Context:
Evaluate the limit: \[ \lim_{n\to\infty} \sum_{r=0}^n \frac{\binom{n}{r}}{(n^r)(r+3)} \]
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Context:
Find the radius of convergence for the power series \( \sum_{n=1}^{\infty}\frac{ x^{6n+2}}{(1+\frac{1}{n})^{n^2}} \).
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Context:
Let $A$ be an $R$-right module, $N$ be a submodule of an $R$-left module $M$, and let $\pi: M \rightarrow M/N$ be the natural epimorphism. Determine $\ker(\pi \otimes 1_A)$. Provide your answer in terms of $M$, $N$, and $A$.
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Context:
Given that \(7^{-1} \equiv 55 \pmod{101}\), find \(49^{-1} \pmod{101}\), as a residue modulo 101. (Answer should be between 0 and 100, inclusive.)
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Context:
In triangle ABC, angle C = 90 degrees and AC = BC = sqrt(2). Triangle ABC is rotated clockwise by 60 degrees around point A to the new position of triangle AB'C'. Connect C' to B. Calculate the length of C'B. Express your answer as a single value.
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Context:
What is the smallest set of points required to intersect all lines in a projective geometry $PG(d,q)$?
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Context:
Find a point on the curve \(r(t) = (1 - 2t)i + (t^2)j + \frac{t^3}{2}k\), where \(t > 0\), such that the tangent line at that point is parallel to the plane \(5x + y + z - 3 = 0\).
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Context:
A specialty store sells three types of products: peach slices, rice crispy treats, and sesame snacks. The cost of each pack of peach slices is twice the cost of sesame snacks. The selling prices of each pack of peach slices, rice crispy treats, and sesame snacks are $20\%$, $30\%$, and $20\%$ higher than their respective costs. In May, the store sold peach slices, rice crispy treats, and sesame snacks in a ratio of $1:3:2$. The total profit from the three products was $25\%$ of the total cost. Find the ratio of the cost of each pack of rice crispy treats to the cost of each pack of sesame snacks.
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Context:
In a game of craps, if the first roll is neither a natural nor craps, the player can make an additional bet equal to their original one, betting that they will make their point before a seven appears. The payoff odds are as follows: 2:1 for a point of four or ten, 3:2 for a point of five or nine, and 6:5 for a point of six or eight. Calculate the expected winnings for the player if they make this additional bet when given the opportunity.
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Context:
Positive integers $a, b$, and $c$ have the property that $a^{b}, b^{c}$, and $c^{a}$ end in 4, 2, and 9, respectively. Compute the minimum possible value of $a+b+c$.
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Context:
Given that the angle bisector of the lines represented by the equation \(a(x-1)^2+2h(x-1)(y-2)+b(y-2)^2=0\) is \(x+2y-5=0\), and the ratio \(\frac{b-a}{h}\) can be expressed as \(\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, find the value of \(p+q\).
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Context:
As shown in the diagram, there are 12 points on the circumference of a circle, dividing the circumference into 12 equal parts. How many rectangles can be formed using these equally divided points as the four vertices?
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Context:
K2) Let $n$ be a natural number. Roger has a square garden of size $(2 n+1) \times(2 n+1)$. He erects fences to divide it into rectangular beds. He wants exactly two horizontal $k \times 1$ beds and exactly two vertical $1 \times k$ beds for every even number $k$ between 1 and $2 n+1$, as well as a single square bed of size $1 \times 1$ when he is finished. How many different ways can Roger divide his garden?
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Context:
Choose positive integers \( b_{1}, b_{2}, \ldots \) satisfying
\[
1=\frac{b_{1}}{1^{2}}>\frac{b_{2}}{2^{2}}>\frac{b_{3}}{3^{2}}>\frac{b_{4}}{4^{2}}>\cdots
\]
and let \( r \) denote the largest real number satisfying \( \frac{b_{n}}{n^{2}} \geq r \) for all positive integers \( n \). What are the possible values of \( r \) across all possible choices of the sequence \( \left( b_{n} \right) \)?
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Context:
Ajay spends 40% of his monthly income on household items, 20% on buying clothes, 10% on medicines, 5% on entertainment, and 10% on transportation. He invests 5% of his monthly income. Over a period of 6 months, his savings percentages are 10%, 15%, 8%, 12%, 7%, and 18% of his monthly income. Given the total savings for the entire 6-month period as Rs. 58,500, calculate his average monthly income during this time.
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Context:
Determine the smallest positive integer \( n \) such that for any \( n \) points \( A_1, A_2, \ldots, A_n \) on the circumference of a circle, there are at least 2007 angles \( \angle A_i O A_j \) (where \( 1 \leq i < j \leq n \)) that do not exceed \( 120^\circ \).
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Context:
Problem 6. Each of the two sides of a triangle is divided into six equal parts; the corresponding division points are connected by segments. Find the sum of the lengths of these segments if the third side of the triangle is 12.
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Context:
There are 4 different books on the bookshelf. Three people, A, B, and C, go to choose books, with each person choosing at least one book. How many different ways are there to choose?
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Context:
设 $a_{1}, a_{2}, a_{3}, a_{4}$ 为四个有理数, 使得: $\left\{a_{i} a_{j} \mid 1 \leq i<j \leq 4\right\}=\left\{-24,-2,-\frac{3}{2},-\frac{1}{8}, 1,3\right\}$, 求 $a_{1}+a_{2}+a_{3}+a_{4}$ 的值。原始答案是 \frac{m}{n} 的形式，请给出 m + n 的值。
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Context:
The triangular prism \( S-ABC \) has three lateral edges that are pairwise perpendicular. \( O \) is a point inside the base triangle \( \triangle ABC \). What is the minimum value of \( W = \tan \angle OSA \cdot \tan \angle OSB \cdot \tan \angle OSC \)?
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Context:
Calculate the product of the non-zero eigenvalues of the matrix \(\begin{pmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{pmatrix}\).
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Context:
Consider an $n$-dimensional sphere of radius $\sqrt{n}$. A point $(x_1, x_2, \ldots, x_n)$ is chosen randomly on this sphere. Determine the probability that the first coordinate $x_1$ is less than or equal to a given value $a$ as $n$ approaches infinity.
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Context:
Calculate the sum \( \sum_{i=1}^{200} f(i) \) where the function \( f(x) \) is defined as follows:
\[
f(x) = \begin{cases}
\left[\frac{1}{\{\sqrt{x}\}}\right] & \text{if } x \neq k^2 \\
0 & \text{if } x = k^2
\end{cases}
\]
Here, \( x, k \in \mathbb{N} \), \([.]\) denotes the greatest integer function, and \(\{.\}\) denotes the fractional part. Determine the value of the sum.
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Context:
How many $k$-simplices in a triangulation of the simplex $\sigma_k:=\{(x_1,...,x_{k+1}):x_1+\cdots+x_{k+1}=1, x_i\ge 0\}$ does a non-boundary $(k-1)$-simplex lie in?
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Context:
While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$.
Clarifications
The problem should read $|a+b+c| = 21$. An earlier version of the test read $|a+b+c| = 7$; that value is incorrect.
$|b|^2+|c|^2$ should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers $m$ and $n$. Find $m+n$.''
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Context:
A teacher received a number of letters from Monday to Friday, which were 10, 6, 8, 5, 6, respectively. The variance $s^2$ of this set of data is \_\_\_\_\_\_.
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Context:
Determine the tangent space of the manifold \( M = \{(x,y,z):z=\sqrt{y-x}\} \) at the point \((x,y,z) = (2,6,*)\).
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Context:
Determine the radius of convergence for the Maclaurin series:
\[
(-1)^{n+1} \frac{\left(\left|\frac{1-x}{1+x}\right|-1\right)^n}{n}
\]
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Context:
Find the infimum of the integral \[ \iint_{x^2+y^2\leqslant1}\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2\,\mathrm{d}x\,\mathrm{d}y \] for \( C^\infty \)-functions \( u \) that vanish at the origin and are equal to 1 on the boundary \( x^2+y^2=1 \).
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Context:
Calculate the expected time until a bank balance, modeled by a Brownian motion with positive drift, first becomes negative.
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Context:
What fraction of regular hexagon \( {ZUMING} \) is covered by both quadrilateral \( {ZUMI} \) and quadrilateral \( {MING} \) ?
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Context:
Find a nontrivial example of a function \( f \) defined on a neighborhood of \( z \in \mathbb{C} \) and a path from \( z \) to \( z \) such that the analytic continuation of \( f \) along the path is \( f' \). Ensure that \( f \neq f' \).
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Context:
In triangle $ABC$, $AX = XY = YB = BC$ and the measure of angle $ABC$ is 120 degrees. What is the number of degrees in the measure of angle $BAC$?

