=2\cos A\sin B$. Find the value of $\cos A$.
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Context:
If a sequence $\{a_n\}$ satisfies: $a_1 < a_2 > a_3 < a_4 > \ldots > a_{2n-1} < a_{2n} > a_{2n+1} \ldots$, then the sequence $\{a_n\}$ is called a "sine sequence". Now, arrange the numbers 1, 2, 3, 4, 5 into a "sine sequence". Let the total number of such arrangements be $a$. The coefficient of the $x^2$ term in the expansion of the binomial $(\sqrt{x} - \frac{a}{\sqrt{x}})^6$ is ______.
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Context:
Find the limit: \[ \lim_{x\to\infty} \frac{\displaystyle \sum\limits_{i = 0}^\infty \frac{x^{in}}{(in)!}}{\displaystyle\sum\limits_{j = 0}^\infty \frac{x^{jm}}{(jm)!}} \] for natural numbers \(n\) and \(m\).
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Context:
Sandy grew 8 carrots and some turnips. Mary grew 6 carrots. They grew 14 carrots in all. How many turnips did Sandy grow?
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Context:
Determine the number of groups of non-negative integers \(b_1, b_2, \ldots, b_m\) such that the bitwise XOR of all the elements in the group equals a given integer \(k\), where \(0 \leq b_i \leq 2^{20}-1\).
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Context:
I have two random variables $X$ and $Y$ with joint PDF $$ f_{x,y}(x,y) = e^{-y}1_{0<x<y} ; 0<x<y<\infty $$ and 0 otherwise. Given $Z=X+Y$, I need to find the PDF of $Z$.
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Context:
A sequence of vertices $v_1,v_2,\ldots,v_k$ in a graph, where $v_i=v_j$ only if $i=j$ and $k$ can be any positive integer, is called a $\textit{cycle}$ if $v_1$ is attached by an edge to $v_2$, $v_2$ to $v_3$, and so on to $v_k$ connected to $v_1$.  Rotations and reflections are distinct: $A,B,C$ is distinct from $A,C,B$ and $B,C,A$.  Supposed a simple graph $G$ has $2013$ vertices and $3013$ edges.  What is the minimal number of cycles possible in $G$?
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Context:
There is a solution of table salt in a flask. From the flask, $\frac{1}{5}$ of the solution is poured into a test tube and evaporated until the salt concentration in the test tube doubles. After that, the evaporated solution is poured back into the flask. As a result, the salt concentration in the flask increases by $3\%$. Determine the initial percentage concentration of salt.
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Context:
The Venusian prophet Zabruberson sent his pupils a $10000$-letter word, with each letter being either $A$ or $E$. This is known as the Zabrubic word. The pupils consider that for $1 \leq k \leq 10000$, each word comprised of $k$ consecutive letters of the Zabrubic word is a prophetic word of length $k$. It is known that there are at most $7$ prophetic words of length $3$. Find the maximum number of prophetic words of length $10$.
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Context:
Determine the smallest positive integer \( m \) such that the symmetric group \( S_n \) can be embedded into the general linear group \( GL_m(F) \), where \( F \) is a field.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the Hausdorff dimension of the set \( F \), where \( F \) consists of numbers \( x \in [0,1] \) with base 3 expansions \( 0.a_1a_2\ldots \) such that there exists an integer \( k \) for which \( a_i \neq 1 \) for all \( i \geq k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(ABCD\) be a trapezoid such that \(AB \parallel CD, \angle BAC=25^{\circ}, \angle ABC=125^{\circ}\), and \(AB+AD=CD\). Compute \(\angle ADC\).
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Context:
A $3 \times 6$ grid is filled with the numbers in the list $\{1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9\}$ according to the following rules:

1. Both the first three columns and the last three columns contain the integers 1 through 9.
2. No numbers appear more than once in a given row.

Let $N$ be the number of ways to fill the grid and let $k$ be the largest positive integer such that $2^k$ divides $N$. What is $k$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the functional equation \( f(f(x))[1+f(x)] = -f(x) \), find the value of \( f(3) \).
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Context:
A set of 8 problems was prepared for an examination. Each student was given 3 of them. No two students received more than one common problem. What is the largest possible number of students?
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Context:
Let $b_n$ be the number obtained by writing the integers 1 to $n$ from left to right, where each integer is squared. For example, $b_3 = 149$ (since $1^2 = 1$, $2^2 = 4$, $3^2 = 9$), and $b_5 = 1491625$. For $1 \le k \le 100$, determine how many $b_k$ are divisible by 4.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $p$, $q$, $r$, and $s$ are distinct integers, and the polynomial equation $(x-p)(x-q)(x-r)(x-s)=16$ has an integral root $t$, find the value of $p+q+r+s$ in terms of $t$.
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Context:
Determine the number of edges in the line graph L(G) of a graph G, given that G is regular, planar, and its line graph is planar but not isomorphic to G.
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Context:
Mike is saving up to buy a house. He puts away 15% of his $120,000 a year salary. He needs to save up 25% of the cost of a $550,000 house for a downpayment. How long will it take?
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Context:
Find the minimum value of the expression:
\[ \frac{1}{|x_1-x_2|\cdots|x_1 - x_{n}|} + \frac{1}{|x_2-x_1||x_2-x_3|\cdots|x_2 - x_n|}+\cdots+\frac{1}{|x_n-x_1|\cdots|x_n-x_{n - 1}|} \]
where \(x_1, x_2, \ldots, x_n \in [-1, 1]\).
Please reason step by step, and put your final answer within \boxed{}.


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

1. each chosen number is not divisible by $6$, by $7$, nor by $8$;
2. the positive difference of any two distinct chosen numbers is not divisible by at least one of the
numbers $6$, $7$, and $8$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_4^6 \sin^4 (\pi x) \cdot \cos^2 ( 2 \pi x) \,\mathrm{d}x \) using trigonometric identities.
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Context:
Evaluate the limit: $$ \lim_{t\to 1^-} (1-t) \sum_{n=0}^\infty \frac{t^n}{1+t^n} $$
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Context:
We say that an integer $a$ is a quadratic, cubic, or quintic residue modulo $n$ if there exists an integer $x$ such that $x^2 \equiv a \pmod{n}$, $x^3 \equiv a \pmod{n}$, or $x^5 \equiv a \pmod{n}$, respectively. Further, an integer $a$ is a primitive residue modulo $n$ if it is exactly one of these three types of residues modulo $n$.

How many integers $1 \le a \le 2015$ are primitive residues modulo $2015$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a list of integers of size \( n \), determine the minimum number of moves required to convert it into a permutation of the numbers from 1 to \( n \). In one move, you may increase or decrease any element of the list by one. For example, if \( n = 3 \) and the list is \((-1, -1, 2)\), the minimum number of moves required is 6 to convert it to the permutation \((1, 3, 2)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \( \alpha > 0 \) for which the integral \( \int_{0}^{\infty}\exp\left ( \frac{1}{2}x-x^\alpha \right )\,\mathrm{d}x \) converges.
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Context:
Given \( x \in \left(0, \frac{\pi}{2}\right) \), what is the integer part of the number \( m = 3^{\cos^2 x} + 3^{\sin^5 x} \)?
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Context:
In $\triangle PQR,$ where $PQ=PR=17$ and $QR=15.$ Points $G,H,$ and $I$ are on sides $\overline{PQ},$ $\overline{QR},$ and $\overline{PR},$ respectively, such that $\overline{GH}$ and $\overline{HI}$ are parallel to $\overline{PR}$ and $\overline{PQ},$ respectively. What is the perimeter of parallelogram $PGHI$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{2\pi}\exp(\ic\cos(\theta))\cos(\theta)\,\dd\theta \).
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Context:
In the Cartesian coordinate system, the vertices of the isosceles trapezoid \(ABCD\) are given by \(A(1,1)\), \(B(2,-1)\), \(C(-2,-1)\), and \(D(-1,1)\). A point \(P(0,2)\) is rotated \(180^\circ\) around point \(A\) to obtain point \(P_1\). Subsequently, point \(P_1\) is rotated \(180^\circ\) around point \(B\) to obtain point \(P_2\), and this process continues cyclically around points \(C\) and \(D\). Determine the coordinates of point \(P_{2021}\) after 2021 such rotations.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the annihilator of the tensor $e_1\wedge e_2 + e_3\wedge e_4$ in the vector space $V$ with basis $\{e_1, e_2, e_3, e_4\}$.
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Context:
If $(3x-1)^7 = a_7x^7 + a_6x^6 + \cdots + a_0$, then $a_7 + a_6 + \cdots + a_0$ equals
Please reason step by step, and put your final answer within \boxed{}.


