Context:
Petya wants to paint several cells of an $8 \times 8$ square such that for any vertex, there is a painted square to which it belongs. What is the minimum number of squares he needs to paint?
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Context:
Seryozha and Lena have several chocolate bars, each weighing no more than 100 grams. No matter how they divide these chocolate bars, the total weight of the chocolate bars for one of them will not exceed 100 grams. What is the maximum possible total weight of all the chocolate bars?
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Context:
Determine the number of ways to select a sequence of 8 sets $A_{1}, A_{2}, \ldots, A_{8}$, such that each is a subset (possibly empty) of \{1,2\}, and $A_{m}$ contains $A_{n}$ if $m$ divides $n$.
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Context:
A package of seeds was passed around a table. The first person took 1 seed, the second person took 2 seeds, the third took 3 seeds, and so forth, with each subsequent person taking one more seed than the previous one. It is known that during the second round a total of 100 more seeds were taken than during the first round. How many people were sitting at the table?
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Context:
A bicycle wheel has a certain diameter. It makes 482.5323296660876 complete revolutions in 1 km. What is the diameter of the wheel?
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Context:
In an arithmetic sequence \(\left\{a_{n}\right\}\), if \(\frac{a_{11}}{a_{10}} < -1\), and its partial sum \(S_{n}\) has a maximum value, then when \(S_{n}\) attains its smallest positive value, \(n = \) \(\qquad\).
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Context:
Determine the number of ways to select a positive number of squares on an $8 \times 8$ chessboard such that no two lie in the same row or the same column and no chosen square lies to the left of and below another chosen square.
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Context:
Jeffrey writes the numbers $1$ and $100000000 = 10^8$ on the blackboard. Every minute, if $x, y$ are on the board, Jeffrey replaces them with \[ \frac{x + y}{2} \text{ and } 2 \left(\frac{1}{x} + \frac{1}{y}\right)^{-1}. \]

After $2017$ minutes, the two numbers are $a$ and $b$. Find $\min(a, b)$ to the nearest integer.
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Context:
In a four-level symbol pyramid, the symbols in the two bottom cells determine the cell above them: if they are the same, the cell above is "+", otherwise, it is "-". How many possible ways are there to fill the bottom row of four cells such that the top of the pyramid results in a "+"? Express your answer as a single integer.
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Context:
What is the cardinality of the class of all rational Cauchy sequences that converge to $0$ in $\mathbb{R}$?
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Context:
Let \( f : \mathbb{C} \rightarrow \mathbb{C} \) be a holomorphic function such that for \( x, y \in \mathbb{R} \), \( \text{Im}f(x + iy) = x^2 - y^2. \) Find the function \( f. \)
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Context:
Let $f(x) = x^4 + ax^3 + bx^2 + cx + d$ be a polynomial with $4$ non-necessarily distinct real roots $x_1, x_2, x_3, x_4$. If $b \geq d + 5$, what is the minimum possible value of the expression $(x_1^2 + 1)(x_2^2 + 1)(x_3^2 + 1)(x_4^2 + 1)$?
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Context:
Ada has a set of identical cubes. She makes solids by gluing together 4 of these cubes. When cube faces are glued together, they must coincide. Each of the 4 cubes must have a face that coincides with a face of at least one of the other 3 cubes. Find the number of unique solids that Ada can make using 4 cubes, considering all possible configurations, rotations, and reflections.
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Context:
Given \( \theta = \frac{\pi}{6} \), evaluate the expression:
\[
\frac{\sin(9\theta)}{\left(1 + \sum_{k=1}^{8}\frac{\cos(k\theta)}{\cos^k(\theta)}\right) \sin\left(\frac{\pi}{6}\right) \left(\cos\left(\frac{\pi}{6}\right)\right)^8}
\]
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Context:
Calculate the value of \( \sup_{f \in \mathscr{H}} \inf_{z \in D} [f_x^2(z) + f_y^2(z)] \), where \( f: \mathbb{R}^2 \to \mathbb{R} \) is continuous, has partial derivatives in \( D = \{(x,y) : x^2 + y^2 \leq 1\} \), and satisfies \( \sup_D |f| \leq 1 \).
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Context:
Let \(ABC\) be an equilateral triangle of side length 15. Let \(A_b\) and \(B_a\) be points on side \(AB\), \(A_c\) and \(C_a\) be points on side \(AC\), and \(B_c\) and \(C_b\) be points on side \(BC\) such that \(\triangle AA_bA_c\), \(\triangle BB_cB_a\), and \(\triangle CC_aC_b\) are equilateral triangles with side lengths 3, 4, and 5, respectively. Compute the radius of the circle tangent to segments \(\overline{A_bA_c}\), \(\overline{B_aB_c}\), and \(\overline{C_aC_b}\).
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Context:
Determine the spectrum of the operator $M_b$ on $L_2 (\mathbb{R}^n)$, where $M_b$ is defined by multiplication by the function $b(x) = e^{|x|^2}$ and $|x|^2 = x_1^2 + \ldots + x_n^2$.
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Context:
Given the recurrence relation \(x_{n+1} = 4x_n - x_{n-1}\) for \(n \geq 1\), with initial conditions \(x_0 = 1\) and \(x_1 = 2\), find an odd prime factor of \(x_{2015}\).
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Context:
Identify the unique degree 2 subfield of the field extension $k(x)$ over the field $k$.
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Context:
Find the equation for the trajectory of the dynamical system defined by \(\dot x = \frac{(x+iy)^3 + (x-iy)^3}{2}\) and \(\dot y = \frac{(x+iy)^3 - (x-iy)^3}{2i}\) in polar coordinates, given that the trajectory passes through the point \((x,y) = (1,0)\).
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Context:
At a party, there are 20 girls and 20 boys. How many dance pairs can be formed if boys may dance with boys and girls with girls?
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Context:
In triangle ABC, let H, G, and O be the orthocenter, centroid, and circumcenter, respectively. Given that the Euler line is parallel to AC and \( \angle HBC = 2 \angle OCA \), calculate the distance GO if AH = a. Express your answer in terms of a.
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Context:
Determine the range of \( \beta \) values for which the random walk \( S_n = \varepsilon_1 + \cdots + \varepsilon_n \), with i.i.d. increments \( \{\varepsilon_n\} \) and distribution:

\[ P(\varepsilon_1 = -1) = 1 - C_{\beta} \quad \text{and} \quad P(\varepsilon_1 > t) = C_{\beta}e^{-t^{\beta}}, \quad t \geq 0, \]

exhibits heavy-tailed behavior, given that \( C_{\beta} \in (0,1) \) and \( \mathbb{E}\varepsilon_1 = -\frac{1}{2} \).
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Context:
Find the sum of the 11th powers of all the roots of the polynomial equation \(x^5 + 5x + 1 = 0\).
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Context:
A circular disk is divided by $2n$ equally spaced radii ($n > 0$) and one secant line. What is the maximum number of non-overlapping areas into which the disk can be divided? The original answer is in the form $kn + m$. Please find the value of $k + m$.
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Context:
Determine the number of equally spaced data points required to uniquely determine a three-harmonic Fourier series representation of a periodic function $f(\theta)$ with a period of $\frac{\pi}{2}$.
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Context:
Given that the function $f(x)$ has a domain of $(0, +\infty)$, $f(1) = e$, and for any $x_{1}$, $x_{2} \in (0, +\infty)$, when $x_{2} > x_{1}$, it holds that $\frac{{f(x_{1}) - f(x_{2})}}{{x_{1}x_{2}}} > \frac{{e^{x_{2}}}}{{x_{1}}} - \frac{{e^{x_{1}}}{{x_{2}}}}$ (where $e$ is the base of the natural logarithm). If $f(\ln a) > 2e - a\ln a$, then the range of real number $a$ is ____.
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Context:
The numbers from 1 to 9 are placed in the cells of a $3 \times 3$ table such that the sum of the numbers on one diagonal is 7 and on the other diagonal is 21. What is the sum of the numbers in five shaded cells?
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Context:
For a fixed positive integer $n \geq 3$ we are given a $n$ $\times$ $n$ board with all unit squares initially white. We define a [i]floating plus [/i]as a $5$-tuple $(M,L,R,A,B)$ of unit squares such that $L$ is in the same row and left of $M$, $R$ is in the same row and right of $M$, $A$ is in the same column and above $M$ and $B$ is in the same column and below $M$. It is possible for $M$ to form a floating plus with unit squares that are not next to it. Find the largest positive integer $k$ (depending on $n$) such that we can color some $k$ squares black in such a way that there is no black colored floating plus.

[i]Proposed by Nikola Velov[/i]
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Context:
Given the function $f(x)=2\sin \omega x \sin(\omega x+\frac{\pi}{3})+\cos 2\omega x-\frac{1}{2}$, find the range of real number $\omega$ such that the function $f(x)$ has exactly two zeros in the interval $\left[0,\pi \right]$. Express your answer as an interval.
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Context:
How many three-digit numbers $\underline{abc}$ have the property that when it is added to $\underline{cba}$, the number obtained by reversing its digits, the result is a palindrome? (Note that $\underline{cba}$ is not necessarily a three-digit number since before reversing, $c$ may be equal to $0$.)
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Context:
What is the largest integer \( n \) such that

$$
\frac{\sqrt{7}+2 \sqrt{n}}{2 \sqrt{7}-\sqrt{n}}
$$

is an integer?
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Context:
Evaluate the integral \( \int_{|x| \geq 1} \frac{1}{|x|^n} \, dx \) where \( x \in \mathbb{R}^n \).
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Context:
Let \( g_{1}(x) = \frac{1}{3} \left(1 + x + x^{2} + \cdots \right) \) for all values of \( x \) for which the right-hand side converges. Let \( g_{n}(x) = g_{1}(g_{n-1}(x)) \) for all integers \( n \geq 2 \). What is the largest integer \( r \) such that \( g_{r}(x) \) is defined for some real number \( x \)?
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Context:
Find all integer/s \( n \) such that \( \frac{{5}^{n} - 1}{3}\) is a prime or a perfect
 square of an integer.
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Context:
Consider a disk with an even number of points $n$ randomly distributed uniformly within it. These points are paired to form $n/2$ line segments, creating a perfect matching. Construct the intersection graph $G$ where each node represents a segment, and an edge exists between two nodes if their corresponding segments intersect properly. Determine the diameter of the graph $G$ as $n$ approaches infinity.
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Context:
Find the dual one-form field of the vector field \( V = x\, \frac{\partial}{\partial z} + y \frac{\partial}{\partial x} \) in \( (x,y,z) \in \mathbb{R}^3 \).
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Context:
Given propositions $P$ and $Q$, where $P$ states that the equation $\frac{x^{2}}{2m} + \frac{y^{2}}{9-m} = 1$ represents an ellipse with its foci on the $y$-axis, and $Q$ states that the eccentricity $e$ of the hyperbola $\frac{y^{2}}{5} - \frac{x^{2}}{m} = 1$ is within the range $(\frac{\sqrt{6}}{2}, \sqrt{2})$. If only one of the propositions $P$ and $Q$ is true, find the range of values for the real number $m$.
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Context:
31. [23] Let
$$
P=\prod_{i=0}^{2016}\left(i^{3}-i-1\right)^{2} .
$$