[asy]
pair A,X,Y,B,C;
X = A + dir(30); Y = X + dir(0); B = Y + dir(60); C = B + dir(-30);
draw(B--Y--X--B--C--A--X);
label("$A$",A,W); label("$X$",X,NW); label("$Y$",Y,S); label("$B$",B,N); label("$C$",C,E);
[/asy]
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Context:
On the island of Truth and Lies, there are knights who always tell the truth, and liars who always lie. One day, 20 inhabitants of the island lined up in order of their height (from tallest to shortest, the tallest being the first) for a game. Each of them had to say one of the following phrases: "There is a liar below me" or "There is a knight above me". As a result, those standing in positions from the third to the seventh said the first phrase, while the others said the second phrase. How many knights were there among these 20 people, if all the inhabitants have different heights?
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Context:
Evaluate the iterated integral: $$\int_0^8\int_{y^{1/3}}^2 4e^{x^4} \,dx\,dy$$ for the region defined by $0 \leq y \leq 8$ and $y^{1/3} \leq x \leq 2$. Determine the value of the integral.
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Context:
In triangle ABC, points A', B', and C' are located on sides BC, AC, and AB, respectively. The cevians AA', BB', and CC' are concurrent at point O. Given that the sum of the ratios AO:OA', BO:OB', and CO:OC' is 92, determine the value of the product AO:OA' × BO:OB' × CO:OC'.
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Context:
Determine the values of \(\alpha\) for which the integral \(\int\int_{D}\frac{1}{(x+y)^{\alpha}}dxdy\) converges, where the region \(D\) is defined by \(0 \le y \le 1 - x\) and \(0 \le x \le 1\).
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Context:
Given that $0 < a \leqslant b \leqslant c$ and $abc = 1$, find the minimum value of $\frac{1}{a^{2}(b+c)} + \frac{1}{b^{3}(a+c)} + \frac{1}{c^{3}(a+b)}$.
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Context:
Zhendarov R.G.