Context:
A magician has a hat that contains $a$ white rabbits and $b$ black rabbits. The magician repeatedly draws pairs of rabbits chosen at random from the hat, without replacement. Call a pair of rabbits *checkered* if it consists of one white rabbit and one black rabbit. Given that the magician eventually draws out all the rabbits without ever drawing out an unpaired rabbit and that the expected value of the number of checkered pairs that the magician draws is $2020$, compute the number of possible pairs $(a,b)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Side $AB$ of triangle $ABC$ has length 8 inches. Line $DEF$ is drawn parallel to $AB$ so that $D$ is on segment $AC$, and $E$ is on segment $BC$. Line $AE$ extended bisects angle $FEC$. If $DE$ has length $5$ inches, then the length of $CE$, in inches, is:

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


Context:
Given that the lengths of the sides $a$, $b$, $c$ opposite to the angles $A$, $B$, $C$ in triangle $ABC$ form an arithmetic sequence, find the maximum value of angle $B$.
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Context:
Side $AB$ of triangle $ABC$ has length 8 inches. Line $DEF$ is drawn parallel to $AB$ so that $D$ is on segment $AC$, and $E$ is on segment $BC$. Line $AE$ extended bisects angle $FEC$. If $DE$ has length $5$ inches, then the length of $CE$, in inches, is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
If p, some digit, and r are distinct positive digits and the product of the two-digit integers pq and pr is 221, then the sum of the digits p, this digit, and r is 11. What is the digit?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the operation $x*y$ be defined as $x*y = (x+1)(y+1)$. The operation $x^{*2}$ is defined as $x^{*2} = x*x$. Calculate the value of the polynomial $3*(x^{*2}) - 2*x + 1$ when $x=2$.
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Context:
Three vehicles, "GAZ", "MAZ", and "KamAZ" simultaneously departed from Zlatoust to Miass. "KamAZ", upon reaching Miass, immediately turned back and met "MAZ" 18 km from Miass and "GAZ" 25 km from Miass. "MAZ", upon reaching Miass, also immediately turned back and met "GAZ" 8 km from Miass. What is the distance from Zlatoust to Miass?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of gain \( K \) at which the root locus of the closed-loop system \( T(s) = \frac{-K(s+1)^2}{(1-K)s^2 + 2(1-K)s + (2-K)} \) intersects with the imaginary axis. Use the provided Routh table:
\[
\begin{bmatrix}
s^2 & (1-K) & (2-K) \\
s^1 & 2(1-K) & 0 \\
s^0 &(2-K) & 0
\end{bmatrix}
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $P(x)=x^3+ax^2+bx+c$ where $a,b,c$ are positive real numbers. If $P(1)\geq 2$ and $P(3)\leq 31$, how many of integers can $P(4)$ take?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( P(x) \) is a polynomial of degree 9 and satisfies \( P(k) = 2^k \) for each integer \( k \) from 1 to 10, determine the value of \( P(12) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the diagram, \(PR, PS, QS, QT,\) and \(RT\) are straight line segments. \(QT\) intersects \(PR\) and \(PS\) at \(U\) and \(V\), respectively. If \(PU = PV\), \(\angle UPV = 24^\circ\), \(\angle PSQ = x^\circ\), and \(\angle TQS = y^\circ\), what is the value of \(x + y\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^1 f(x) \, dx \), where \( f(x) = \sum_{n=0}^{\infty} \frac{[2^n x] - 2[2^{n-1} x]}{3^n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a right triangle \(ABC\) with angle \(A\) equal to \(30^{\circ}\), a circle with radius \(R\) is inscribed. A second circle, lying outside the triangle, touches the side \(BC\) and the extensions of the other two sides. Find the distance between the centers of these circles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence of the series: \[ f(z) = \frac{1}{2z^3} + \frac{1}{12z} - \frac{z}{240}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that $f(x) = \frac{x}{x^2 - 2x + 2}$ and $g(x_1, x_2, \ldots , x_7) = f(x_1) + f(x_2) + \ldots + f(x_7)$. If $x_1, x_2, \ldots, x_7$ are non-negative real numbers with sum $5$, determine for how many tuples $(x_1, x_2, \ldots , x_7)$ does $g(x_1, x_2, \ldots , x_7)$ obtain its maximal value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
how long does a train 100 m long running at the speed of 65 km / hr takes to cross a bridge 145 m length ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a finite abelian group \( G \), the indices of two subgroups \( H_1 \) and \( H_2 \) are both \( m \), and the index of \( H_1 \) in \( H_2 \) is \( n \). Determine the index of \( H_1 \cap H_2 \) in \( G \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the Radon-Nikodym derivative \( \frac{d\mu}{dm} \) where \( m \) is the Lebesgue measure on \([0,1]\), \( f(x)=x^2 \), and \( \mu(E)=m(f(E)) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
I am trying to draw a diagonal from a vertex of an icosikaipentagon. How many diagonal lines can you draw?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many different seven-digit natural numbers exist such that all digits are distinct, the number does not start with the digit 2, and the number 2018 can be formed by removing some of its digits?
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Context:
Evaluate the expression for any positive integer $k$:
\[ T = 1\frac{\left( k \right)! \left( k \right)! }{(2k)!}[ (C_0)^2 -2(C_1)^2 + 3(C_2)^2 +\cdots + (-1)^{2k}(2k+1)(C_{2k})^2 ] \]
where $C_r$ denotes ${}^{n}C_r$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest integer \( n \) such that the system of equations \( \sin x_{1} + \sin x_{2} + \cdots + \sin x_{n} = 0 \) and \( \sin x_{1} + 2 \sin x_{2} + \cdots + n \sin x_{n} = 100 \) has a solution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify the group $G$ generated by the permutations $(1234)$ and $(3456)$ in the symmetric group $S_6$. The order of $G$ is 120. What is the group $G$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the sequence \( \{y_n\} \) defined recursively by:

\[ y_n = y_{n-1} \cdot \left(3-\frac{\cos^{-1}(n) \cdot n^2 - 2}{4 + n^2}\right) \]