The remainder when $P$ is divided by the prime 2017 is not zero. Compute this remainder.
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Context:
Find the total length of the intervals on the number line where the inequalities \( |x| < 1 \) and \( \operatorname{tg} \log_{5} |x| < 0 \) hold.
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Context:
Three natural numbers are written on the board: two ten-digit numbers \( a \) and \( b \), and their sum \( a + b \). What is the maximum number of odd digits that could be written on the board?
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Context:
The lengths of the sides of triangle \(ABC\) are \(a\), \(b\), and \(c\) (with \(AB = c\), \(BC = a\), \(CA = b\) and \(a < b < c\)). Points \(B_1\) and \(A_1\) are marked on rays \(BC\) and \(AC\) respectively such that \(BB_1 = AA_1 = c\). Points \(C_2\) and \(B_2\) are marked on rays \(CA\) and \(BA\) respectively such that \(CC_2 = BB_2 = a\). Find the ratio \(A_1B_1 : C_2B_2\).
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Context:
Find the integral \(\int \operatorname{ch}^{3} x \operatorname{sh} x \, dx\).
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Context:
A fly trapped inside a cubical box with side length \(1\) meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. The maximum possible length of its path is in the form of \(k\sqrt{2} + m\sqrt{3}\). Please find the value of \(k + m\).
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Context:
Given that tetrahedron $ABCD$ is inscribed in sphere $O$, and $AD$ is the diameter of sphere $O$. If triangles $\triangle ABC$ and $\triangle BCD$ are equilateral triangles with side length 1, what is the volume of tetrahedron $ABCD$?
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Context:
A crazy physicist has discovered a new particle called an emon. He starts with two emons in the plane, situated a distance $1$ from each other. He also has a crazy machine which can take any two emons and create a third one in the plane such that the three emons lie at the vertices of an equilateral triangle. After he has five total emons, let $P$ be the product of the $\binom{5}{2} = 10$ distances between the $10$ pairs of emons. Find the greatest possible value of $P^2$.
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Context:
Four new students are to be assigned to three classes: A, B, and C, with at least one student in each class. Student A cannot be assigned to class A. How many different assignment plans are there?
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Context:
Let P be a polynomial with each root real and each coefficient either 1 or -1. What is the maximum possible degree of P? Express your answer as a whole number.
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Context:
一个含有底面的半球形容器内放置有三个两两外切的小球。若这三个小球的半径均为 1 ，且每个小球都与半球的底面和球面相切，则该半球的半径 $R=$ 的原始答案是 $\frac{m+\sqrt{n}}{k}$，请给出m + n + k的值。
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Context:
Construct spheres that are tangent to 4 given spheres. If we accept the point (a sphere with zero radius) and the plane (a sphere with infinite radius) as special cases, how many such generalized spatial Apollonian problems exist?
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Context:
Define a sequence \( a_{i, j} \) of integers such that \( a_{1, n} = n^n \) for \( n \geq 1 \) and \( a_{i, j} = a_{i-1, j} + a_{i-1, j+1} \) for all \( i, j \geq 1 \). Find the last (decimal) digit of \( a_{128, 1} \).
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Context:
The monotonic decreasing interval of the function $y = -x^2 + 2x + 1$ is __ .
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Context:
Find the greatest seven-digit integer divisible by $132$ whose digits, in order, are $2, 0, x, y, 1, 2, z$ where $x$, $y$, and $z$ are single digits.
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Context:
Evaluate the integral \( K = \iiint_{P}(x^2+\sin(xy^2)+ y^2) \,dx\,dy\,dz \) where \( P = \{(x,y,z) : 0 \leq z \leq 2 -|x| -|y|\} \).
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Context:
There are 150 'zeroes' and some 'ones' written on a blackboard. A step involves choosing 3 digits and replacing them with a single digit. If all the 3 digits are identical, they are replaced with the same digit. Otherwise, the digit that appears twice replaces the 3 chosen digits. The minimum number of steps after which there will be no zero written on the blackboard is 76. How many 'ones' are initially written on the blackboard?
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Context:
已知正四棱雉 $\Gamma$ 的高 $h=3$, 侧面与底面所成角 $\theta=\frac{\pi}{3}$ 。先在正四棱雉 $\Gamma$ 内放入一个内切球 $O_{1}$, 然后依次放入球 $O_{2}, O_{3}, \cdots$, 使得后放入的各球均与前一个球及正四棱雉 $\Gamma$的四个侧面都相切。则放入所有球的体积之和的原始答案为$\frac{m\pi}{n}$的格式，请给出m + n的值。
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Context:
Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of maximum length?
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Context:
Given the functions $f(x, y)$ and $g(x, y)$ where $f(1, 2) = 2$, $g(1, 2) = 1$, and their differentials at $(1, 2)$ are $df = -2dx + 5dy$ and $dg = 4dx - 3dy$, respectively. An implicitly defined function $z(x, y)$ satisfies the equation $z^2f + zg + x - y = 0$ with $z(1, 2) = -1$. Determine $\frac{\partial z}{\partial x}(1, 2)$. \(\boxed{\text{Answer}}\)
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Context:
8、A digital clock uses two digits to display hours, minutes, and seconds (such as $10: 09: 18$). Between $05: 00: 00 \sim 22: 59: 59$ on the same day, the probability that all six digits on the clock are different is $\qquad$ -
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Context:
Find the invariant factor decomposition of the tensor product \( \mathbb{Z}_6 \otimes_{\mathbb{Z}} \mathbb{Z}_9 \).
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Context:
Consider the set \(X=\{2^m3^n|0 \le m, \ n \le 9 \}\). Determine the number of quadratic equations of the form \(ax^2+2bx+c\) that have equal roots, where \(a, b, c\) are distinct elements of \(X\).
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Context:
Calculate the maximum probability that the random variable \( \xi \) exceeds 0.5, where \( \xi \) is defined as follows:

\[\xi\overset{\circ}{=}\begin{cases} \frac{p_{1}\cdot0.3}{p_{1}\cdot0.3+p_{2}\cdot0.7} & \text{with}\;p=p_{1}\cdot0.3+p_{2}\cdot0.7\\ \frac{\left(1-p_{1}\right)\cdot0.3}{\left(1-p_{1}\right)\cdot0.3+\left(1-p_{2}\right)\cdot0.7} & \text{with}\;1-p=\left(1-p_{1}\right)\cdot0.3+\left(1-p_{2}\right)\cdot0.7 \end{cases}\]

where \( p_{1},p_{2}\in\left[0,1\right] \). Find \( \max_{p_{1},p_{2}}\mathbf{P}\left(\xi>0.5\right) \).
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Context:
Determine the cardinality of the set of functions \( F: \mathbb{N} \longrightarrow \{0,1,2\} \).
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Context:
Using the formula for relativistic mass, \( m = m_0 \left(1 - \frac{v^2}{c^2}\right)^{-\frac{1}{2}} \), estimate the percentage increase in mass when the velocity of an object increases from \( v = 0.90c \) to \( v = 0.92c \) using differentials.
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Context:
What is the minimum number of cells that need to be colored on a $6 \times 6$ board so that, no matter how a 4-cell L-shaped figure is placed on the board (it can be rotated or flipped), there is at least one colored cell in that figure?
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Context:
Given a machine that transforms a positive integer \( N \) based on the rule: if \( N = 7 \), the machine outputs \( 3 \times 7 + 1 = 22 \).

By inputting the result back into the machine and repeating five times, the output sequence is:
\[ 7 \rightarrow 22 \rightarrow 11 \rightarrow 34 \rightarrow 17 \rightarrow 52 \rightarrow 26 \]

If there exists some positive integer \( N \) whereby six iterations of the process results in the final output being 1,
\[ N \rightarrow_{-} \rightarrow_{-} \rightarrow_{-} \rightarrow_{-} \rightarrow 1 \]

What is the sum of all such positive integers \( N \)?
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Context:
8. Two Go teams, Team A and Team B, each have 5 players who compete in a relay match according to a predetermined order. The No. 1 players of both teams start the match, and the loser is eliminated; then the No. 2 player of the losing team competes against the winner, and the loser is eliminated. This continues until all players of one team are eliminated, at which point the other team is declared the winner. Assuming each player has equal strength, the expected number of Team A's players who have not played when the match ends, $E(X)=$ $\qquad$