In the cells of a $4 \times 4$ table, numbers are written such that the sum of the neighbors of each number is 1 (cells are considered neighbors if they share a side).

Find the sum of all the numbers in the table.

#
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Context:
Kelvin the Frog lives in the 2-D plane. Each day, he picks a uniformly random direction (i.e., a uniformly random bearing $\theta \in [0, 2\pi]$) and jumps a mile in that direction. Let $D$ be the number of miles Kelvin is away from his starting point after ten days. Determine the expected value of $D^4$.
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Context:
Consider a  $6 \times 6$  grid of squares. Edmond chooses four of these squares uniformly at random. What is the probability that the centers of these four squares form a square?
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Context:
In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake?
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Context:
Given the sequence $\{a_n\}$ where $a_n = (\sqrt{2}+1)^n - (\sqrt{2}-1)^n \quad (n \in \mathbf{N})$, and let $[x]$ denote the greatest integer not exceeding the real number $x$, compute the units digit of $[a_{2017}]$. Express your answer as a single digit.
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Context:
Given that the function $f(x)$ is an even function on $\mathbb{R}$, and $f(x)$ is monotonically increasing on $[0, +\infty)$, let $m=f(-1)$, $n=f(a^2+2a+3)$, then the relationship between $m$ and $n$ is.
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Context:
Let \( A = \{1, -1, \mathrm{i}, -\mathrm{i} \} \) (where \( \mathrm{i} \) is the imaginary unit), and \( f(x) \) be a function whose domain and range are both \( A \). Given that for any \( x, y \in A \), \( f(x y) = f(x) f(y) \), how many such functions \( f(x) \) satisfy this condition?
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Context:
Determine the number of conjugacy classes in a non-abelian group of order $p^3$, where $p$ is a prime number.
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Context:
We write integers into an $n \times k$ table in such a way that every integer from 1 to $k$ appears in each of the $n$ rows. Let $S$ denote the largest of the $k$ column sums. For every $n$ and $k$, determine the smallest possible value of $S$!
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Context:
Glue two small cuboids, each with a length of 3cm, a width of 2cm, and a height of 1cm, into one large cuboid. Then, cut it into two small cuboids of the same size. The surface area of the final small cuboid could be at most how many square centimeters larger than the surface area of the original small cuboid.
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Context:
The positions of cyclists in the race are determined by the total time across all stages: the first place goes to the cyclist with the shortest total time, and the last place goes to the cyclist with the longest total time. There were 500 cyclists, the race consisted of 15 stages, and no cyclists had the same times either on individual stages or in total across all stages. Vasya finished in seventh place every time. What is the lowest position (i.e., position with the highest number) he could have taken?
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Context:
现有 7 把钥匙，用这些钥匙随机开锁，则 $D_{1}, D_{2}, D_{3}$ 这三把钥匙不能打开对应的锁的概率是 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Compute the double integral \( \iint\limits_{G} \sqrt{x^{2} - y^{2}} \, dx\, dy \), where the region \( G \) is defined as \( G=\{(x,y) \in \mathbb{R_{+}^{2}}: 0 \leq x + y \leq 2, 0 \leq x - y \leq 2\} \).
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Context:
Evaluate the double integral: \[ J = \int\limits_0^1 \int\limits_0^1 \dfrac{x-y}{(x+y)^3} \, dy \, dx. \]
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Context:
Find the maximum value of
$$
\int_{0}^{y} \sqrt{x^{4}+\left(y-y^{2}\right)^{2}} d x
$$
for $0 \leq y \leq 1$.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
$4 \cdot 35$ Find the smallest positive integer $n$, such that in any two-coloring of the complete graph $K$ with $n$ vertices, there always exist $m$ edges of the same color, which are pairwise non-adjacent.