with the initial condition \( y_0 = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest number among those $A$ values for which the following statement holds. No matter in what order we write down the first 100 positive integers, there will always be ten consecutive numbers in the sequence whose sum is at least $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the lower bound for the sum: \[\sum_{n=N^\delta}^N\sqrt{\ln(n)}\] where \(\delta\) is a positive constant less than 1.
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Context:
Determine the limit of the sequence defined by the recursive logarithms:
\[ t_1 = \ln(n) \]
\[ t_2 = \ln(n - t_1) = \ln(n - \ln(n)) \]
\[ t_3 = \ln(n - t_2) = \ln(n - \ln(n - \ln(n))) \]
\[ \ldots \]
\[ t_k = \ln(n - t_{k-1}) \]
Find the value to which this sequence converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Six soccer teams are competing in a tournament in Waterloo. Every team is to play three games, each against a different team. How many different schedules are possible?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two matrices, $\mathrm{M}$, a $3 \times 2$ matrix, and $\mathrm{N}$, a $2 \times 3$ matrix, such that their product $\mathrm{MN}$ equals the matrix $\begin{pmatrix}8&2&-2\\2&5&4\\-2&4&5\end{pmatrix}$ and $\mathrm{NM}$ is invertible, find the determinant of $\mathrm{NM}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The distances from point A to point B by the river and by the channel are both equal to 1 km. The speed of the current in the channel is $V$ km/h, while in the river it is $(2V+1)$ km/h. The current in both the river and the channel flows from A to B. If the difference between the travel times of the boat through the channel from B to A and back is added to the travel time of a raft along the river from A to B, the sum is exactly 1 hour. By how many kilometers per hour is the speed of the boat greater than the speed of the current in the channel? The value of $V$ is not given. The answer should be a number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 $a, b \in \mathbf{R}$ .若关于 $z$ 的方程
$$
\left(z^{2}+a z+b\right)\left(z^{2}+a z+2 b\right)=0
$$
有 4 个互不相等的复数根 $z_{1}, z_{2}, z_{3}, z_{4}$, 且它们在复平面上对应的点恰是一个边长为 1 的正方形的四个顶点，求 $\left|z_{1}\right|+\left|z_{2}\right|+\left|z_{3}\right|+\left|z_{4}\right|$ 的值。原始答案为\sqrt{m} + n\sqrt{p}的形式，请给出m + n + p的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p,q$ be two-digit number neither of which are divisible by $10$. Let $r$ be the four-digit number by putting the digits of $p$ followed by the digits of $q$ (in order). As $p,q$ very, a computer prints $r$ on the screen if $\gcd(p,q) = 1$ and $p+q$ divides $r$. Suppose that the largest number that is printed by the computer is $N$. Determine the number formed by the last two digits of $N$ (in the same order).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple integral \( \iiint_D (4-x^2) \, dV \) over the solid region \( D \) bounded by the cylinder \( x^2 + y^2 = 4 \) and the planes \( z = 0 \) and \( z = 4-x^2 \). Use cylindrical coordinates to find the volume of the region.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a diagnostic math test for high school seniors at a certain school, the scores (in points) follow a normal distribution $N(100, 10^{2})$. Let $X$ be the score of a randomly selected student. Let event $A$ be defined as the student's score being $80 < X \leqslant 100$, and event $B$ be defined as the student's score being $70 < X \leqslant 90$. Find the conditional probability of event $B$ given that event $A$ has occurred, denoted as $P(B|A)=\_\_\_\_\_\_$ (express the result as a fraction).

Given data: $P(\mu -\sigma < X \leqslant \mu + \sigma) = 0.68$, $P(\mu -2\sigma < X \leqslant \mu + 2\sigma) = 0.95$, $P(\mu -3\sigma < X \leqslant \mu + 3\sigma) = 0.99$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are five students, A, B, C, D, and E, arranged to participate in the volunteer services for the Shanghai World Expo. Each student is assigned one of four jobs: translator, guide, etiquette, or driver. Each job must be filled by at least one person. Students A and B cannot drive but can do the other three jobs, while students C, D, and E are capable of doing all four jobs. The number of different arrangements for these tasks is _________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \int_C \frac{dw}{e^w-1} \) over a loop \( C \) contained in the annulus \( 0<|z|<2\pi \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A mouse named Jerry decided to gift a cat named Tom a birthday cake in the shape of an $8 \times 8$ square. He placed fish in three pieces marked with the letter 'P', sausage in two pieces marked with the letter 'K', and both fish and sausage in one piece, which was not marked. All other pieces are without filling. Jerry also informed Tom that in any $6 \times 6$ square, there are at least 2 pieces with fish, and in any $3 \times 3$ square, there is at most one piece with sausage.

What is the minimum number of cake pieces Tom needs to eat to guarantee that he gets the piece with both fish and sausage?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the intersection number \( \iota^*\mathcal{O}(2,3) \cdot \iota^*\mathcal{O}(2,3) \) for the smooth subvariety \( \iota:X=V_+(f)\subset\mathbb{P}^2\times\mathbb{P}^1 \) where \( f\in H^0(\mathcal{O}(1,2)) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two people play a math game. Player A selects a group of single-digit integers \( x_{1}, x_{2}, \cdots, x_{n} \) as the secret numbers, which can be positive or negative. Player B can ask questions about the sum \( a_{1} x_{1} + a_{2} x_{2} + \cdots + a_{n} x_{n} \), where \( a_{1}, a_{2}, \cdots, a_{n} \) can be any set of numbers. Determine the minimum number of questions Player B needs to ask in order to guess the secret numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Triangles $ABC$ and $AFG$ have areas $3012$ and $10004$, respectively, with $B=(0,0),$ $C=(335,0),$ $F=(1020, 570),$ and $G=(1030, 580).$ Find the sum of all possible $x$-coordinates of $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected number of times the number 2 is rolled when repeatedly rolling a fair die until the sequence "2, 4, 6" appears.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $O$ is the origin of the coordinate system, the focus of the parabola $y^{2}=4x$ is $F$, and $P$ is a point on the parabola. If $|PF|=3$, find the area of $\triangle OPF$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all four-digit numbers $\overline{abcd}$ such that

$$a(a+b+c+d)(a^{2}+b^{2}+c^{2}+d^{2})(a^{6}+2b^{6}+3c^{6}+4d^{6})=\overline{abcd}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
a can run 2 km distance in 2 min . , while b can run this distance in 2 min 40 sec . by how much distance can a beat b ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the remainder when the expression \( S = \sum_{i=0}^{99} (n+i)^6 + 2^{2^{2558}} + 1 \) is divided by 100, given that \( 2^{2^{2558}} \equiv 16 \pmod{100} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point D on the south edge of the field. Cao arrives at point D at the same time that Ana and Bob arrive at D for the first time. The ratio of the field's length to the field's width to the distance from point D to the southeast corner of the field can be represented as $p : q : r$, where $p$, $q$, and $r$ are positive integers with p and q relatively prime. Find $p + q + r$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A student sits for 6 examination papers, each worth 100 marks. In how many ways can the student score a total of 240 marks across all papers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Exercise 11. A group of 20 dwarves coming out of the mine sit around a round table to count the gold nuggets each has mined. They make the following observations:

$\triangleright$ The difference in the number of nuggets between two neighboring dwarves is always 2 or 3.

$\triangleright$ All the dwarves have mined a different number of nuggets.