---

I hope this translation meets your needs. If you have any further requests or need additional assistance, feel free to let me know!
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Context:
Given the equations:
\[ xw^3 + yz^2 + z^3 = -1 \]
\[ zw^3 - xz^3 + y^2w = 1 \]
find the partial derivative \( \frac{\partial w}{\partial y} \) at the point \((x, y, z, w) = (1, -1, -1, 1)\).
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Context:
Let $S_n$ be the sum of the first $n$ terms of a geometric series $\{a_n\}$ with positive terms. If $S_4 - 2S_2 = 2$, find the minimum value of $S_6 - S_4$.
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Context:
One standard balloon can lift a basket with contents weighing not more than 80 kg. Two standard balloons can lift the same basket with contents weighing not more than 180 kg. Find the weight of the basket in kilograms. Express your answer as a whole number.
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Context:
Compute the expected value \( \mathbb{E}[ \tau C^2_{\tau}] \) where \( \tau = \inf \{t \geq 0 : |C_t| = 1 \} \) and \( C \) is a Standard Brownian Motion.
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Context:
King Arthur has two advisors, Merlin and Percival, each providing correct answers with probability \( p \). If both advisors agree, the king follows their advice. If they disagree, the king flips a coin to decide. Arthur wonders if he should dismiss one advisor, believing the probability of making correct decisions won't decrease. If true, he will dismiss one advisor by coin flip. Percival answers truthfully, while Merlin answers to minimize his dismissal probability. What is the probability that Merlin will be dismissed?
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Context:
Let $\Gamma_1$ be a circle with radius $\frac{5}{2}$. Points $A$, $B$, and $C$ are on $\Gamma_1$ such that $\overline{AB} = 3$ and $\overline{AC} = 5$. Let $\Gamma_2$ be a circle tangent to $AB$ and $BC$ at $Q$ and $R$, and also internally tangent to $\Gamma_1$ at $P$. $\Gamma_2$ intersects $AC$ at $X$ and $Y$. The area $[PXY]$ can be expressed as $\frac{a\sqrt{b}}{c}$. Find $a+b+c$.
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Context:
The population of a town is 15000. It decreases by 20% in the first year due to job losses. In the second year, the population increases by 10% due to new job opportunities, but also experiences a 5% decrease due to migration. In the third year, the population increases by 8% as a result of increased birth rates, but also decreases by 25% due to an economic recession. What will be the net population of the town after these three years?
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Context:
Given the Markov chain $\{X_n\}$ with transition probability matrix $P$:

$$P=\left( \begin{array}{ccccccc}
\frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 & 0 & 0 \\
0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 & 0 \\
\frac{1}{4} & 0 & 0 & \frac{1}{2} & \frac{1}{4} & 0 & 0 \\
0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 \\
0 & 0 & 0 & 0 & \frac{1}{2} & \frac{1}{4} & \frac{1}{4} \\
0 & 0 & 0 & 0 & \frac{1}{4} & \frac{1}{2} & \frac{1}{4} \\
0 & 0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} & 0 \\
\end{array} \right),$$

where $X_0 = 1$, calculate the expected value of the sum $\sum_{k=1}^{T} \mathbb{I}_{\{X_k=3\}}$, where $T = \inf\{k\geq 0: X_k \in \{5,6,7\}\}$. Here, $\mathbb{I}_A$ denotes the indicator function that is $1$ when the event $A$ occurs and $0$ otherwise.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the permutations \( \theta = (13)(14)(567) \), \( \tau = (137)(2456) \), and \( \sigma = (13456)(27) \) in the symmetric group \( S_7 \), find the intersection of the left cosets \( \sigma \langle \theta \rangle \) and \( \tau \langle \theta \rangle \).
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Context:
Determine the total number of 7-bit strings such that any pair of these strings differ in exactly 4 bits.
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Context:
Jeremy is driving 600 miles to visit his parents. His average speed for this trip is 50 miles per hour. Every two hours of driving, he makes a rest stop for 15 minutes. His car gets 18 miles per gallon of gas, and his gas indicator comes on when he's used 15 gallons, at which point he stops to refill, which takes 10 minutes. How long, in minutes, does he take to get to his parents' house, if he starts out with a full tank of gas?
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Context:
A ring x of diameter 16 inches is placed inside another ring y of diameter 18 inches. Additionally, a third ring z with a diameter of 20 inches is placed outside of ring y. Calculate the combined uncovered fractions of the surface area of ring x and the surface area of ring z not covered by the rings y and z, respectively.
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Context:
An unpainted cone has radius \( 3 \mathrm{~cm} \) and slant height \( 5 \mathrm{~cm} \). The cone is placed in a container of paint. With the cone's circular base resting flat on the bottom of the container, the depth of the paint in the container is \( 2 \mathrm{~cm} \). When the cone is removed, its circular base and the lower portion of its lateral surface are covered in paint. The fraction of the total surface area of the cone that is covered in paint can be written as \( \frac{p}{q} \) where \( p \) and \( q \) are positive integers with no common divisor larger than 1. What is the value of \( p+q \)?

(The lateral surface of a cone is its external surface not including the circular base. A cone with radius \( r \), height \( h \), and slant height \( s \) has lateral surface area equal to \( \pi r s \).)
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Context:
Given a positive integer \( n\left( {n > {2004}}\right) \) , we put 1, 2, 3, …, \( {n}^{2} \) into squares of an \( n \times  n \) chessboard with one number in a square. A
 square is called a "good square" if the square satisfies following conditions:
 There are at least 2004 squares that are in the same row with the square such that any number within these 2004 squares is
 less than the number within the square.
 There are at least 2004 squares that are in the same column with the square such that any number within these 2004 squares
 is less than the number within the square.
 Find the maximum value of the number of the "good square".
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Context:
Find the slope of the curve of intersection of the surface \(z = x^{3} + xy^{2}\) and the plane \(2x-2y = 1\) at the point \((1, \frac{1}{2}, \frac{5}{4})\).
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Context:
Find a complex function that outputs only real values, similar to the function \( f(z) = |z| \).
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Context:
Determine the order of the poles of the function \( f(z)= \dfrac{z}{ (1-e^{z}) \sin z} \) at \( z=2\pi ik \), where \( k \in \mathbb{Z} \) and \( k \neq 0 \).
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Context:
The integer part and decimal part of $(\sqrt{10}+3)^{2n+1}$ for $n \in \mathbf{N}$ are denoted as $I$ and $F$, respectively. What is the value of $P(I + F)$?
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Context:
In triangle $ABC$, where all sides are different, the bisector of angle $BAC$ intersects side $BC$ at point $D$.

It is known that $\angle AB - BD = a$ and $AC + CD = b$.

Find $AD$.
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Context:
A dog is being treated for fleas. The effectiveness of each flea treatment varies as follows: the first treatment eliminates 60% of fleas, the second treatment eliminates 45% of the remaining fleas, the third treatment eliminates 30% of the remaining fleas, and the fourth treatment eliminates 20% of the remaining fleas. After four treatments, the dog has 20 fleas left. How many more fleas did the dog have before the flea treatments than after?
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Context:
Every month Ivan pays a fixed amount of his salary for a mortgage, and the remaining portion of his salary is spent on current expenses. In December, Ivan paid 40% of his salary for the mortgage. In January, Ivan's salary increased by 9%. By what percentage did the amount spent on current expenses increase in January compared to December?
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Context:
Given point \( A \) has coordinates \( (0,3) \), and points \( B \) and \( C \) are on the circle \( O: x^{2}+y^{2}=25 \), satisfying \(\angle B A C=90^{\circ}\). Find the maximum area of \(\triangle A B C\).
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Context:
对于一个实数 x, 令 $\{x\}=x-[x]$. 记 $S=\int_{0}^{2024} \min \left(\left\{\frac{x}{8}\right\},\left\{\frac{x}{4}\right\}\right) \mathrm{d} x$, 则 $[S]=$ $\qquad$.
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Context:
The first doughnut is priced at $1 and then if you purchase additional doughnuts as dozens, the price is a certain amount per dozen. The total number of doughnuts purchased for $24 is 48. What is the price per dozen for additional doughnuts?
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Context:
Simplify the expression: \[ \frac{(8+2\sqrt{15})^{3/2}+(8-2\sqrt{15})^{3/2}}{(12+2\sqrt{35})^{3/2}-(12-2\sqrt{35})^{3/2}} \]
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Context:
Compute the Hausdorff measure \( \mathcal{H}_{2} \) of the unit cube \( A = \{(x,y,z) \in \mathbb{R}^{3}: x\in [0,1], y \in[0,1], z\in[0,1]\} \).
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Context:
Find the largest constant $C$ so that for all real numbers $x$, $y$, and $z$,
\[x^2 + y^2 + z^3 + 1 \ge C(x + y + z).\]
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Context:
Find the distance between the foci of the ellipse
\[\frac{x^2}{48} + \frac{y^2}{12} = 3.\]
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Context:
On March 12, 2016, the fourth Beijing Agriculture Carnival opened in Changping. The event was divided into seven sections: "Three Pavilions, Two Gardens, One Belt, and One Valley." The "Three Pavilions" refer to the Boutique Agriculture Pavilion, the Creative Agriculture Pavilion, and the Smart Agriculture Pavilion; the "Two Gardens" refer to the Theme Carnival Park and the Agricultural Experience Park; the "One Belt" refers to the Strawberry Leisure Experience Belt; and the "One Valley" refers to the Yanshou Ecological Sightseeing Valley. Due to limited time, a group of students plans to visit "One Pavilion, One Garden, One Belt, and One Valley." How many different routes can they take for their visit? (Answer with a number)
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Context:
In an arithmetic sequence \(a_1, a_2, \ldots, a_{1990}\) with a non-zero common difference, where \(a_i a_j > 0\) for all \(i, j = 1, 2, \ldots, 1990\), let \(b_1 = a_1 \cdot a_{1990}\), \(b_2 = a_2 \cdot a_{1989}\), \(\ldots\), \(b_k = a_k \cdot a_{1990 - k + 1}\) for \(k = 1, 2, \ldots, 1990\). Determine the largest term among \(\{b_k\}\).
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Context:
Given four independent random variables $X_1, X_2, X_3, X_4$ each uniformly distributed on the interval $(0,1)$, let $(X_{(1)}, X_{(2)}, X_{(3)}, X_{(4)})$ denote the order statistics. Compute the probability $P(X_{(2)} \leq 3X_{(1)})$. Use integration techniques to find your answer.
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Context:
Given the differential equations \( f'(x) = -\frac{f(x)}{\sqrt{(f(x))^2+(g(x))^2}} \) and \( g'(x) = 1 - \frac{g(x)}{\sqrt{(f(x))^2+(g(x))^2}} \) for all \( x \in \mathbb{R} \), with initial conditions \( g(0) = 0 \) and \( f(0) = 10 \), and knowing that \( \lim_{x\to\infty}f(x) = 0 \), find the value of \( \lim_{x\to\infty}g(x) \).
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Context:
A child builds towers using identically shaped cubes of different colors. Determine the number of different towers with a height of 6 cubes that can be built with 3 yellow cubes, 3 purple cubes, and 2 orange cubes (Two cubes will be left out).
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Context:
Determine the set of limit points of the set \( A = \left\{ \frac{\sqrt{m} - \sqrt{n}}{\sqrt{m} + \sqrt{n}} \mid m, n \in \mathbb{N} \right\} \).
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Context:
Evaluate the limit: $$\lim\limits_{x \to 0^{\large +}}\mrm{Li}_{x}\pars{1}$$
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Context:
Let $A=\{1,2,\ldots,2012\}, \: B=\{1,2,\ldots,19\}$ and $S$ be the set of all subsets of $A.$ Find the number of functions $f : S\to B$ satisfying $f(A_1\cap A_2)=\min\{f(A_1),f(A_2)\}$ for all $A_1, A_2 \in S.$
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Context:
Given a semicircle of diameter $AB$ and center $O$. Let $CD$ be the chord of the semicircle tangent to two circles of diameters $AO$ and $OB$. If $CD=120$ cm,, caclulate area of the semicircle.
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Context:
Calculate \(142^{381} \mod 773\) using the given modular arithmetic relations: \(142^1 \equiv 142 \pmod{773}\), \(142^2 \equiv 66 \pmod{773}\), and \(142^3 \equiv 96 \pmod{773}\).
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Context:
Provide an example of a continuous function $f(x)$ such that the integral $\int_{0}^{\infty}f(x)\,dx$ converges, but the limit $\lim_{x\to\infty}f(x)$ does not exist.
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Context:
Evaluate the integral \( \int_0^5 (x^2+1) \, d \lfloor x \rfloor \), where \( \lfloor \cdot \rfloor \) is the greatest integer function.
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Context:
Lei Lei bought some goats and sheep. If she had bought 2 more goats, the average price of each sheep would increase by 60 yuan. If she had bought 2 fewer goats, the average price of each sheep would decrease by 90 yuan. Lei Lei bought $\qquad$ sheep in total.
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Context:
Add the square of the smallest area from squares of size $1 \times 1, 2 \times 2,$ and $3 \times 3,$ such that the number of squares of each size is the same.
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Context:
Let $R$ be the set of points $(x, y)$ such that $x$ and $y$ are positive, $x + y$ is at most 2013, and
\[
  \lceil x \rceil \lfloor y \rfloor = \lfloor x \rfloor \lceil y \rceil.
\]
Compute the area of set $R$.  Recall that $\lfloor a \rfloor$ is the greatest integer that is less than or equal to $a$, and $\lceil a \rceil$ is the least integer that is greater than or equal to $a$.
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Context:
Let $I$ be an uncountable directed set, and let $T = \{E_{\alpha}: \alpha \in I\}$ be a collection of countably infinite sets such that $E_{\alpha} \subset E_{\beta}$ whenever $\alpha \le \beta$. The collection $T$ is totally ordered with respect to set inclusion. Determine the cardinality of $\bigcup_{\alpha \in I} E_{\alpha}$.
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Context:
In triangle $ABC$, $B=\frac{π}{3}$, the angle bisector of $\angle BAC$ intersects $BC$ at $D$, and $AD=3$, $BD=2$, then $\cos C=$____.
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Context:
Find the smallest natural number $k$ such that for all sufficiently large $n$, less than 1% of all permutations of $[n]$ have at least $k$ fixed points. Assume that the number of derangements $D(n)$ for a set of $n$ elements satisfies $D(n) = n! \sum_{k=0}^n \frac{(-1)^k}{k!}$ and $\sum_{k=0}^{\infty} \frac{(-1)^k}{k!} = \frac{1}{e}$. Provide your answer as a single integer $k$. \(\boxed{}\)
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Context:
Let $A B C D$ be an isosceles trapezoid with parallel bases $A B=1$ and $C D=2$ and height 1. Find the area of the region containing all points inside $A B C D$ whose projections onto the four sides of the trapezoid lie on the segments formed by $A B, B C, C D$ and $D A$.
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Context:
A store purchased 3 different types of electronics: Radios, TVs, and Mobile Phones in the following quantities and cost prices:

- 5 Radios at a cost price of Rs. 8,000 each.
- 3 TVs at a cost price of Rs. 20,000 each.
- 4 Mobile Phones at a cost price of Rs. 15,000 each.

The store offers different discounts and sales tax rates on each type of electronic:

- Radios: 10% discount and 6% sales tax on the price after discount.
- TVs: 15% discount and 7% sales tax on the price after discount.
- Mobile Phones: 8% discount and 5% sales tax on the price after discount.

The store sells all the electronic items and receives the following sale amounts post-discount and including sales tax:

- Radios: Rs. 7,200 each.
- TVs: Rs. 18,000 each.
- Mobile Phones: Rs. 14,500 each.

Calculate the combined loss percentage of the store on selling all these electronic items.
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Context:
In Vila Par, all the truth coins weigh an even number of grams, while the false coins weigh an odd number of grams. The electronic device only provides the parity (odd or even) of the total weight of a set of coins. If there are $2020$ truth coins and $2$ false coins, determine the smallest integer $k$ such that there exists a strategy to identify the two false coins using the electronic device at most $k$ times.
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Context:
Find the area of the portion of the plane \( z = x + 3y \) that lies inside the elliptical cylinder given by the equation \( \frac{x^2}{4} + \frac{y^2}{9} = 1. \)
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Context:
Determine the number of subgroups of index $p$ in the additive group $\mathbb{Z}^2$, where $p$ is a prime number.
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Context:
Given a positive integer $k$, let \|k\| denote the absolute difference between $k$ and the nearest perfect square. For example, \|13\|=3 since the nearest perfect square to 13 is 16. Compute the smallest positive integer $n$ such that $\frac{\|1\|+\|2\|+\cdots+\|n\|}{n}=100$.
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Context:
From a solid regular tetrahedron with edge length of one unit, a smaller regular tetrahedron with an edge length of $\frac{1}{n}$ is cut off from each of its vertices, where $n$ is an integer greater than 1. For what value of $n$ will the ratio of the surface area to the volume of the remaining solid be minimized?
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Context:
Which prime numbers can be expressed as the sum of two positive composite numbers?
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Context:
A caravan of camels, 1 km long, moves uniformly through the desert. A rider travels from the end of the caravan to the front and back to the end. During this time, the caravan covers 1 km. How far did the rider travel?
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Context:
Let  $ABCD$  be a rectangle with  $AB = 6$  and  $BC = 6 \sqrt 3$ .  We construct four semicircles  $\omega_1$ ,  $\omega_2$ ,  $\omega_3$ ,  $\omega_4$  whose diameters are the segments  $AB$ ,  $BC$ ,  $CD$ ,  $DA$ . It is given that  $\omega_i$  and  $\omega_{i+1}$  intersect at some point  $X_i$  in the interior of  $ABCD$  for every  $i=1,2,3,4$  (indices taken modulo  $4$ ). Compute the square of the area of  $X_1X_2X_3X_4$ .

*Proposed by Evan Chen*
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Context:
14 Labeled as $1,2, \cdots, 100$, there are some matches in the matchboxes. If each question allows asking about the parity of the sum of matches in any 15 boxes, then to determine the parity of the number of matches in box 1, at least how many questions are needed?
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Context:
1. In how many ways can a $3 \times 3$ square table be filled with the numbers $2,2,3,3,3,4,4,4,4$ so that the sum of the numbers in each $2 \times 2$ square of this table is equal to 14?
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Context:
Find all positive integers $k$ such that for any positive numbers $a, b, c$ that satisfy the inequality $k(a b+b c+c a)>5\left(a^{2}+b^{2}+c^{2}\right)$, there always exists a triangle with side lengths $a, b, c$.
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Context:
Consider two fictional states: Alpha and Beta. Alpha issues license plates with a format of two letters followed by four numbers, and then ending with one letter (LLNNNNL). Beta issues plates with three letters followed by three numbers and lastly one letter (LLLNNNL). Assume all 10 digits and 26 letters are equally likely to appear in the respective slots. How many more license plates can state Alpha issue than state Beta?
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Context:
The area of the region bounded by the graph of $x^2+y^2 = 3|x-y| + 3|x+y|$ is $m+n\pi$, where $m$ and $n$ are integers. What is the value of $m + n$?
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Context:
B4. On each of the 10,000 fields of a $100 \times 100$ chessboard, there is a number. On the top row, from left to right, the numbers are 0 to 99. In the left column, from top to bottom, the numbers are 0 to 99. The sum of four numbers in a $2 \times 2$ block is always 20. What number is in the bottom right corner of the board?
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Context:
Distribute 5 students into dormitories A, B, and C, with each dormitory having at least 1 and at most 2 students. Among these, the number of different ways to distribute them without student A going to dormitory A is \_\_\_\_\_\_.
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Context:
10 people -- liars and knights -- sat around a round table. Liars always lie, and knights always tell the truth. Each of them was given a coin. Then each person passed their coin to one of their two neighbors. After that, 5 people said, "I have one coin," and the remaining 5 people said, "I have no coins." What is the maximum number of knights that could have been sitting at the table?
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Context:
Given that A, B, C, D are distinct integers from 1 to 6, and the curves y = Ax^2 + B and y = Cx^2 + D intersect, calculate the number of distinct ways to choose values for A, B, C, D such that the curves intersect, considering that the intersection is independent of the order of A, B, C, D. Express your answer as a single integer.
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Context:
In the plane Cartesian coordinate system $xOy$, $AB$ is a chord of the parabola $y^{2}=4x$ passing through the point $F(1,0)$, and the circumcircle of triangle $\triangle AOB$ intersects the parabola at point $P$ (different from points $O, A, B$). If $PF$ bisects the angle $\angle APB$, find all possible values of $|PF|$.
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Context:
In quadrilateral \(ABCD\), it is known that \(AB = BC\) and \(\angle ABC = \angle ADC = 90^{\circ}\). From vertex \(B\), a perpendicular \(BH\) is dropped to side \(AD\). Find the area of quadrilateral \(ABCD\) given that \(BH = h\).
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Context:
If the lengths of the major axis, minor axis, and focus of an ellipse form a geometric sequence, then the eccentricity of this ellipse is ___.
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Context:
Given proposition $p$: "$1\leqslant x\leqslant 5$ is a sufficient but not necessary condition for $x^{2}-(a+1)x+a\leqslant 0$", and proposition $q$: "There are two triangles $\triangle ABC$ that satisfy $AC=6$, $BC=a$, and $\angle CAB=30^{\circ}$". If the proposition $\lnot p\land q$ is true, find the range of values for the real number $a$.
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Context:
A rope can make 70 rounds of the circumference of a cylinder whose radius of the base is 14 cm. How many times can the rope go around a cylinder having a different radius if it can go around the second cylinder 49 times?
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Context:
9 - 3 ÷ 1 / 3 + what number equals 3?
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Context:
Determine the smallest value of $(a+5)^2 + (b-2)^2 + (c-9)^2$ for all real numbers $a, b, c$ satisfying $a^2 + b^2 + c^2 - ab - bc - ca = 3$.
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Context:
A regular tetrahedron has two vertices on the body diagonal of a cube with side length 12. The other two vertices lie on one of the face diagonals not intersecting that body diagonal. Find the side length of the tetrahedron.
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Context:
In how many ways every unit square of a $2018$ x $2018$ board can be colored in red or white such that number of red unit squares in any two rows are distinct and number of red squares in any two columns are distinct.
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Context:
Find the least prime number greater than $1000$ that divides $2^{1010} \cdot 23^{2020} + 1$.
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Context:
Each pan of a balance has k weights, numbered from 1 to k. The left pan is heavier. It turns out that if we interchange the places of any two weights with the same number, then either the right pan becomes heavier or the two pans reach an equilibrium. For what k this is possible? Please provide the sum of all possible values of k.
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Context:
Kartik wants to shade three of the squares in a grid blue, and Lucy wants to shade the remaining two squares red. How many possible finished grids are there where Lucy's red squares are next to each other? Express your answer as a whole number.
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Context:
What is the least positive integer with the property that the product of its digits is $9! ?$
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Context:
The data of sample A are as follows: 52, 54, 54, 56, 56, 56, 55, 55, 55, 55. If the data of sample B are exactly the data of sample A each increased by 6, then what statistical measure remains the same for both samples? Express your answer as a single word or phrase representing a statistical measure.
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Context:
A regular octahedron $A B C D E F$ is given such that $A D, B E$, and $C F$ are perpendicular. Let $G, H$, and $I$ lie on edges $A B, B C$, and $C A$ respectively such that \frac{A G}{G B}=\frac{B H}{H C}=\frac{C I}{I A}=\rho. For some choice of $\rho>1, G H, H I$, and $I G$ are three edges of a regular icosahedron, eight of whose faces are inscribed in the faces of $A B C D E F$. Find $\rho$.
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Context:
Let  $A(x)=\lfloor\frac{x^2-20x+16}{4}\rfloor$ ,  $B(x)=\sin\left(e^{\cos\sqrt{x^2+2x+2}}\right)$ ,  $C(x)=x^3-6x^2+5x+15$ ,  $H(x)=x^4+2x^3+3x^2+4x+5$ ,  $M(x)=\frac{x}{2}-2\lfloor\frac{x}{2}\rfloor+\frac{x}{2^2}+\frac{x}{2^3}+\frac{x}{2^4}+\ldots$ ,  $N(x)=\textrm{the number of integers that divide }\left\lfloor x\right\rfloor$ ,  $O(x)=|x|\log |x|\log\log |x|$ ,  $T(x)=\sum_{n=1}^{\infty}\frac{n^x}{\left(n!\right)^3}$ , and  $Z(x)=\frac{x^{21}}{2016+20x^{16}+16x^{20}}$  for any real number  $x$  such that the functions are defined. Determine   $$ C(C(A(M(A(T(H(B(O(N(A(N(Z(A(2016)))))))))))))). $$  *2016 CCA Math Bonanza Lightning #5.3*
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Context:
Given that the volume of the tetrahedron \(ABCD\) is 40, \(AB = 7\), \(BC = 8\), \(AC = 9\), and the orthogonal projection of vertex \(D\) onto the plane \(ABC\) is precisely the incenter \(H\) of triangle \(ABC\), what is the surface area of the tetrahedron?
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Context:
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. What time does she finish the third task? Express your answer in time format (HH:MM PM).
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Context:
Suppose a regular tetrahedron \( P-ABCD \) has all edges equal in length. Using \(ABCD\) as one face, construct a cube \(ABCD-EFGH\) on the other side of the regular tetrahedron. Determine the cosine of the angle between the skew lines \( PA \) and \( CF \).
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Context:
Let $n \geq 5$ be a positive integer and let $A$ and $B$ be sets of integers satisfying the following conditions:

i) $|A| = n$, $|B| = m$ and $A$ is a subset of $B$
ii) For any distinct $x,y \in B$, $x+y \in B$ iff $x,y \in A$

Determine the minimum value of $m$.
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Context:
How many of the numbers $1\cdot 2\cdot 3$, $2\cdot 3\cdot 4$,..., $2020 \cdot 2021 \cdot 2022$ are divisible by $2020$?
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Context:
A number of women proposed once each, of whom one-eighth were widows. Consequently, a number of men were to be married, of whom one-eleventh were widowers. Of the proposals made to widowers, one-fifth were declined. All the widows were accepted. Thirty-five forty-fourths of the widows married bachelors. One thousand two hundred and twenty-one spinsters were declined by bachelors. The number of spinsters accepted by bachelors was seven times the number of widows accepted by bachelors. How many women proposed?
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Context:
In $\triangle ABC$, $AB > AC$, $\angle BAC = 45^\circ$. Point $E$ is the intersection of the external angle bisector of $\angle BAC$ with the circumcircle of $\triangle ABC$. Point $F$ is on $AB$ such that $EF \perp AB$. Given $AF = 1$ and $BF = 5$, find the area of $\triangle ABC$.
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Context:
Each of two teams has seven players numbered 1 to 7. In the first game, the two players numbered 1 play each other. The loser of each game is eliminated and replaced by the next player of the same team, until all players from one team have been eliminated. Let $N$ be the number of possible sequences of games. Find the remainder when $N$ is divided by 1000.
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Context:
Given that there are \(2^n\) coins distributed among a group of kids, an interchange occurs when one of the kids has at least half of all the coins. During an interchange, the kid with at least half of the coins gives each of the other kids the same number of coins as they currently have. If all the coins are with one kid, no interchange is possible. What is the greatest possible number of consecutive interchanges? (\(n\) is a natural number)
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Context:
Define a sequence of sets \( \{S_i\}_{i=0}^{\infty} \) such that \( S_0 = \{0\} \) and \( S_i = \bigcup\limits_{k=0}^{i-1} \{S_k\} \). Find the number of brackets in the power set of \( S_4 \) if the empty set is written as \( \{\} \).
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Context:
Find a real number \(a\) such that the function \(g_a(x) = \begin{cases} x^a \sin \left(\frac{1}{x}\right) & x \neq 0 \\ 0 & x = 0 \end{cases}\) is differentiable on \(\mathbb{R}\) and its derivative \(g_a'(x)\) is unbounded on the interval \([0,1]\).
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Context:
Suppose we have a nonnegative sequence $b_m \in \ell^1$. Define a new sequence $d_n$ as:

$$d_n = \alpha^n \sum\limits_{m=n}^{\infty} b_m \binom{m}{n}\beta^{m-n}.$$

where $0 < \alpha < 1$ and $0 \leq \beta < 1$ satisfy $\alpha + \beta \leq 1$. Find a constant $L$ such that $\|d_n\|_{\ell^1} \leq L \|b_n\|_{\ell^1}$. Determine the value of $L$.
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Context:
Given that the random variable $\xi$ follows the normal distribution $N(1, \sigma^2)$, if $P(\xi > 3) = 0.023$, then $P(-1 \leq \xi \leq 3)$ equals to?
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Context:
记 I=\ds{m=1}{\infty}\ds{n=1}{\infty}\df{1}{n!n!+m!n!}，则 [10I]=__________.
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Context:
Given an equilateral triangle $\triangle ABC$. Points $D$ and $E$ are taken on side $BC$ such that $BC = 3DE$. Construct an equilateral triangle $\triangle DEF$ and connect point $A$ to point $F$. Draw $DG$ parallel to $AF$ intersecting side $AB$ at point $G$, and draw $EH$ parallel to $AF$ intersecting side $AC$ at point $H$. Draw perpendiculars $GI \perp AF$ and $HJ \perp AF$. Given that the area of $\triangle BDF$ is $45$ and the area of $\triangle DEF$ is $30$, find the ratio $GI \div HJ$.
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Context:
There are 8 different books and 10 different movies in the 'crazy silly school' series. If you read a certain number of the books and watched 61 of the movies, there are 2 more movies than books in the 'crazy silly school' series. How many books did you read?
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Context:
Let \( D \) be a regular ten-sided polygon with edges of length 1. A triangle \( T \) is defined by choosing three vertices of \( D \) and connecting them with edges. How many different (non-congruent) triangles \( T \) can be formed?
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Context:
A digit is inserted between the digits of a two-digit number to form a three-digit number. Some two-digit numbers, when a certain digit is inserted in between, become three-digit numbers that are $k$ times the original two-digit number (where $k$ is a positive integer). What is the maximum value of $k$?
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Context:
The Wolf and Ivan Tsarevich are 20 versts away from a source of living water, and the Wolf is taking Ivan Tsarevich there at a speed of 3 versts per hour. To revive Ivan Tsarevich, one liter of water is needed, which flows from the source at a rate of half a liter per hour. At the source, there is a Raven with unlimited carrying capacity; it must gather the water, after which it will fly towards the Wolf and Ivan Tsarevich at a speed of 6 versts per hour, spilling a quarter liter of water every hour. After how many hours will it be possible to revive Ivan Tsarevich?
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Context:
Calculate the variance of the random variable \( X = \int_{0}^{1} W^2(t) \, dt \), where \( W(t) \) is a Wiener process.
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Context:
Let $[x]$ denote the greatest integer less than or equal to $x$. When $0 \leqslant x \leqslant 10$, find the number of distinct integers represented by the function $f(x) = [x] + [2x] + [3x] + [4x]$.
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Context:
Eight red boxes and eight blue boxes are randomly placed in four stacks of four boxes each. The probability that exactly one of the stacks consists of two red boxes and two blue boxes is  $\frac{m}{n}$  , where m and n are relatively prime positive integers. Find  $m + n$ .
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Context:
Find the maximum possible value of the sum \(\sum_{1 \leq i < j \leq n} |x_i-x_j|\), where \(x_1,...,x_n\) are integers in the interval \([1,35]\).
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Context:
四面体 $A B C D$ 中, $A B \perp B C, C D \perp B C, B C=2$, 且异面直线 $A B$ 与 $C D$ 所成的角为 $60^{\circ}$ 。若四面体 $A B C D$ 的外接球半径为 $\sqrt{5}$, 则四面体 $A B C D$ 体积的最大值为 $\qquad$.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。
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Context:
A school has 9 English teachers, some history, and 6 geography teachers. Each teacher can teach 2 subjects max. The minimum number of teachers required is 11. How many history teachers are there in the school?
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Context:
Find the range of real number k such that the inequality kx^2 + 2kx - (k+2) < 0 always holds true. Express your answer in interval notation, including any relevant mathematical notation (e.g., parentheses, brackets).
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Context:
We wish to color the integers $1,2,3, \ldots, 10$ in red, green, and blue, so that no two numbers $a$ and $b$, with $a-b$ odd, have the same color. (We do not require that all three colors be used.) In how many ways can this be done?
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Context:
Tim and Allen are playing a match of tennis. In a match of tennis, the two players play a series of games, each of which is won by one of the two players. The match ends when one player has won exactly two more games than the other player, at which point the player who has won more games wins the match. In odd-numbered games, Tim wins with probability \( \frac{3}{4} \), and in the even-numbered games, Allen wins with probability \( \frac{3}{4} \). What is the expected number of games in a match?
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Context:
Example 6 Given
$$
\left\{\begin{array}{l}
\frac{x^{2}}{2^{2}-1^{2}}+\frac{y^{2}}{2^{2}-3^{2}}+\frac{z^{2}}{2^{2}-5^{2}}+\frac{w^{2}}{2^{2}-7^{2}}=1 \\
\frac{x^{2}}{4^{2}-1^{2}}+\frac{y^{2}}{4^{2}-3^{2}}+\frac{z^{2}}{4^{2}-5^{2}}+\frac{w^{2}}{4^{2}-7^{2}}=1 \\
\frac{x^{2}}{6^{2}-1^{2}}+\frac{y^{2}}{6^{2}-3^{2}}+\frac{z^{2}}{6^{2}-5^{2}}+\frac{w^{2}}{6^{2}-7^{2}}=1 \\
\frac{x^{2}}{8^{2}-1^{2}}+\frac{y^{2}}{8^{2}-3^{2}}+\frac{z^{2}}{8^{2}-5^{2}}+\frac{w^{2}}{8^{2}-7^{2}}=1
\end{array}\right.
$$