 untranslated parts:
求最小正整数 $n$, 使在任何二染色的 $n$ 个顶点的完全图 $K$,中,总存在同色的 $m$ 条线段,它们两两之间没有公共端点。

translated:
Find the smallest positive integer $n$, such that in any two-coloring of the complete graph $K$ with $n$ vertices, there always exist $m$ edges of the same color, which are pairwise non-adjacent.
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Context:
Compute the number of ordered quadruples of positive integers $(a, b, c, d)$ such that
\[ a! \cdot b! \cdot c! \cdot d! = 24! \]
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Context:
On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise was $6:57\textsc{am}$, and the sunset as $8:15\textsc{pm}$. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set?
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Context:
A city bus moving at a certain speed covers a distance of 600.048 meters in 30 seconds. What is the speed of the bus in kmph?
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Context:
Find the derivative of the expression $b^tAc$ with respect to the matrix $A$ in matrix notation. What is the resulting derivative?
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Context:
Let $S = \{ A = (a_1, \ldots, a_s) \mid a_i = 0 \text{ or } 1, \ i = 1, \ldots, 8 \}$. For any two elements of $S$, $A = \{ a_1, \ldots, a_8 \}$ and $B = \{ b_1, \ldots, b_8 \}$. Let $d(A,B) = \sum_{i=1}^{8} |a_i - b_i|$. Call $d(A,B)$ the distance between $A$ and $B$. At most, how many elements can $S$ have such that the distance between any two sets is at least 5?
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Context:
Calculate the value of the Eisenstein series $G_2(\Lambda)$ for the lattice $\Lambda=\mathbb{Z}[i]$ in the complex plane when $k=1$. Provide your answer as a complex number.
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Context:
A line passing through the left focus $F_1$ of a hyperbola at an inclination of 30° intersects with the right branch of the hyperbola at point $P$. If the circle with the diameter $PF_1$ just passes through the right focus of the hyperbola, determine the eccentricity of the hyperbola.
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Context:
Find the minimum value of the expression \( \sqrt{x^2+1}+ \sqrt {4+(y-z)^2} + \sqrt{1+ (z-x)^2} + \sqrt{9+(10-y)^2} \) where \( x, y, z \in \mathbb{R} \).
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Context:
Find the number of automorphisms of the group $\mathbb{Z}_4 \oplus \mathbb{Z}_2$ that map the element $(1, 0)$ to an element of order 4 and the element $(0, 1)$ to an involution.
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Context:
In a building with 1,800 light bulbs, the lifetime of each bulb (in days) is modeled as a Poisson process with an expected lifetime of 200 days. Calculate the probability that at most 10 bulbs fail on any given day.
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Context:
Given a quadratic function \( f(x) \), define \( F(x) = \int_0^x f(t) \, dt \). Assume \( a \) is a positive number and \( F(x) \) has extreme values at \( x = -2a \) and \( x = 2a \). Find the local maximum value of the function \( \frac{F(x)}{F'(0)} \).
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Context:
397. What is the maximum number of queens that can be placed on an $8 \times 8$ chessboard so that each queen is attacked by at least one other queen?
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Context:
There are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $\gcd(a, b, c, d) = 77$ and $\operatorname{lcm}(a, b, c, d) = n$. What is the smallest possible value for $n$? Please provide your answer as an integer without any commas.
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Context:
The ant starts at point A and travels $1+\frac{1}{10}$ cm north, then $2+\frac{2}{10}$ cm west, then $3+\frac{3}{10}$ cm south, then $4+\frac{4}{10}$ cm east, then $5+\frac{5}{10}$ cm north, then $6+\frac{6}{10}$ cm west, and so on. After 1000 steps, the ant reaches point B. Find the straight-line distance between points A and B in centimeters. Record the square of the found distance as your answer.
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Context:
已知正方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 的棱长为 1, 以顶点 $A$ 为球心， $\frac{2 \sqrt{3}}{3}$ 为半径作一个球，则球面与正方体的表面相交所得到的曲线的长等于 $\frac{m \sqrt{k}}{n} \pi$，请给出 m + n + k 的值。
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Context:
Alex starts with a rooted tree with one vertex (the root). For a vertex  $v$ , let the size of the subtree of  $v$  be  $S(v)$ . Alex plays a game that lasts nine turns. At each turn, he randomly selects a vertex in the tree, and adds a child vertex to that vertex. After nine turns, he has ten total vertices. Alex selects one of these vertices at random (call the vertex  $v_1$ ). The expected value of  $S(v_1)$  is of the form  $\tfrac{m}{n}$  for relatively prime positive integers  $m, n$ . Find  $m+n$ .**Note:** In a rooted tree, the subtree of  $v$  consists of its indirect or direct descendants (including  $v$  itself).
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Context:
A shipment of 250 smartphones contains 3 different types: Type A, Type B, and Type C. There are 100 Type A smartphones with 30 defective ones, 80 Type B smartphones with 25 defective ones, and 70 Type C smartphones with 21 defective ones. If a customer buys two smartphones at random from the shipment, what is the approximate probability that both phones are defective?
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Context:
Assume the Generalized Continuum Hypothesis (GCH). Calculate \( \aleph_{\omega_1}^{\aleph_0} \).
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Context:
Determine the dimension of the ring \( \mathbb{K}[x,y,z]/(xy, xz, yz) \), where \( \mathbb{K} \) is a field.
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Context:
Bookman purchased some copies of a new book released recently, 10 of which are hardback and sold for $20 each, and the rest are paperback and sold for $10 each. If 14 copies were sold and the total value of the remaining books was $360, how many total copies did Bookman purchase initially?
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Context:
The perimeter of triangle $APM$ is $180$, and angle $PAM$ is a right angle. A circle of radius $20$ with center $O$ on line $\overline{AP}$ is drawn such that it is tangent to $\overline{AM}$ and $\overline{PM}$. Let $\overline{AM} = 2\overline{PM}$. Find the length of $OP$.
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Context:
已知数列 $\left\{a_{n}\right\}$ 满足: $a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right]\left(n \in \mathbf{N}^{*}\right)$ ，其中 $[x]$ 表示不超过实数 $x$ 的最大整数. 设 $C$ 为实数, 且对任意的正整数 $n$, 都有 $\sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C$, 则 $C$ 的最小值是 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.