What is the greatest possible difference in nuggets between the dwarf who found the most and the one who found the least?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the parallelogram \(ABCD\), side \(AB\) is equal to 6, and the height dropped to the base \(AD\) is 3. The angle bisector of \(\angle BAD\) intersects side \(BC\) at point \(M\), with \(MC = 4\). Point \(N\) is the intersection of the bisector \(AM\) and diagonal \(BD\). Find the area of triangle \(BNM\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mary uses plastic grocery bags that can hold a maximum of twenty pounds each. She needs to distribute the weight of her groceries evenly between two bags. She buys 4 pounds of green beans, 6 pounds of milk, twice the amount of carrots as green beans, 3 pounds of apples, and 1 pound of bread. Additionally, she purchases a 5-pound bag of rice, but the rice must be placed in one of the bags without any of the other items. How many more pounds of groceries can Mary fit in the remaining bag?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$ \lim_{n\rightarrow\infty}\frac{1+n+\frac{n^{2}}{2!}+\cdots +\frac{n^{n}}{n!}}{e^{n}} $$
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Context:
The numbers $1, 2, \ldots, 33$ are written on the board. A student performs the following procedure: choose two numbers from those written on the board such that one of them is a multiple of the other. After choosing, the student deletes the two numbers and writes on the board their greatest common divisor (GCD). The student repeats this procedure until only numbers without multiples remain on the board. Determine how many numbers remain on the board when the student can no longer repeat the procedure.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An ant moves on the following lattice, beginning at the dot labeled $A$. Each minute he moves to one of the dots neighboring the dot he was at, choosing from among its neighbors at random. What is the probability that after 5 minutes he is at the dot labeled $B$? [asy]
draw((-2,0)--(2,0));
draw((0,-2)--(0,2));
draw((1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle);
dot((0,0)); dot((1,0)); dot((2,0)); dot((-1,0)); dot((-2,0)); dot((0,1)); dot((0,2)); dot((0,-1)); dot((0,-2)); dot((1,1)); dot((1,-1)); dot((-1,-1)); dot((-1,1));
label("$A$",(0,0),SW);
label("$B$",(0,1),NE);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a constant \( d \) such that the inequality \( \prod_{p < x} \left(1 - \frac{n}{p} \right) \geq d(\log(x))^{-n} \) holds for all \( x \) and \( n \), using Merten's Theorem and the provided inequality.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\Phi$ be a homogeneous Poisson point process (PPP) of intensity $\lambda$ on $\mathbb{R}$. Consider the ordered points $\{x_0, x_1, \dotsc\} = \Phi \cap \mathbb{R}^+$, such that $0 < x_0 < x_1 < \dotsc$. Find the complementary cumulative distribution function (CCDF) of the ratio $x_0/x_k$, where $x_k$ is Erlang distributed with probability density function $f_{x_k}(x) = \frac{\lambda^{k+1}x^k}{k!}e^{-\lambda x}$. Provide your answer in terms of $\lambda$ and $k$.
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Context:
Let \( g(t) = t \) for all \( t \in [-1,1] \) and \( f(t) = t^{1/n} \), where \( n \) is a large odd integer. Consider the operator \( T: C[-1,1] \rightarrow \mathbb{R} \), defined by \( T_g(f) = \int_{-1}^1 g(t)f(t)dt \). Given the norm \( \|f\| = \max_{t \in [-1,1]} |f(t)| \), find \( \|T\| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a local election, 6 initial candidates received the following number of votes: 15236, 20689, 12359, 30682, 25213, and 18492. Before the final tally, the candidates who received 12359 and 18492 votes had to withdraw due to personal reasons. The votes for these two candidates were redistributed to the remaining candidates based on the following rules: 60% of their votes were given to the candidate who initially had 15236 votes, 25% were given to the candidate with 30682 votes, and the remaining 15% were distributed to the candidate with 20689 votes. What % of the total votes did the winning candidate get in the final tally of the election?
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Context:
A and B play a game with the following rules: In the odd-numbered rounds, A has a winning probability of \(\frac{3}{4}\), and in the even-numbered rounds, B has a winning probability of \(\frac{3}{4}\). There are no ties in any round, and the game ends when one person has won 2 more rounds than the other. What is the expected number of rounds played until the game ends?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given circle $O$: $x^{2}+y^{2}=4$, if a line $l$ that does not pass through the origin $O$ intersects the circle $O$ at points $P$ and $Q$, and the slopes of lines $OP$, $PQ$, and $OQ$ form a geometric sequence, then the slope of line $l$ is _______.
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Context:
Find a polynomial \( p(x) \) of degree \( \le 2 \) that approximates \( f(x) = x^{1/3} \) and minimizes the error \[ E = (f(-1)-p(-1))^2 + \int_{-1}^1 (f(x)-p(x))^2 \, dx + (f(1)-p(1))^2. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the positive integer \( n \) be expressed as the sum of the reciprocals of four different positive integers. How many such \( n \) exist? Express your answer as a single integer.
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Context:
Find the largest possible value of the product $pqr$, where $p$, $q$, and $r$ are prime numbers satisfying the conditions $p+q<111$ and $\frac{p+q}{r}=p-q+r$. What is the largest possible value of $pqr$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are relatively prime positive integers $p$ and $q$ such that $\frac{p}{q} = \sum_{n=3}^{\infty} \frac{1}{n^5 - 5n^3 + 4n}$. Find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an acute triangle \( {ABC} \) ,let \( D \) be a point on \( \left\lbrack  {AC}\right\rbrack \) and \( E \) be a point on \( \left\lbrack  {AB}\right\rbrack \) such that
 \( \angle {ADB} = \angle {AEC} = {90}^{ \circ  } \) . If perimeter of triangle \( {AED} \) is 9, circumradius of \( {AED} \) is
 \( \frac{9}{5} \) and perimeter of triangle \( {ABC} \) is 15 , then \( \left| {BC}\right| \) is
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $n \geq 2$  be an integer. Lucia chooses  $n$  real numbers  $x_1,x_2,\ldots,x_n$  such that  $\left| x_i-x_j \right|\geq 1$  for all  $i\neq j$ . Then, in each cell of an  $n \times n$  grid, she writes one of these numbers, in such a way that no number is repeated in the same row or column. Finally, for each cell, she calculates the absolute value of the difference between the number in the cell and the number in the first cell of its same row. Determine the smallest value that the sum of the  $n^2$  numbers that Lucia calculated can take.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a blackboard with 128 ones, in each step you can erase any two numbers \(a\) and \(b\) on the blackboard and replace them with \(a \cdot b + 1\). After performing this operation 127 times, leaving only one number, find the last digit of the maximum possible value of this number, denoted as \(A\).
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Context:
Find the sum of all values of $n$ such that the function \( g(x) = 100|f(x)| - \sum_{k=1}^n |f(x^k)| \) is differentiable over \( \mathbb{R} \), where \( f(x) = e^{x+1} - 1 \) and \( n \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the area enclosed by the curve $g(x)$, the lines $x=-3$ and $x=5$, and the $x$-axis, where $g(x)$ is the inverse function of $f(x)=x^3+3x+1$. Use techniques of definite integrals to find the area.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the simple closed form of the integral given by $$\int_{0}^{\infty} \frac{\cos\left(\frac{2x}{\pi}\right) - \cos^2\left(\frac{2x}{\pi}\right)}{x^2} \cdot \ln(x) \, \mathrm dx \tag1$$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\sigma_1 : \mathbb{N} \to \mathbb{N}$ be a function that takes a natural number $n$, and returns the sum of the positive integer divisors of $n$. For example, $\sigma_1(6) = 1 + 2 + 3 + 6 = 12$. What is the largest number $n$ such that $\sigma_1(n) = 1854$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an analytical function $f(z)$ such that $f(z + \sin i) = f(z)$ and $f(z + \cos i) = f(z)$. Provide the function that satisfies these conditions.
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Context:
An urn initially contains 5 white and 7 black balls. Each time a ball is drawn, its color is noted and it is placed back in the urn along with two balls of the same color. Compute the probability that the first two balls selected are black given that the next two balls drawn are white.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Create a six-digit number using the digits $1$, $2$, $3$, $4$, $5$, $6$ without repetition, such that the digit $1$ is not at either end, and among the three even digits $2$, $4$, $6$, exactly two are adjacent. Find the number of such six-digit numbers.
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Context:
We color all the sides and diagonals of a regular polygon $P$ with 43 vertices either red or blue. Each vertex is an endpoint of 20 red segments and 22 blue segments. A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same color. Suppose there are 2022 blue monochromatic triangles. How many red monochromatic triangles are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the quadrilateral $ABCD$, the points $E$, $F$, and $K$ are midpoints of the sides $AB$, $BC$, and $AD$, respectively. It is known that $KE \perp AB$, $KF \perp BC$, and the angle $\angle ABC = 118^\circ$. Find $\angle ACD$ (in degrees).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$\int_0^\infty J_0(x) \cdot \text{sinc}(\pi x) \cdot e^{-x} \, \mathrm dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In which acute-angled triangle is the value of the product \(\operatorname{tg} \alpha \cdot \operatorname{tg} \beta \cdot \operatorname{tg} \gamma\) minimized?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of the parameter $p$ for which the improper integral \( \int_{0}^{\pi/2}\tan^{p}(x) \, dx \) is convergent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the last two digits of the number \( n = 2017^{(2018)^{(2019)^{(\cdots)}}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Yesterday, Robert drove from City A to City B. Due to the heavy traffic, he drove at an average speed of 50 miles per hour. Today, he drove back from City B to City A along the same route. Because the traffic was lighter, he drove at an average speed of 60 miles per hour. On his way back, he also spent 20 minutes at a rest stop. If the total time he spent driving (excluding the rest stop) corresponds to an average speed of 55 miles per hour, what is the distance between City A and City B?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Vasya replaced the same digits in two numbers with the same letters, and different digits with different letters. The number ZARAZA is divisible by 4, and ALMAZ is divisible by 28. What are the last two digits of the sum ZARAZA + ALMAZ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For an odd prime $p$, determine the minimum number of distinct primitive roots that must be multiplied together to ensure that the product is not a primitive root.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $C$ be the set of real numbers $c$ such that there are exactly two integers n satisfying $2c < n < 3c$. Find the expected value of a number chosen uniformly at random from $C$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \(\lim_{{n \to \infty}} \left(n - 1 - 2\left(\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n - 1}{2}\right)}\right)^2\right)\) using Stirling's approximation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\triangle ABC$ satisfy $AB = 17$, $AC = \frac{70}{3}$, and $BC = 19$. Let $I$ be the incenter of $\triangle ABC$ and $E$ be the excenter of $\triangle ABC$ opposite $A$. (Note: this means that the circle tangent to ray $AB$ beyond $B$, ray $AC$ beyond $C$, and side $BC$ is centered at $E$.) Suppose the circle with diameter $IE$ intersects $AB$ beyond $B$ at $D$. If $BD = \frac{a}{b}$ where $a$ and $b$ are coprime positive integers, find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a square defined by \( \{(x,y): 0\le x,y \le 1\} \) divided into \( n^2 \) small squares by the lines \( x = i/n \) and \( y = j/n \). For \( 1\le i \le n \), let \( x_i = i/n \) and define \( d_i = \min_{0\le j\le n} \left| \sqrt{1 - x_i^2 } - j/n\right| \). Determine the value of \( \lim_{n\to\infty} \sum_{i=1}^n d_i \) if it exists.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the rectangular coordinate system, a point (-16, 0) lies on circle C, and the maximum possible value of the radius of C is 16. What is the other point on the circle that lies on the x-axis?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let p be the perfect number 496, and q be the perfect number 8128. A positive integer is a perfect number provided that the sum of all its positive factors, including 1 and the number itself, is equal to 2 times the number. What is the sum of the reciprocals of all the positive factors of the perfect numbers p and q?
Please reason step by step, and put your final answer within \boxed{}.