Find the value of $x^{2}+y^{2}+z^{2}+w^{2}$.
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Context:
Determine the maximum distance from the origin to a point on the curve given by the equation \(|z+e^{i\theta}w|=r_z+r_w\), where \(z\) and \(w\) are fixed complex numbers and \(\theta\) varies over the interval \([0,2\pi]\).
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Context:
How many total strikes do the clocks make in a day if they strike once every half an hour and strike $1,2,3 \ldots 12$ times each hour?
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Context:
Determine the number of integral solutions for the equation \(x^4 - ax^3 - bx^2 - cx - d = 0\), given that \(a \geq b \geq c \geq d\) and \(a, b, c, d \in \mathbb{N}\).
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Context:
Find \( F^{(n+2)}(x) \) for \( F(x) = \int_0^x (x-t)^n u(t) \, dt \), where \( u(t) \) is a function of \( t \), using the Leibniz rule.
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Context:
How many circles of radius \(\frac{\pi}{4}\) can be drawn within a circle of radius \(\pi\) such that they do not intersect one another?
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Context:
Evaluate \( \lim_{x\to\infty }x\int_1^{x}\frac{e^t}{t}dt-e^x \).
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Context:
点集 $S=\left\{(x, y) \mid x \leq 5, y \leq 4, x, y \in \mathrm{~N}^{*}\right\}$, 则由 S 中的点可组成 $\qquad$个不同的三角形。
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Context:
What is the minimum order of a differential equation whose solution can be expressed as a linear combination of three trigonometric functions with different arguments?
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Context:
Given that the number of Master's degrees awarded by faculty F of university U follows a Poisson distribution, and the funding for faculty F in a given year is proportional to the sum of the number of degrees awarded in the three preceding years, find the variance of the funding for year \((2000 + i)\), where \(i \ge 3\).
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Context:
Let $x, y$ be complex numbers such that \frac{x^{2}+y^{2}}{x+y}=4$ and \frac{x^{4}+y^{4}}{x^{3}+y^{3}}=2$. Find all possible values of \frac{x^{6}+y^{6}}{x^{5}+y^{5}}$.
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Context:
In which dimension does a Riemannian manifold with constant scalar curvature also have constant sectional curvature?
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Context:
Find the perimeter of a rectangle if the sum of the lengths of its three different sides can be equal to 6 or 9.
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Context:
Let $Q$ be a smooth hyperquadric in $\mathbb{P}^5_{\mathbb{C}}$. Determine the dimension of the variety $F_2(Q)$, which parametrizes all 2-planes contained in $Q$. Assume that each hyperquadric contains at least one plane and that the dimension of the Grassmannian $G(k,n)$ is given by $(n-k)(k+1)$. Find $\dim(F_2(Q))$. \( \text{Provide your answer as a single integer.} \)
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Context:
Evaluate the limit: \[ \lambda = \lim_{z \to \infty} \left[ 2^z - \left(\frac{4}{3}\right)^z - \zeta(\zeta(z)) \right] \]
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Context:
Determine the values of $n$ for which it is possible to collect all the water into a single glass in a finite number of steps, given that each of the $n$ glasses initially contains an equal amount of water and you can pour as much water from any glass into any other as the latter contains.
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Context:
Calculate the definite integral:
$$
\int_{0}^{\pi / 4} \frac{5 \operatorname{tg} x+2}{2 \sin 2 x+5} d x
$$
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Context:
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{1-\cos(2x)}{x^2}dx \) using the Residue Theorem.
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Context:
Determine the cardinality of the set of subrings of \(\mathbb{Q}\).
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Context:
What is the molecular weight of 4 moles of K2Cr2O7?
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Context:
Tom Brady is shy of the passing yards record in a season, which is 5999 yards. He has some passing yards with 6 games left in the season and needs to average 300 passing yards per game to beat the record. How many passing yards does he currently have?
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Context:
The product \( 29 \cdot 11 \), and the numbers 1059, 1417, and 2312, are each divided by \( d \). If the remainder is always \( r \), where \( d \) is an integer greater than 1, what is \( d - r \) equal to?
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Context:
Consider the following infinite geometric series: $$\frac{7}{8}-\frac{14}{27}+\frac{28}{81}-\dots$$ Find the common ratio of this series.
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Context:
Let \(\{a_{n}\}\) be an integer sequence such that for any \(n \in \mathbf{N}^{*}\), the condition \((n-1) a_{n+1} = (n+1) a_{n} - 2 (n-1)\) holds. Additionally, \(2008 \mid a_{2007}\). Find the smallest positive integer \(n \geqslant 2\) such that \(2008 \mid a_{n}\).
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Context:
Keesha wants to get her hair, nails, and makeup done for prom, and also receive a pedicure and facial. She decides to go to three different boutiques for these services. For hair, updos cost $50 and come with a 10% student discount. Manicures cost $30 and have a buy-one-get-one 50% off deal on pedicures, which cost $35 each. Makeup applications cost $40, with a 7% tax applied. Pedicure services are regularly $45 but are included in the buy-one-get-one deal on manicures. Facial treatments cost $60 and have a special 15% off promo. Keesha decides to tip each beautician 20%. Calculate the total cost Keesha will pay for these services, taking into account service costs, discounts, taxes, and tips.
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Context:
After watching a movie, viewers rated the movie sequentially with an integer score from 0 to 10. At any given moment, the movie rating was calculated as the sum of all given scores divided by their number. At a certain moment $T$, the rating turned out to be an integer, and then with each new voting viewer, it decreased by one unit. What is the maximum number of viewers who could have voted after moment $T ?$
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Context:
Evaluate the limit: \( \lim_{n\to\infty}n\cdot \sin(2\pi e(n!)) \)
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Context:
Use the divergence theorem to evaluate the surface integral \( I = \iint_S \mathbf{G} \cdot \mathbf{n} \, dS \), where \( \mathbf{G} = (x+y) \mathbf{i} +(y+z) \mathbf{j} +(z+x) \mathbf{k} \) and \( S \) is the surface of the sphere \( x^2+y^2+z^2 =9 \) between \( -2<z<2 \).
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Context:
A train 400 m long can cross an electric pole in 40 sec. What is the speed of the train?
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Context:
Let $\Omega = D(0,2) \setminus \{\frac{1}{2}\}$, where $D(0,2)$ is a disc, and let $f$ be holomorphic in $\Omega$. Given that $\frac{1}{2}$ is a simple pole for $f$ with residue $1$, calculate the limit:
\[ \lim_{n \to \infty} \frac{f^{(n)}(0)}{2^{n}n!} \]
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Context:
Calculate the variance \( \mathrm{Var}(e^{-X}) \) given that the random variable \( X \) has the probability density function \( f_\theta(x) = \exp(-(x-\theta) - \exp(-(x-\theta))) \).
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Context:
设集合 $T=\{\left(a,b,c\right)|a,b,c\in\mathbf{Z}_{+}$,且 $a、b、c$
可构成某个三角形的三边长}.则$\sum_{(a,b,c)\in T}\frac{2^a}{3^b\times5^c}$的值为$\_$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:
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O.$ Let the intersection points of the perpendicular bisector of $CH$ with $AC$ and $BC$ be $X$ and $Y$ respectively. Lines $XO$ and $YO$ cut $AB$ at $P$ and $Q$ respectively. If $XP+YQ=AB+XY,$ determine $\measuredangle OHC.$
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Context:
Determine the image of the imaginary axis under the function \( f(z) = \frac{z+1}{z-1} \), where \( f: \mathbb{C} \to \mathbb{C} \).
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Context:
Find the norm of the linear operator $K(f)$, where $K(f)(x) = f(x) + \int_{0}^{x} f(t)dt$.
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Context:
Calculate the flux of the vector field \( \bar{F} = \langle x, y, x \rangle \) over the surface \( z = 1 - x - y \) in the first octant using the upward unit normal vector \( \bar{n} = \frac{1}{\sqrt{3}} \langle -1, -1, 1 \rangle \). The flux is given by the integral \( \int\int_S \bar{F} \cdot \bar{n} \, dA \), where \( dS = \sqrt{3} \, dA \).
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Context:
If \((2 x+4)^{2 n}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{2 n} x^{2 n}\) for \(n \in \mathbf{N}^{+}\), then the remainder when \(a_{2}+a_{4}+\cdots+a_{2 n}\) is divided by 3 is .
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Context:
A number, when divided by 44, gives 432 as quotient and a certain remainder. When dividing the same number by 38, the remainder is 8. What is the remainder in the first division?
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Context:
Matrices  $A$ ,  $B$  are given as follows.
\[A=\begin{pmatrix} 2 & 1 & 0  1 & 2 & 0  0 & 0 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 2 & 0  2 & 4 & 0  0 & 0 & 12\end{pmatrix}\]
Find volume of  $V=\{\mathbf{x}\in\mathbb{R}^3 : \mathbf{x}\cdot A\mathbf{x} \leq 1 < \mathbf{x}\cdot B\mathbf{x} \}$ .
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Context:
Evaluate