Context:
Problem 11.8. In a $28 \times 35$ table, some $k$ cells are painted red, another $r-$ in pink, and the remaining $s-$ in blue. It is known that

- $k \geqslant r \geqslant s$
- each boundary cell has at least 2 neighbors of the same color;
- each non-boundary cell has at least 3 neighbors of the same color.

What is the smallest value that the quantity $k-s$ can take?

(A cell is called boundary if it is adjacent to the edge of the table. Neighbors are cells that share a common side.)
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Context:
Jessy plans to finish a 140-page book in a week. She initially decides to read twice daily at a speed of 10 pages an hour, spending 30 mins each time, every day of the week. By mid-week, she realizes she is behind. She decides to increase her reading speed to 15 pages per hour and add more reading time to her schedule. How many extra minutes should she read per day for the rest of the week to finish the book on time?
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Context:
Evaluate the limit $$\lim_{x \to 0}\frac{\sin(\tan x)-\tan(\sin x)}{x-\sin x}$$ using Taylor series or L'Hôpital's rule.
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Context:
Calculate the gradient with respect to the matrix $\pmb{A}$ of the function $\log(\det(\pmb{A}\pmb{X}))$, where both $\pmb{X}$ and $\pmb{A}$ are positive definite matrices.
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Context:
An individual contracted influenza and after two rounds of transmission, a total of 100 people were infected. Calculate the average number of people who contracted the flu from one infected person per round of transmission.
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Context:
Calculate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \), where \( S \) is the boundary of the solid \( E \) enclosed by the paraboloids \( z = x^2 + y^2 \) and \( z = 12 - 2x^2 - 2y^2 \), and \( \mathbf{F}(x, y, z) = \langle x^3 + y^2, 2yz + e^z, y^2 - z^2 \rangle \). Use the Divergence Theorem to find the value of the integral.
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Context:
In a circle, there are two mutually perpendicular chords $AB$ and $CD$. Determine the distance between the midpoint of segment $AD$ and the line $BC$, given that $AC=6$, $BC=5$, and $BD=3$. If necessary, round the answer to two decimal places.
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Context:
In the Tenth Kingdom, there are 17 islands, each with 119 inhabitants. The inhabitants are divided into two castes: knights, who always tell the truth, and liars, who always lie. During a population census, each person was first asked, "Not including yourself, are there an equal number of knights and liars on your island?" It turned out that on 7 islands, everyone answered "Yes," while on the rest, everyone answered "No." Then, each person was asked, "Is it true that, including yourself, people of your caste are less than half of the inhabitants of the island?" This time, on some 7 islands, everyone answered "No," while on the others, everyone answered "Yes." How many liars are there in the kingdom?
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Context:
Thirty girls - 13 in red dresses and 17 in blue dresses - were dancing around a Christmas tree. Afterwards, each was asked if the girl to her right was in a blue dress. It turned out that only those who stood between two girls in dresses of the same color answered correctly. How many girls could have answered affirmatively?
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Context:
Determine the number of 19th power residues modulo 229.
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Context:
There is a moving point \( M \) on the base \( A_{1}B_{1}C_{1}D_{1} \) of the cube \( ABCD - A_{1}B_{1}C_{1}D_{1} \), and \( BM \parallel \) plane \( ADC \). Find the maximum value of \( \tan \angle D_{1}MD \).
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Context:
Let \( r = \frac{1 + \sqrt{5}}{2} \). Calculate
\[
7 \arctan ^{2} r + 2 \arctan ^{2} r^{3} - \arctan ^{2} r^{5} =
\]
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Context:
## Task B-3.7.