Context:
在极坐标系中，圆 $\rho=3$ 被直线 $\theta=\frac{\pi}{3}$ 分成两部分的面积之比是多少？
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Context:
We will call a natural number interesting if all its digits, except for the first and the last, are less than the arithmetic mean of their two neighboring digits. Find the largest interesting number.
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Context:
For every $n \ge 2$, $a_n = \frac{\sqrt[3]{n^3 + n^2 - n - 1}}{n}$. Find the least value of positive integer $k$ satisfying $a_2a_3\cdots a_k > 3$.
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Context:
Consider the $100 \times 100$ grid of points with integer coordinates $S=\{(x,y) \in \mathbb{Z}^2\mid$ $1 \le x \le 100,$ $1 \le y$ $\le$ $100\}.$ A set $C$ is formed by selecting each $p \in S$ with probability $\tfrac{1}{2}$ uniformly at random. The expansion of $C$ is defined as the set of points $q \in S$ such that $\min_{p \in C} d(q,p) \le 1,$ where $d(q,p)$ denotes the Euclidean distance between $q,p.$ If the expected size of the expansion of $C$ can be written as $\tfrac{A}{B}$ for relatively prime positive integers $A,B,$ find $A+B.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In ten years, I'll be twice my brother's age, three times my sister's age, and four times my youngest brother's age. The sum of our ages will then be 88 years old. My brother is 3 years older than my sister, and my youngest brother is exactly half my sister's age at the moment. If my youngest brother is currently 4 years old, how old am I now?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the expression:
\[ \sum_{i=1}^n \frac{1}{\prod_{j \ne i} |x_j - x_i|} \]
where \(x_1, x_2, \ldots, x_n\) are real numbers in the interval \([-1, 1]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the tetrahedron \(ABCD\), the ratios of the lengths are:
\[ BD : CD : AB : AC : AD : BC = \sqrt{3} : \sqrt{2} : 1 : 1 : 1 : 1 \]

Find the angle between \(AD\) and \(BC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct ways to arrange 3 red, 2 yellow, 5 green, and 2 blue indistinguishable balls around a square table with three balls on each side, considering rotations as identical.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \int_{C} \frac{y\,dx - (x+1)\,dy}{x^2 + y^2 + 2x + 1} \) where \( C \) is the curve \( |x| + |y| = 4 \) oriented counterclockwise for a full revolution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
求使$n ^ { 2 } + 2 0 0 7 n$为完全平方数的自然数n的最大值.
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Context:
Let \(\xi\) be a Gaussian vector in \(\mathbb{R}^{d}\) with a unit covariance matrix. Let the matrix \(B=\left(b_{ij}\right)_{i, j=1}^{d}\) be symmetric and positive definite (\(B>0\)), i.e., such that \(z^{\top} B z > 0\) for all non-zero vectors \(z \in \mathbb{R}^{d}\). For which \(d \geq 1\) does \(\mathrm{E}\left(\xi^{\top} B \xi\right)^{-1}\) exist?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of real $x$ for which the series \(\sum_{n=1}^{+\infty} \left (\frac{x^2n^2-2|x|^3n}{1+2xn^2} \right)^{7n}\) is convergent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the integral \( \int_{-\pi/2}^{\pi/2} f(x) \, dx \) where \( f(x) \) satisfies the functional equation \( f(x) = \cos(x + f(x)) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two positive acute angles \( A \) and \( B \) satisfy the equations \( 3\cos^2 A + 2\cos^2 B = 4 \) and \( \dfrac {3\sin A}{\sin B} = \dfrac {2\cos B}{\cos A} \). What is the value of \( A + 2B \)?
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Context:
Let \( A \) be a \( 7 \times 7 \) matrix with rank \( 3 \), and \( a \) be a \( 7 \times 1 \) column vector. Determine the smallest possible rank of the matrix \( A + (aa^T) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the fraction with the smallest denominator in the open interval \(\left(\frac{47}{245}, \frac{34}{177}\right)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that Carl has 24 fence posts and places one on each of the four corners, with 3 yards between neighboring posts, where the number of posts on the longer side is three times the number of posts on the shorter side, determine the area, in square yards, of Carl's lawn.
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Context:
Find the least positive integer   $n$   such that there are at least  $1000$   unordered pairs of diagonals in a regular polygon with  $n$   vertices that intersect at a right angle in the interior of the polygon.
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Context:
A Beaver-number is a positive 5 digit integer whose digit sum is divisible by 17. Call a pair of Beaver-numbers differing by exactly $1$ a Beaver-pair. The smaller number in a Beaver-pair is called an MIT Beaver, while the larger number is called a CIT Beaver. Find the positive difference between the largest and smallest CIT Beavers (over all Beaver-pairs).
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Context:
Find the maximum number of linearly independent homogeneous quadratic forms in 4 variables over \( \mathbb{C} \) such that their intersection has a connected component that is positive dimensional and does not lie on a hyperplane.
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Context:
There are three boxes of stones. Sisyphus moves stones one by one between the boxes. Whenever he moves a stone, Zeus gives him the number of coins that is equal to the difference between the number of stones in the box the stone was put in, and that in the box the stone was taken from (the moved stone does not count). If this difference is negative, then Sisyphus returns the corresponding amount to Zeus (if Sisyphus cannot pay, generous Zeus allows him to make the move and pay later).
After some time all the stones lie in their initial boxes. What is the greatest possible earning of Sisyphus at that moment?
*I. Izmest’ev*
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{n \to \infty} \int_1^\infty \frac{2n}{e^{2x}+n^2x}dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The villages Arkadino, Borisovo, and Vadimovo are connected by straight roads in pairs. Adjacent to the road between Arkadino and Borisovo is a square field, one side of which completely coincides with this road. Adjacent to the road between Borisovo and Vadimovo is a rectangular field, one side of which completely coincides with this road, and the other side is 4 times longer. Adjacent to the road between Arkadino and Vadimovo is a rectangular forest, one side of which completely coincides with this road, and the other side is 12 km. The area of the forest is 45 square km more than the sum of the areas of the fields. Find the total area of the forest and fields in square km.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f(x) = |1 - 2x| \) for \( x \in [0, 1] \). Define \( f_1(x) = f(x) \), \( f_2(x) = f(f_1(x)) \), \( f_3(x) = f(f_2(x)) \), and generally \( f_{n+1}(x) = f(f_n(x)) \). How many solutions does the equation \( f_n(x) = \frac{1}{2} x \) have in the interval \([0, 1]\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \( n > 2 \) natural numbers such that there are no three equal numbers among them, and the sum of any two of these numbers is a prime number. What is the largest possible value of \( n \)?
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Context:
Given the quadratic inequality about $x$, $ax^{2}+bx+2 > 0$, the solution set is $(- \dfrac{1}{2} , \dfrac{1}{3} )$. Find the value of $a+b$.
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Context:
An advertising board is illuminated by 100W bulbs. The life of each bulb follows an exponential distribution with an average life of 600 hours. Using the central limit theorem, determine the minimum number of bulbs required to ensure the board is lit for at least 20,000 hours with a probability of at least 0.9.
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Context:
Calculate the focal distance of the ellipse given by the equation $\frac{x^2}{5} + \frac{y^2}{4} = 1$, using the relationship $c^2 = a^2 - b^2$ where $a^2 = 5$ and $b^2 = 4$. Express your answer as a numerical value.
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Context:
Evaluate the limit: \[ \lim_{x\rightarrow0}\frac{6x\cdot\sin x-6\cdot\sin\left(x^{2}\right)+x^{4}}{x^{5}\left(e^{x}-1\right)} \]
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Context:
In a cage, there are some chickens and rabbits. When counted from the top, there are 35 heads; when counted from the bottom, there are 94 legs. Chickens have 2 legs and rabbits have 4 legs. Calculate the number of chickens and rabbits in the cage. Express your answer as two integers, the number of chickens and the number of rabbits.
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Context:
 $m$  and  $n$  are two nonnegative integers. In the Philosopher's Chess, The chessboard is an infinite grid of identical regular hexagons and a new piece named the Donkey moves on it as follows:

Starting from one of the hexagons, the Donkey moves  $m$  cells in one of the  $6$  directions, then it turns  $60$  degrees clockwise and after that moves  $n$  cells in this new direction until it reaches it's final cell.

At most how many cells are in the Philosopher's chessboard such that one cannot go from anyone of them to the other with a finite number of movements of the Donkey?

*Proposed by Shayan Dashmiz*
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Context:
Determine the number of ring homomorphisms from \( \mathbb{Z}[x,y] \) to \( \mathbb{F}_2[x]/(x^3+x^2+x+1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the variance of the stochastic integral \( Y_t = \int^2_0 W_t^2 \, dW_t \), where \( W_t \) is a Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知非负实数$a、b、c、x、y、z$满足$a+b+c=x+y+z=1$。
求$\left(a-x^{2}\right)\left(b-y^{2}\right)\left(c-z^{2}\right)$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:
Given the joint density function \( f_{(x,y)} = \begin{cases} x+y, & \text{if } 0<x<1, 0<y<1 \\ 0, & \text{elsewhere}. \end{cases} \), find the density function of the random variable \( Z = XY \).
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Context:
Evaluate the complex integral \( \int_C z^3\exp{\left(\dfrac{-1}{z^2}\right)} dz \) where \( C \) is the contour \( |z|=5 \). Use the Residue Theorem to find the value of the integral.
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Context:
Evaluate the lambda calculus expression: \[ (\;(\lambda f.\lambda x. f(f(f(x))))\;(\lambda g.\lambda y.g(g(y))) \;)(\lambda x.x+1)(0) \]
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Context:
Some of $100$ towns of a kingdom are connected by roads.It is known that for each two towns $A$ and $B$ connected by a road there is a town $C$ which is not connected by a road with at least one of the towns $A$ and $B$. Determine the maximum possible number of roads in the kingdom.
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Context:
Bob is coloring lattice points in the coordinate plane. Find the number of ways Bob can color five points in $\{(x, y) \mid 1 \leq x, y \leq 5\}$ blue such that the distance between any two blue points is not an integer.
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Context:
Let $k$ be an integer greater than 1. Suppose $a_0 > 0$, and define \[a_{n+1} = a_n + \frac{1}{\sqrt[k]{a_n}}\] for $n > 0$. Evaluate \[\lim_{n \to \infty} \frac{a_n^{k+1}}{n^k}.\]
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Context:
Find the limit of the function \( y(x) \) as \( x \to \infty \), where \( y(x) \) is defined by the integral equation: \[ y(x) = 7 + \int_0^x \frac{4(y(t))^2}{1+t^2} \, dt. \]
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Context:
Find the maximum value of $a$ such that the differential equation $y' = -\frac{1}{4} + y - \frac{y^2}{a}$ results in an unstable negative population of crabs.
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Context:
Compute the flux of \( \nabla \times G \) exiting from the surface \( S = \{(x,y,z): x^2+y^2+z^2 = 4, x^2+y^2 \leq 2, z>0 \} \), where \( G(x,y,z)=(xy,zx,x^2+y^2) \).
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Context:
Evaluate the limit: \[ \lim_{n\to \infty}\left(\sum_{k=1}^{2n} \frac{1}{\binom{2n}{k}}\right)^n \]
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Context:
Find the derivative \( \frac{dy}{dx} \) if \( y = \frac{1}{1 + x^{n-m} + x^{p-m}} + \frac{1}{1 + x^{m-n} + x^{p-n}} + \frac{1}{1 + x^{m-p} + x^{n-p}} \).
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Context:
In triangle ABC, angle C is a right angle, and CD is the altitude. Find the radius of the circle inscribed in triangle ABC if the radii of the circles inscribed in triangles ACD and BCD are 6 and 8, respectively.
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Context:
# Task № 8.3

## Condition:

Donkey Eeyore thinks that his clock is 20 minutes slow, but in reality, it is 15 minutes fast. He agreed to meet Piglet on the meadow at 10:00, but then realized he had more things to do and would leave 30 minutes later than planned. What time will Eeyore actually arrive at the meeting? Write the answer in the format HH:MM.
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Context:
Given a map \( f: T^2 \rightarrow T^2 \), where the induced homomorphism on the first homology group \( f^{**}: H_1(T^2) \rightarrow H_1(T^2) \) is given by \( f(x+y) = mx + ny \), calculate the degree of the map \( f \).
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Context:
Evaluate the integral \( \int\int\int_E z \, dV \), where \( E \) is the region above the paraboloid \( z = x^2 + y^2 \) and below the plane \( z = 6y \). Express the integral in cylindrical coordinates and compute the volume, rounding your answer to the nearest hundredth.
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Context:
Given the sets $A={z|z=i+i^{2}+i^{3}+…+i^{n},n∈N^{*}}$ and $B={z|z=z_{1}⋅z_{2},z_{1}∈A,z_{2}∈A}$, determine the number of elements in set $B$.
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Context:
Evaluate the contour integral \( \oint_{\gamma} \frac{d z}{z^{2}+4} \), where \( \gamma = |z-2i|=1 \) is a closed loop counterclockwise, using the Cauchy integral formula.
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Context:
Given a hyperbola \( x^{2} - y^{2} = t \) (where \( t > 0 \)), the right focus is \( F \). Any line passing through \( F \) intersects the right branch of the hyperbola at points \( M \) and \( N \). The perpendicular bisector of \( M N \) intersects the \( x \)-axis at point \( P \). When \( t \) is a positive real number (not equal to zero), find the value of \( \frac{|F P|}{|M N|} \).
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Context:
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\cos(x)-1}{x^2}\,dx \) using contour integration and the residue theorem.
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Context:
# 11.3. (7 points)