\[ \int_e^{e^{2009}} \frac{1}{x}\left\{1+\frac{1-\ln x}{\ln x\cdot \ln \frac{x}{\ln (\ln x)}}\right\}\ dx\]
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Context:
Find a positive integer \( n \) less than 2006 such that \( 2006n \) is a multiple of \( 2006 + n \).
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Context:
Frieda the frog begins a sequence of hops on a \(3 \times 3\) grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge. For example, if Frieda begins in the center square and makes two hops "up", the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
What is the molecular weight of 7 moles of a certain compound if the molecular weight of that compound is 1176?
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Context:
Find the kernel of the evaluation map \( g: \mathbb{Z}[x] \longrightarrow \mathbb{Z}[3^{1/4}] \) where \( x = 3^{1/4} \).
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Context:
Given the sequences \(a_n=\int_0^1 (1-x^2)^n \,dx\) and \(b_n=\int_0^1 (1-x^3)^n \,dx\) where \(n \in \mathbb{N}\), find the limit:
\[ \lim_{n\to \infty}(10\sqrt[n]{a_n}+5\sqrt[n]{b_n}). \]
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Context:
Find the set of integer values of $n$ for which the inequality $3 \uparrow^{n} n < 2\uparrow^{n+2} 2$ is true.
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Context:
Determine the maximum value of \( \lambda \) such that for any distinct positive real numbers \( a, b, c \), the inequality \[ \frac{a^{3}}{(b-c)^{2}}+\frac{b^{3}}{(c-a)^{2}}+\frac{c^{3}}{(a-b)^{2}} \geqslant \lambda(a+b+c) \] holds.
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Context:
Consider a convex polygon $\mathcal{P}$ in space with perimeter $20$ and area $30$. What is the volume of the locus of points that are at most $1$ unit away from some point in the interior of $\mathcal{P}$?
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Context:
Compare the values of $P(28, 19, 1, 2)$ and $P(30, 10, 2, 7)$, where $P(n, k, a, b)$ denotes the number of arrangements of $n$ indistinguishable balls into $k$ distinct buckets such that there are no less than $a$ and no more than $b$ balls in each bucket. Which value is higher?
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Context:
A cube with dimensions \(20 \times 20 \times 20\) is divided into 8000 unit cubes. Each unit cube has a number written on it. In every row and column of 20 unit cubes, parallel to any edge of the cube, the sum of the numbers is 1. In one of these unit cubes, the number 10 is written. Three \(1 \times 20 \times 20\) layers pass through this unit cube, parallel to the faces of the cube. Calculate the sum of all the numbers outside of these three layers.
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Context:
Side $BC$ of quadrilateral $ABCD$ is the diameter of the circle circumscribed around this quadrilateral. Find side $AB$ if $BC=8$, $BD=4\sqrt{2}$, and $\angle DCA : \angle ACB = 2 : 1$.
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Context:
Find the derivative of the implicit function $y(x)$ defined by the equation $$ y^x = x^y - x $$ at the point $(1,1)$. Express your answer as a single value.
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Context:
Determine the number of equally spaced data points needed to uniquely specify a three harmonic Fourier series for a periodic function with a period of \(\frac{\pi}{2}\).
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Context:
Determine the smallest integer $k$ such that a $k$-vector with binary entries can be used to communicate any integer between 1 and 1000 to a friend, even after one of its components is altered by a function $f_i$, which sets the $i$-th component to 1.
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Context:
Dragoons take up \(1 \times 1\) squares in the plane with sides parallel to the coordinate axes such that the interiors of the squares do not intersect. A dragoon can fire at another dragoon if the difference in the \(x\)-coordinates of their centers and the difference in the \(y\)-coordinates of their centers are both at most 6, regardless of any dragoons in between. For example, a dragoon centered at \((4,5)\) can fire at a dragoon centered at the origin, but a dragoon centered at \((7,0)\) cannot. A dragoon cannot fire at itself. What is the maximum number of dragoons that can fire at a single dragoon simultaneously?
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Context:
Given that the complex number $z$ and $(z-3)^2 + 5i$ are both purely imaginary, find $z=$ ______.
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Context:
Let $G$ be a graph with 100 vertices numbered from 1 to 100. Two vertices $i$ and $j$ are adjacent if $|i-j|=8$ or $|i-j|=12$. What is the number of connected components in the graph $G$?
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Context:
Let $T:L^p[0,1]\to L^p[0,1]$ be a linear operator defined by $T(f(x))=x^kf(x)$. Determine the operator norm of $T$. \(1 \leq p < \infty\).
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Context:
Five positive integers are listed in increasing order. The difference between any two consecutive numbers in the list is three. The fifth number is a multiple of the first number. How many different such lists of five integers are there? Express your answer as a single integer.
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Context:
For which values of $k$ is the blow-up $X_k = Bl_{p_1,...,p_k}\mathbb{P}^2$ of $\mathbb{P}^2$ at $k$ general points a Mori Dream Space?
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Context:
Let \( \omega_1 \) and \( \omega_2 \) be circles with radii 3 and 12, respectively, and externally tangent at point \( P \). A common external tangent intersects \( \omega_1 \) and \( \omega_2 \) at points \( S \) and \( T \), respectively, and the common internal tangent at point \( Q \). Define \( X \) to be the point on \( \overrightarrow{QP} \) such that \( QX = 10 \). If \( XS \) and \( XT \) intersect \( \omega_1 \) and \( \omega_2 \) a second time at points \( A \) and \( B \), respectively, determine \( \tan\angle APB \).
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Context:
My frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, what is the expected score of the losing team?
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Context:
在$\Delta ABC$中，角$A,B,C$所对的边分别为$a,b,c$，若$b\cos C=\left( 2a-c \right)\sin \left( B+\frac{\pi }{2} \right)$，且$b=\sqrt{3}$，记$h$为$AC$边上的高，则$h$的取值范围为__________．
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Context:
Let $A B C$ be a triangle and $D, E$, and $F$ be the midpoints of sides $B C, C A$, and $A B$ respectively. What is the maximum number of circles which pass through at least 3 of these 6 points?
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Context:
The mass of the first cast iron ball is $1462.5\%$ greater than the mass of the second ball. By what percentage less paint is needed to paint the second ball compared to the first ball? The volume of a ball with radius $R$ is $\frac{4}{3} \pi R^{3}$, and the surface area of a ball is $4 \pi R^{2}$.
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Context:
3. Given a cube and 12 colors. Find the number of ways to paint the faces of this cube using these colors (each face in one color) such that adjacent faces are of different colors. Colorings that differ by a rotation are considered different.
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Context:
Determine the order of the Galois group of the residue field extension $k_i/\mathbf{F}_q$ in the context of the theorem relating the inertia index $f$ to the order of $q$ modulo $p$. Provide your answer as a single integer.
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Context:
Let \( S = \{ 1, 2, 3, \ldots, 280 \} \). Find the smallest integer \( n \) such that each \( n \)-element subset of \( S \) contains five numbers which are pairwise relatively prime.
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Context:
Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?
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Context:
Let $(b_1, b_2, b_3, \ldots, b_{10})$ be a permutation of $(1, 2, 3, \ldots, 10)$ such that $b_1 > b_2 > b_3 > b_4 > b_5$ and $b_5 < b_6 < b_7 < b_8 < b_9 < b_{10}$. An example of such a permutation is $(5, 4, 3, 2, 1, 6, 7, 8, 9, 10)$. Find the number of such permutations.
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Context:
What shape is impossible to obtain as a cross-section when a plane intersects a cylinder? Provide your answer as the name of the shape.
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Context:
Find the smallest possible value of \(x^2 + y^2\) given that \(x\) and \(y\) are positive integers such that \(x(x + 1)\mid y(y + 1)\), but neither \(x\) nor \(x + 1\) divides either \(y\) or \(y + 1\).
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Context:
Today is Barbara's birthday, and Alberto wants to give her a gift by playing the following game. The numbers $0, 1, 2, \ldots, 1024$ are written on a blackboard.

1. First, Barbara erases $2^{9}$ numbers.
2. Then, Alberto erases $2^{8}$ numbers.
3. Next, Barbara erases $2^{7}$ numbers.
4. This process continues until there are only two numbers $a$ and $b$ left.

Barbara earns $|a-b|$ euros.