In a right-angled triangle, the sum of the lengths of the legs and three times the length of the height on the hypotenuse is 12. The triangle rotates around the line on which the hypotenuse lies, thus forming a rotational body of maximum surface area. What is the volume of the resulting body?
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Context:
Determine the smallest integer \( n \) such that \( |e^x - p_n(x)| < 10^{-15} \) for all \( x \) in the interval \([-1, 1]\).
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Context:
Given a square \(ABCD\) with point \(P\) inside such that \(PA = 1\), \(PB = 2\), and \(PC = 3\), calculate the angle \(\widehat{APB}\).
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Context:
Let \( ABC \) be a triangle such that \(\angle CAB = 20^\circ\). Let \( D \) be the midpoint of segment \([AB]\). It is given that \(\angle CDB = 40^\circ\). What is the measure of \(\angle ABC\)?
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Context:
A point is randomly dropped onto the interval $[8, 13]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-2k-35\right)x^{2}+(3k-9)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$.
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Context:
已知函数$f\left( x \right)$，任取两个不相等的正数${{x}_{1}}$， ${{x2}}$，总有$\left[ f\left( {{x}_{1}} \right)-f\left( {{x2}} \right) \right]\left( {{x}_{1}}-{{x2}} \right)>0$，对于任意的$x>0$，总有$f\left[ f\left( x \right)-\text{ln}x \right]=1$，若$g\left( x \right)=f'\left( x \right)+f\left( x \right)-{{m}^{2}}+m$有两个不同的零点，则正实数$m$的取值范围为__________．
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Context:
Several sets of prime numbers, such as $\{7, 83, 421, 659\}$, use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?
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Context:
During summer break, 644997 kids from Lawrence county stayed home and the other 893835 kids went to camp. An additional 78 kids from outside the county attended the camp. How many kids are in Lawrence county?
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Context:
How many positive integers $x$ less than $10000$ are there such that $2^x-x^2$ is divisible by $7$ ?
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Context:
A fair six-sided die is rolled repeatedly until an odd number appears. Calculate the probability that each even number (2, 4, 6) appears at least once before any odd number appears. Express your answer as a fraction.
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Context:
Compute the limit $$ \lim_{x\to 0}\frac{e^{f(x)}-e^x}{2x-\sin(f(2x))}, $$ where $f:\mathbb{R}\longrightarrow\mathbb{R}$ is a function defined as $f(x) = x - x^3 + o(x^3)$.
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Context:
Given that $m > 0$ and for any $x \in (1, +\infty)$, the inequality $2{e}^{2mx}-\frac{\ln x}{m} \geq 0$ always holds, find the minimum value of the real number $m$. Express your answer as a single real number.
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Context:
Jack Sparrow needed to distribute 150 piastres into 10 purses. After putting some amount of piastres in the first purse, he placed more in each subsequent purse than in the previous one. As a result, the number of piastres in the first purse was not less than half the number of piastres in the last purse. How many piastres are in the 6th purse?
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Context:
Alice and Bob are playing a game with a string of 2015 beads. The player whose turn it is cuts the string between two beads. The other player decides which of the two pieces will be used for the rest of the game, and the other piece is discarded.

Alice starts, and they take turns cutting the string. The player who has to make a move when only one bead is left loses the game since no more cuts can be made.