The base of the pyramid with vertex $P$ is a quadrilateral $A B C D$, where the sum of angles $A$ and $D$ is five times smaller than the sum of angles $B$ and $C$. Find the angle between the planes of the faces $P A B$ and $P C D$, if both are perpendicular to the base.
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Context:
There are several solid and hollow circles arranged in a certain pattern as follows:
●○●●○●●●○●○●●○●●●…
Among the first 2001 circles, how many hollow circles are there? Provide your answer as a whole number.
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Context:
A line is drawn through a vertex of a triangle and cuts two of its middle lines (i.e. lines connecting the midpoints of two sides) in the same ratio. Determine this ratio.
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Context:
Find the number of positive integers that are equal to or equal to 1000 that have exactly 6 divisors that are perfect squares
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Context:
The diameter divides a circle into two parts. In one of these parts, a smaller circle is inscribed so that it touches the larger circle at point \( M \) and the diameter at point \( K \). The ray \( MK \) intersects the larger circle a second time at point \( N \). Find the length of \( MN \), given that the sum of the distances from point \( M \) to the ends of the diameter is 6.
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Context:
Let $F$ be a ring with characteristic $p$, and let $f(x) = a_0 + a_1x + \cdots + a_nx^n$ be a polynomial in $F[x]$. Define the derivative of $f(x)$ as $f'(x) = a_1 + 2a_2x + \cdots + na_nx^{n-1}$. Consider the homomorphism of abelian groups $D : F[x] \to F[x]$ given by $D(f(x)) = f'(x)$. Determine the kernel of $D$.
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Context:
Given the function $f(x)=\sin (\omega x+\varphi)$ $(0 < \omega < 3,0 < \varphi < \pi)$, if $x=-\frac{\pi}{4}$ is a zero of the function $f(x)$, and $x=\frac{\pi}{3}$ is an axis of symmetry for the graph of the function $f(x)$, then the value of $\omega$ is \_\_\_\_.
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Context:
One hundred students each send messages to 50 different students. What is the least number of pairs of students who send messages to each other? Express your answer as a whole number.
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Context:
Find the greatest positive integer $A$ with the following property: For every permutation of \{1001, 1002, \ldots, 2000\}, the sum of some ten consecutive terms is greater than or equal to $A$.
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Context:
In square \(ABCD\) with side length 9, point \(P\) is on \(AB\) such that \(AP: PB = 7:2\). A quarter circle is drawn inside the square with center \(C\) and radius \(CB\). The tangent from \(P\) meets the circle at \(E\) and \(AD\) at \(Q\). Segments \(CE\) and \(DB\) intersect at \(K\), and \(AK\) and \(PQ\) intersect at \(M\). Determine the length of \(AM\).
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Context:
Suppose we have $n$ independent normal random variables $X_1, X_2, \ldots, X_n$ and another independent normal random variable $Y$. What is the probability that $Y$ is less than all of $X_1, X_2, \ldots, X_n$?
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Context:
Find the greatest $a$ such that $\frac{7\sqrt{(2a)^2+(1)^2}-4a^2-1}{\sqrt{1+4a^2}+3}=2$.
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Context:
大润发超市用6800元购进A、B两种计算器各40、80只，这两种计算器的进价、标价如下表。A型：进价(元/只)：30，标价(元/只)：50；B型：进价(元/只)：70，标价(元/只)：100；元旦活动期间，超市决定将A型计算器按标价的9折（标价的90%）出售，为保证这批计算器全部售出后盈利不低于1400元，则B型计算器最多按照标价的多少出售？
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Context:
A line with slope $2$ passes through the focus $F$ of the parabola $y^2 = 2px$ $(p > 0)$ and intersects the parabola at points $A$ and $B$. The projections of $A$ and $B$ on the $y$-axis are $D$ and $C$ respectively. If the area of trapezoid $\triangle BCD$ is $6\sqrt{5}$, then calculate the value of $p$.
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Context:
A finite sequence of numbers satisfies the following conditions: The sum of any three consecutive terms is negative, and the sum of any four consecutive terms is positive. What is the maximum number of terms in this sequence?
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Context:
Natural numbers $1, 2, 3, \cdots$ are written consecutively to form a number $123456789101112 \cdots$. At what point does the formed number first become divisible by 72? This number is $\qquad$ _. $\qquad$
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Context:
Find the determinant of the matrix representing the linear transformation $x \mapsto \beta x$ in the vector space $\mathbb{Q}(\sqrt{d})$ with basis $\{1,\sqrt{d}\}$, where $\beta$ is an element of $\mathbb{Q}(\sqrt{d})$. Express your answer as a single number.
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Context:
If $f:\mathbb{R}\rightarrow \mathbb{R}$ and $f(x)=x+\cos x$, find$$\int^{\pi}_{0}f^{-1}(x) \,\d x.$$
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Context:
Charles Dickens' 200th anniversary was celebrated on February 7, 2012, which was a Tuesday. A year is a leap year if it is divisible by 400 or divisible by 4 but not by 100. Using this rule, calculate the day of the week on which Charles Dickens was born. Express your answer as the full day of the week.
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Context:
Consider a sphere inscribed in a right cone with the base radius of 10 cm and height of 40 cm. The radius of the inscribed sphere can be expressed as $b\sqrt{d} - b$ cm. Determine the value of $b+d$.
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Context:
During summer break, 610769 kids from Lawrence county go to camp and the other 590796 kids stay home. An additional 22 kids from outside the county attended the camp. About how many kids are in Lawrence county?
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Context:
What condition must be imposed on a measure space $(X, \Sigma, \mu)$ to ensure that every function $f \in L_\infty$ satisfies $\sup_{p\ge 1} ||f||_p < \infty$, where $||f||_p = \left(\int_X |f|^p d\mu \right)^{\frac{1}{p}}$ for all $p \ge 1$?
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Context:
Given that $\angle BAC = 80^\circ$, $|AB| = |AC|$, $K \in [AB]$, $L \in [AB]$, $|AB|^2 = |AK| \cdot |AL|$, and $|BL| = |BC|$, find $\angle KCB$. If the answer is $x^\circ$, what is the value of $x$?
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Context:
Euhan and Minjune are playing tennis. The first one to reach $25$ points wins. Every point ends with Euhan calling the ball in or out. If the ball is called in, Minjune receives a point. If the ball is called out, Euhan receives a point. Euhan always makes the right call when the ball is out. However, he has a $\frac34$ chance of making the right call when the ball is in, meaning that he has a $\frac14$ chance of calling a ball out when it is in. The probability that the ball is in is equal to the probability that the ball is out. If Euhan won, determine the expected number of wrong callsmade by Euhan.
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Context:
Find the greatest integer \( k \) such that \( \left\lfloor \frac{n}{\sqrt{3}} \right\rfloor + 1 > \frac{n^2}{\sqrt{3n^2 - k}} \) holds for all positive integers \( n \geq 2 \).
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Context:
Alex has 2 cakes - the first cake is cut into 8 slices and the second cake is cut into 12 slices. Alex gives away a fourth of the slices from the first cake and a third of the slices from the second cake to his friends. After that, he gives half of the remaining slices from the first cake and a sixth of the remaining slices from the second cake to his family. Alex decides to store a quarter of the remaining slices of each cake in the freezer for later. He then eats 2 slices from the first cake and 1 slice from the second cake. How many slices of the cake are left including the slices in the freezer?
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Context:
3. (CZS) A tetrahedron $A B C D$ is given. The lengths of the edges $A B$ and $C D$ are $a$ and $b$, respectively, the distance between the lines $A B$ and $C D$ is $d$, and the angle between them is equal to $\omega$. The tetrahedron is divided into two parts by the plane $\pi$ parallel to the lines $A B$ and $C D$. Calculate the ratio of the volumes of the parts if the ratio between the distances of the plane $\pi$ from $A B$ and $C D$ is equal to $k$.  Second Day
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Context:
What are the number of ways in which we can place $7$ apples in $5$ containers such that each container contains at least $1$ apple, given neither apples nor containers are identical?
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Context:
If Jack walked 9 miles in 1 hour and 15 minutes with an elevation gain of 600 feet and a descent of 400 feet, what was his adjusted pace in miles per hour, considering that on average, an additional 2 minutes should be added for every 100 feet of elevation gain and 1 minute should be subtracted for every 200 feet of elevation loss?
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Context:
Let $f(x)$ be an odd function that is differentiable on $\mathbb{R}$, and $f'(x)$ is the derivative of $f(x)$. It is known that for $x > 0$, $f(x) < f'(x)$, and $f(1) = e$. The solution set of the inequality $0 < f(\ln(x+ \sqrt{1+x^{2}})) \leqslant e^{\ln(x+ \sqrt{1+x^{2}})}$ is denoted as $M$. Find the number of zeros of $g(x) = \sin 6x$ in the set $M$.
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Context:
How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and/or reflection.
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Context:
For each nonnegative integer $n$, we define $A_n = 2^{3n} + 3^{6n+2} + 5^{6n+2}$. Find the greatest common divisor of the numbers $A_0, A_1, \ldots, A_{1999}$.
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Context:
Consider the model \( Y_i = \beta + U_i \) for \( i = 1, 2, 3 \), where \( E(U_1) = E(U_2) = E(U_3) = 0 \) and \( \text{Var}(U_1) = 1, \text{Var}(U_2) = 2, \text{Var}(U_3) = 4 \). Determine the best linear unbiased estimate of the variance of \( \beta \).
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Context:
A quadrilateral \(A B C D\) is inscribed in a circle with radius 6 and center at point \(O\). Its diagonals \(A C\) and \(B D\) are mutually perpendicular and intersect at point \(K\). Points \(E\) and \(F\) are the midpoints of \(A C\) and \(B D\), respectively. The segment \(O K\) is equal to 5, and the area of the quadrilateral \(O E K F\) is 12. Find the area of the quadrilateral \(A B C D\).
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Context:
已知$a>0,b>0,c>1$，且$a+b=1$，则$\left( \frac{2a+b}{ab}-3 \right)\cdot c+\frac{\sqrt{2}}{c-1}$的最小值为___________.
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Context:
3. In triangle $A B C$, points $D$ and $E$ lie on the interior of segments $A B$ and $A C$, respectively, such that $A D=1, D B=2, B C=4, C E=2$ and $E A=3$. Let $D E$ intersect $B C$ at $F$. Determine the length of $C F$.
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Context:
Several young men and women gathered around a round table. It is known that exactly 7 women have other women sitting to their left, and 12 women have young men sitting to their left. It is also known that for 75% of the young men, there are women sitting to their right. How many people are sitting at the table?
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Context:
Given a connected simple graph $G$, it is known that the number of its edges is $e$. Some chess pieces are placed on each vertex of $G$, and a chess piece can only be placed on one of the vertices of $G$. The only operation you can perform is that if the number of chess pieces on a vertex $v$ of $G$ is not less than the number of its adjacent vertices $d$, then you can choose $d$ chess pieces from $v$ and distribute them to the adjacent vertices, with each vertex receiving one piece. If the number of chess pieces on each vertex of $G$ is less than the number of its adjacent vertices, then you cannot perform any operation.