Find the maximum number of euros that Barbara can always win, regardless of Alberto's strategy.
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Context:
Determine the smallest integer \( n \) such that the Chebyshev interpolation \( p_n(x) \) of the function \( f(x) = \sin(x) \) on the interval \([0, \pi/2]\) satisfies \( ||f - p_n|| < 10^{-16} \).
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Context:
David has a unit triangular array of 10 points, 4 on each side. A looping path is a sequence \( A_{1}, A_{2}, \ldots, A_{10} \) containing each of the 10 points exactly once, such that \( A_{i} \) and \( A_{i+1} \) are adjacent (exactly 1 unit apart) for \( i=1,2, \ldots, 10 \). Here \( A_{11}=A_{1} \). Find the number of looping paths in this array.
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Context:
Consider positive integers $a \leq b \leq c \leq d \leq e$. There are $N$ lists $a, b, c, d, e$ with a mean of 2023 and a median of 2023, in which the integer 2023 appears more than once, and in which no other integer appears more than once. What is the sum of the digits of $N$?
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Context:
How many ordered triples (x,y,z) of integers satisfy the system of equations below?
\begin{array}{l} x^2-3xy+2y^2-z^2=31 \ -x^2+6yz+2z^2=44 \ x^2+xy+8z^2=100\ \end{array}
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Context:
What is the maximum number of squares that can be marked on a chessboard so that from each marked square, it is possible to reach any other marked square in exactly two moves of a chess knight?
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Context:
Simplify: $\frac{5 \sqrt[4]{5 r^{4}}}{\sqrt[4]{8 r^{2}}}$
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Context:
Let $a_n$ denote the remainder when $(n+1)^3$ is divided by $n^3$; in particular, $a_1=0$. Compute the remainder when $a_1+a_2+\dots+a_{2013}$ is divided by $1000$.
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Context:
设方程 $x y z=900$ 的全部正整数解为 $\left(x_{i}, y_{i}, z_{i}\right)(1 \leqslant i \leqslant n)$. 则 $\sum_{k=1}^{n}\left(x_{k c}+y_{k}+z_{k}\right)=\qquad$.
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Context:
A tournament with 100 competitors, each with different skill levels, will take place. The most skilled competitor always wins against the less skilled one. Each participant plays exactly twice, with two randomly chosen opponents (once with each). A competitor who wins both matches receives a medal. Determine the minimum number of medals that can be distributed in the tournament.
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Context:
The distance traveled by the center \( P \) of a circle with radius 1 as it rolls inside a triangle with side lengths 6, 8, and 10, returning to its initial position.
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Context:
Let $F$ be the focus of the parabola $C$: $y^{2}=3x$. A line passing through $F$ with an inclination angle of $30^{\circ}$ intersects $C$ at points $A$ and $B$. Let $O$ be the origin of the coordinates, then the area of $\triangle OAB$ is \_\_\_\_\_.
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Context:
There are real numbers  $a, b, c, d$  such that for all  $(x, y)$  satisfying  $6y^2 = 2x^3 + 3x^2 + x$ , if  $x_1 = ax + b$  and  $y_1 = cy + d$ , then  $y_1^2 = x_1^3 - 36x_1$ . What is  $a + b + c + d$ ?
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Context:
In parallelogram $ABCD$, diagonal $BD$ is equal to 2, angle $C$ is $45^\circ$, and line $CD$ is tangent to the circumcircle of triangle $ABD$. Find the area of the parallelogram $ABCD$.
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Context:
Find the number of $4$-digit numbers (in base $10$) that have non-zero digits and are divisible by $4$ but not by $8$.
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Context:
A $4 \times 4$ square piece of paper is cut into two identical pieces along its diagonal. The resulting triangular pieces of paper are each cut into two identical pieces. Each of the four resulting pieces is cut into two identical pieces. Each of the eight new resulting pieces is finally cut into two identical pieces. Calculate the length of the longest edge of one of these final sixteen pieces of paper.
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Context:
设定点 $A 、 B 、 C 、 D$ 是以 O 点为中心的正四面体的顶点，用 $\sigma$ 表示空间以直线 $O A$ 为轴满足条件 $\sigma(B)=C$ 的旋转，用 $\tau$ 表示空间关于 $O C D$ 所在平面的镜面反射，设 l 为过 $A B$ 中点与 $C D$ 中点的直线，用 $\omega$ 表示空间以 l 为轴的 $180^{\circ}$ 旋转，设 $\sigma \circ \tau$ 表示变换的复合，先作 $\tau$ ，再作 $\sigma$ ，则 $\omega$ 可以表示为什么？答案的形式为 k \circ m \circ n \circ p \circ m \circ n，其中 k、m、n、p 是待定系数，请给出 k + m + n + p 的值。
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Context:
As shown in the figure, $AD$ is the altitude of $\triangle ABC$. From point $D$, draw $DE \perp AB$ at point $E$. Let $F$ be a point on the extension line of $ED$ such that $AF \perp CF$. Given that $AB = 13$, $BC = 14$, and $AC = 15$, find the perimeter of $\triangle ACF$.
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Context:
We take $100$ consecutive natural numbers $a_{1}, a_{2}, \ldots, a_{100}$. Determine the last two digits of the number $a_{1}^{8} + a_{2}^{8} + \cdots + a_{100}^{8}$.
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Context:
Let $q(x) = 1 - (1-x) + (1-x)^2 - \cdots + (1-x)^{10}$. Let the roots of $q(x)$ be $\beta_i, i \in \{1, 2, \ldots, 10\}$. Find the value of $\sum_{i=1}^{10} \frac{1}{\beta_i}$.
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Context:
Let \( h_{n} = \sum_{k=0}^{n} \binom{n}{k} \frac{2^{k+1}}{k+1} \). If \( S= \sum_{n=0}^{\infty} \frac{h_{n}}{n!} \), find \( \lfloor S \rfloor \).
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Context:
James is a freelancer who works with international clients. In January, he made $4000 from a US client. In February, he earned 50% more than he made in January from a European client when the exchange rate was 1 Euro = 1.2 Dollars. In March, his earnings decreased by 20% from his February earnings from a Japanese client, where the exchange rate was 110 Yen = 1 Dollar. In what he earned each month, how much has James made so far this year in Dollars?
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Context:
Determine the relationship between the functions \(f(n) = \frac{\log n}{\log\log n}\) and \(g(n) = \log(n - \log n)\) using the notations \(\Omega, \Theta, O\). Express the relationship as \(f(n) = K \cdot g(n)\), where \(K\) represents one of these notations.
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Context:
Derek finally gets his own allowance. He puts $2 away in January, some money away in February, $8 away in March, and follows this savings pattern through to December. He has $4096 to spare and save by December. How much money did he put away in February?
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Context:
求最大的正整数 $n$ ，使得存在 $n$ 个正整数 $x_{1}<x_{2}<\cdots<x_{n}$, 满足:
$$
x_{1}+x_{1} x_{2}+\cdots+x_{1} x_{2} \cdots x_{n}=2021
$$
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Context:
Mike earns $12 per hour, Phil earns $6 per hour, and Olivia earns $10 per hour. What is the average percentage difference in earnings per hour between all three individuals?
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Context:
There are three segments of each length $2^{n}$ for $n=0,1, \cdots, 1009$. How many non-congruent triangles can be formed using these 3030 segments? (Answer in numerical form)
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Context:
If $\sin 2x \sin 3x = \cos 2x \cos 3x$, then one value for $x$ is $x^\circ$. What is the value of $x$?
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Context:
Spencer is making burritos, each of which consists of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos; in how many ways can he make exactly five burritos?
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Context:
For which initial values $x_0$ does a fixed point iteration exhibit third-order convergence as described by the error relation $|\alpha - x_{n+1}| = \frac{1}{4} (|\alpha - x_n|)^3$? Express the range of these initial values as an interval around $\alpha$.
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Context:
The length and breadth of the floor of a room are 20 feet and some feet respectively. Square tiles of 2 feet length of different colors are to be laid on the floor. Black tiles are laid in the first row on all sides. White tiles are laid in one-third of the remaining and blue tiles in the rest. There are 16 blue tiles. What is the breadth of the room?
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Context:
The volume of a hemispherical soup bowl is 8 liters. How much soup fills the bowl up to half its height?
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Context:
For the Burgers' equation \( u_t + uu_x = 0 \), determine the breaking time \( t_B \) for the initial condition:
\[
 u(x,0) = \phi(x) = \begin{cases}
 1, & x \leq \pi \\
 \cos x + 2, & \pi \leq x \leq 2\pi \\
 0, & x \geq 2\pi
\end{cases}.
\]
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Context:
Determine the smallest value of $a$ such that the function \( f_a(x) = \frac{(2x)!}{(x!)^a} \) forms a bell curve shape.
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Context:
Arrange the numbers 1 to 2001 in a clockwise direction on a circle. Starting from the number 2, eliminate every second number in a clockwise direction until only one number remains. Calculate the last remaining number.
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Context:
Mitya is 11 years older than Shura. When Mitya was as old as Shura is now, he was twice as old as she was. How old is Mitya?
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Context:
From the 205 positive integers 1, 2, ..., 205, what is the maximum number of integers that can be selected so that for any three selected numbers $a$, $b$, $c$ ($a < b < c$), it holds that $ab \neq c$?
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Context:
The natural numbers \(a, b, c\) are chosen such that \(a < b < c\). Given that the system of equations \(2x + y = 2023\) and \(y = |x - a| + |x - b| + |x - c|\) has exactly one solution, determine the minimum possible value of \(c\).
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Context:
10 is 6% of a, and a certain percentage of b is 6. c equals b / a. What is the value of the percentage that represents the relationship between 6 and b?
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Context:
If the graph of the function $f(x)=ax^{3}-\frac{1}{3}$ ($a>0$) and the function $g(x)=x^{2}-\frac{2}{3}cx$ have exactly three distinct intersection points, and the abscissas of the intersection points form an arithmetic sequence, then the range of real number $a$ is ______.
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Context:
Find the minimal value of integer n that guarantees: Among n sets, there exist at least three sets such that any of them does not include any other; or there exist at least three sets such that any two of them include the other. Express your answer as a single integer.
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Context:
Given a sequence $\{a_n\}$ with the first term $a_1=1$, the function $f(x)=x^4+a_{n+1}\cos 2x-(2a_n+1)$ has a unique zero. Find the sum of the first $n$ terms of the sequence $\{n(a_n+1)\}$.
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Context:
Masha has 2 kg of "Lastochka" candies, 3 kg of "Truffle" candies, 4 kg of "Ptichye Moloko" candies, and 5 kg of "Citron" candies. What is the maximum number of New Year gifts she can make if each gift must contain 3 different types of candies, with 100 grams of each type?
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Context:
A computer component has a lifetime (in years) that follows a Gamma distribution with a mean of 6 years and a variance of 18 years squared. Calculate the probability that the component will last at least 9 years.
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Context:
We selected the vertices, the centroid, and the trisection points of the sides of an equilateral triangle. How many points can we keep among them such that no three of them form an equilateral triangle?
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Context:
A person can swim in still water at 16 km/h. If the speed of water is 8 km/h, how many hours will the man take to swim back against the current for a certain distance if it takes him 1.5 hours?
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Context:
Determine the set of limit points of the set \( T = \{ m + \frac{1}{n^2} \mid m, n \in \mathbb{N} \} \).
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Context:
Calculate \( h'(2) \) given that \( h(x) = g(f(x, x)) \), where \( g(x) = f(x, f(x, x)) \). The function \( f(x, y) \) is differentiable, and the following values are provided: \( f(2, 2) = 4 \), \( f_x(2, 2) = 3 \), and \( f_y(2, 2) = 7 \).
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Context:
Given a function \( f(u) \le 2\sqrt{u}f(\sqrt{u}) + 1 \), determine the asymptotic growth of \( f(u) \) in terms of Big O notation.
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Context:
Determine the number of homomorphisms from the group \( G = \langle x,y \mid x^{-1}yx = y^6 \rangle \) to the symmetric group \( S_5 \).
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Context:
Evaluate the integral \( I = \iint_R |xy|\,dx\,dy \) using polar coordinates, where \( R \) is the region bounded by \( x^2+4y^2 \ge 1 \) and \( x^2+y^2 \le 1 \).
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Context:
Determine the number of non-isomorphic semidirect products between the groups \( \mathbb{Z}_2 \) and \( SL(2,3) \).
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Context:
Evaluate the triple integral \( \int_{-1}^{-2}\int_{-1}^{-2}\int_{-1}^{-2}\frac{x^2}{x^2+y^2+z^2}dxdydz \).
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Context:
Determine the Krull dimension of the algebra \( \mathbb{Q}[X,Y,Z]/(X^{2}-Y,Z^{2}) \).
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