Who has a winning strategy?
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Context:
Dad's birthday is May 1st, and Chunchun's birthday is July 1st. Starting from December 26, 2012 (Day 1), until the 2013th day, how many birthdays have Dad and Chunchun celebrated in total?
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Context:
For positive numbers $a$ and $b$ such that $2a^2 + 7ab + 3b^2 = 7$, find the maximum value of $a + \sqrt{3ab} + 2b$. Provide your answer as a single number.
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Context:
Given three composite numbers \( A, B, C \) that are pairwise coprime and \( A \times B \times C = 11011 \times 28 \). What is the maximum value of \( A + B + C \)?
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Context:
已知椭圆 $\Gamma: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1, F$ 为 $\Gamma$ 的右焦点。过 $F$ 作直线交 $\Gamma$ 于 $M 、 N$ 两点，直线 $l$ 过点 $F$ 且与直线 $M N$ 垂直, 并交直线 $x=4$ 于点 $P$, 直线 $O P$ 交 $\Gamma$ 于 $A 、 B$ 两点. 设直线 $M P 、 M N 、 M A 、 M B$ 的斜率依次为 $m 、 n 、 p$ 、 $q$ (四条直线的斜率均存在且均不为 0 ). 记 $m+n=s, p+q=t, m n=r, s t=f(r)$. 求 $f(\frac{3}{4})$。
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Context:
Find the smallest positive integer $n$ for which the polynomial \[x^n-x^{n-1}-x^{n-2}-\cdots -x-1\] has a real root greater than $1.999$.
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Context:
$\triangle ABC$ is a triangle and $D,E,F$ are points on $BC$, $CA$, $AB$ respectively. It is given that $BF=BD$, $CD=CE$ and $\angle BAC=48^{\circ}$. If the answer is $x^\circ$, find the value of $x$ for the angle $\angle EDF$.
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Context:
Let $P$ be the parabola in the plane determined by the equation $y = x^2$. Suppose a circle $C$ in the plane intersects $P$ at four distinct points. If three of these points are $(-28, 784)$, $(-2, 4)$, and $(13, 169)$, find the sum of the distances from the focus of $P$ to all four of the intersection points.
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Context:
Example 6 Find the simplest fraction (with the smallest denominator) that approximates $\sqrt{8}$, with an error $\leqslant 10^{-6}$.
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Context:
If the simplest quadratic radical $2\sqrt{4m-1}$ is of the same type as $\sqrt{2+3m}$, then $m=$____.
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Context:
设四边形$ABCD$中$\overrightarrow{DC}=\frac{1}{2}\overrightarrow{AB}$且$|\overrightarrow{AD}|=|\overrightarrow{BC}|$，则这个四边形是_______．
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Context:
Find the smallest $x \in \mathbb{N}$ for which $\frac{7x^{25}-10}{83}$ is an integer.
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Context:
Given that p and q are natural numbers that do not end in zero and their product is always a power of 10, find the last digit that p - q cannot be when p > q. Express your answer as a single digit.
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Context:
At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. How many pairs are there in which both students are wearing yellow shirts?
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Context:
Determine the maximum distance between two points within the truncated cone defined by $2x^2 + 2y^2 \leq z^2$ for $z \in [1,2]$.
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Context:
Given the real numbers \( x, y, \) and \( z \) that satisfy \( xy \neq 0 \) and \( x+y \neq 0 \), find the minimum value of the expression
$$
\left(\frac{y+z}{x}+2\right)^{2}+\left(\frac{z}{y}+2\right)^{2}+\left(\frac{z}{x+y}-1\right)^{2}.
$$
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Context:
If 4 people stand in a row again, and exactly one person stands in their original position, calculate the total number of different ways to stand. Express your answer as a single integer.
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Context:
The square-shaped cards are regularly arranged without gaps to form a large square. If the number of cards laid at the perimeter was 240, how many total cards in total were used?
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Context:
Let $ABC$ be a triangle with $AB=5$, $BC=6$, and $AC=7$. Let its orthocenter be $H$ and the feet of the altitudes from $A, B, C$ to the opposite sides be $D, E, F$ respectively. Let the line $DF$ intersect the circumcircle of $AHF$ again at $X$. Find the length of $EX$.
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Context:
Mustafa bought a big rug. The seller measured the rug with a ruler that was supposed to measure one meter. As it turned out to be $30$ meters long by $20$ meters wide, he charged Rs $120,000$. When Mustafa arrived home, he measured the rug again and realized that the seller had overcharged him by $9,408$ Rs. How many centimeters long is the ruler used by the seller?
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Context:
$n ^ { 2 } ( n \geq 4 )$个正数排列$n$行$n$列：