Find the minimum value of $m$ such that there exists an initial way to place the chess pieces, with a total of $m$ pieces, and you can start from this placement and perform the operation infinitely many times.
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Context:
Find the minimum size of a subgroup \( H \leq S_{10} \) that contains at least one 9-cycle, one 2-cycle, and acts transitively on the set \( \{1,2,...,10\} \).
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Context:
Let $a, b \in \mathbb{R}$, $c \in [0, 2\pi)$. Find the number of sets of $a, b, c$ that satisfy the equation $2\sin(3x- \frac{\pi}{3}) = a\sin(bx+c)$ for any real number $x$.
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Context:
Let $x=-2016$. What is the value of $\left| \ \bigl \lvert { \ \lvert x\rvert -x }\bigr\rvert -|x|{\frac{}{}}^{}_{}\right|-x$?
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Context:
What is the remainder when the sum of factorials from $1!$ to $45!$ is divided by $47$?
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Context:
Evaluate the Taylor series of \(f(t)=\exp(t A)\, B \exp(-t A)\) up to the second-order term, given that \(f(0) = B\), \(f'(0) = [A,B]\), and \([A,[A,B]]=0\).
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Context:
Evaluate the limit: $$\lim \limits_{x \to 0} \int_0^1 \frac{\cos(xt)}{t^2}\,dt$$
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Context:
Evaluate the line integral \( \int_C x \, dy - y \, dx \), where \( C \) is the curve composed of a straight line segment from \((-2, 0)\) to \((0, 0)\), a straight line segment from \((0, 0)\) to \((0, -2)\), and the part of the circle of radius 2, centered at the origin, traversed counterclockwise starting from \((0,-2)\) and ending at \((-2, 0)\).
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Context:
How many infinite arithmetic sequences of natural numbers are there in which the ratio of the sum of the first $2n$ terms to the sum of the first $n$ terms does not depend on the choice of $n$, and one of the terms in the sequence is the number 1971?
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Context:
Two congruent right circular cones each with base radius  $3$  and height  $8$  have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance  $3$  from the base of each cone. A sphere with radius  $r$  lies inside both cones. The maximum possible value for  $r^2$  is  $\frac mn$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .
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Context:
In a spelling contest held in her school, Drew got 20 questions correct, winning her the competition. She got some questions wrong, and her competitor Carla got 14 questions correct, and twice as many questions wrong as the number of questions Drew got wrong. There were 52 questions asked in the competition. How many questions did Drew get wrong?
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Context:
A spinner is divided into 4 sectors with areas corresponding to probabilities of 10%, 20%, 30%, and 40%. Calculate the expected number of spins required for the spinner to land on each sector at least once.
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Context:
Divide the set of numbers \(\{1, 2, \cdots, 10\}\) into two groups such that the product of the numbers in the first group, \(p_{1}\), is divisible by the product of the numbers in the second group, \(p_{2}\). Find the minimum value of \(\frac{p_{1}}{p_{2}}\).
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Context:
Determine the largest three-digit prime number that is a factor of \( \binom{2000}{1000} \).
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Context:
Let the lengths of the sides of $\triangle A_nB_nC_n$ be $a_n$, $b_n$, $c_n$, for $n=1,2,3,\ldots$. If $b_1 > c_1$, $b_1+c_1=2a_1$, $a_{n+1}=a_n$, $b_{n+1}=\frac{c_n+a_n}{2}$, $c_{n+1}=\frac{b_n+a_n}{2}$, then the maximum value of $\angle A_n$ is \_\_\_\_\_\_.
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Context:
Penelope the pig eats 20 pounds of food per day, which is 10 times more than Greta the goose eats per day. Milton the mouse eats 1/100 as much as Greta the goose eats per day, but Elmer the elephant eats 4000 times more than Milton the mouse does per day. Rosie the rhino eats 3 times more than Greta and twice less than Elmer. On the other hand, Carl the cow eats half as much as Penelope and 5 times more than Greta. If the total amount of food consumed by all animals cannot exceed 1000 pounds per day, and Greta eats x pounds of food per day, what percentage of the total food consumption, rounded to the nearest whole percentage, does Elmer the elephant eat per day, compared to all the other animals combined?
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Context:
求所有使不等式成立的λ：

$2( x ^ { 3 } + y ^ { 3 } + z ^ { 3 } ) + 3 ( 1 + 3 \lambda )xyz≥(1+λ)(x+y+z)(yz+zx+xy)$这里x,y,z是非负数.
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Context:
Determine the discriminant of the cubic number field $M=\mathbb{Q}(20^{1/3})$, using an integral basis.
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Context:
Given the cost function \( C(x) = 125 \sqrt{x^2 + 5^2} + 100|k-x| \), where the cost of underwater cable is \( \alpha \) per kilometer and the cost on land is \( \beta \) per kilometer with \( \alpha > \beta \), determine the largest value of \( k > 0 \) such that the most efficient path is the hypotenuse \( \overline{AF} \).
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Context:
Determine the values of $b$ for which the set \( F_b = \left\{ \begin{pmatrix} x & by \\ y & x \end{pmatrix} \mid (x,y)\in \mathbb{R}^2 \right\} \) forms a field under matrix addition and multiplication.
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Context:
How many polynomials of degree exactly 5 with real coefficients send the set \{1,2,3,4,5,6\} to a permutation of itself?
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