$a_{11}$ $a_{12}$ $a_{13}$ $a _ { 1 4 }$ $\cdots a _ { 1 n } ,$

$a _ { 2 1 }$ $a _ { 2 2 }$ $a _ { 2 3 }$ $a _ { 2 4 } \cdots a _ { 2 n } ,$

$a _ { 3 1 }$ $a _ { 3 2 }$ $a _ { 3 3 }$ $a _ { 3 4 } \cdots a _ { 3 n } ,$

$a _ { 4 1 }$ $a _ { 4 2 }$ $a _ { 4 3 }$ $a _ { 4 4 } \cdots a _ { 4 n } ,$

·.-

$a _ { n _ { 1 } } a _ { n _ { 2 } } a _ { n _ { 3 } } a _ { n _ { 4 } } \cdots a _ { n _ { n } }$

其中每一行的数成等差数列，每一列的数成等比数列，并且所有公比相等.已知

$a _ { 2 4 } = 1 , a _ { 4 2 } = \frac { 1 } { 8 }$ $, a _ { 4 3 } = \frac { 3 } { 1 6 } ,$

求 $a _ { 1 1 } + a _ { 2 2 } + \cdots + a _ { n n }$的值.
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Context:
In a square, point \( M \) is located on side \( CD \) such that \( CM:MD = 1:3 \). Line \( AM \) intersects the circle circumscribed about the square at point \( E \). Given that the area of triangle \( ACE \) is 14, determine the side length of the square.
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Context:
A ship sails eastward at a speed of 15 km/h. At point A, the ship observes a lighthouse B at an angle of 60° northeast. After sailing for 4 hours, the ship reaches point C, where it observes the lighthouse at an angle of 30° northeast. At this time, the distance between the ship and the lighthouse is ______ km.
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Context:
Approximate the function $f(x) = e^x\cos x$ using a Taylor polynomial centered at $x_0 = 0$. Determine the degree of the polynomial such that the remainder in the interval $[-\frac{1}{100},\frac{1}{100}]$ is less than $10^{-8}$.

Given the function's derivatives:
- $f'(x) = e^x(\cos x - \sin x)$
- $f''(x) = -2e^x\sin x$
- $f'''(x) = -2e^x(\sin x + \cos x)$

How many terms of the Taylor series are needed to meet the error condition?
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Context:
Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\frac{1}{2}$ is $\frac{a-b\pi}{c}$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. Find the value of $a+b+c$.
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Context:
Boys and girls. Nine boys and three girls decided to share their pocket money equally. Each boy gave the same amount to each girl, and each girl gave the same amount (but a different one) to each boy. After this, all children ended up with an equal amount of money. What is the smallest amount that each of them could have had initially?
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Context:
Let $\varphi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Let $S$ be the set of positive integers $n$ such that $\frac{2 n}{\varphi(n)}$ is an integer. Compute the sum $\sum_{n \in S} \frac{1}{n}$.
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Context:
Find all positive integers $n \geq 1$ such that there exists a pair $(a, b)$ of positive integers, such that $a^2 + b + 3$ is not divisible by the cube of any prime, and
$$ n = \frac{ab + 3b + 8}{a^2 + b + 3}. $$
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Context:
Evaluate the double integral \( \int_0^2 \left( \int_{\frac{y}{2}}^1 e^{x^2} \, dx \right) dy. \)
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Context:
2015 people are sitting around a round table, each of whom is either a knight or a liar. Knights always tell the truth, liars always lie. They were each given a card, with a unique number written on it. After looking at the cards of their neighbors, each person at the table said: "My number is greater than that of each of my two neighbors." After this, $k$ of the people said: "My number is less than that of each of my two neighbors." What is the maximum possible value of $k?
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Context:
Write the integers from 1 to 6 on the six faces of a cube so that any pair of consecutive numbers - including the pair 6 and 1 - are on adjacent faces. How many different arrangements are possible? (Two arrangements are not considered different if one can be transformed into the other by some symmetry of the cube - such as rotation or reflection on a plane through the center - or by replacing each number with the next one, with 6 replaced by 1.)
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Context:
Prince Zrychlený invited Princess Zpomalená to his castle. When she took too long, he set out to meet her. After two days of travel, he met her at one fifth of her journey. Together, they continued their journey twice as fast as when the princess was traveling alone. They arrived at Prince's castle the second Saturday after their meeting. Which day did they meet, if the princess set out from her castle on Friday?

(M. Dillingerová)
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Context:
Evaluate the integral \( \iint_{R} \lfloor x+y \rfloor \, \mathrm{d}x\, \mathrm{d}y \) where \( R= \{(x,y)| 1\leq x\leq 3, 2\leq y\leq 5\} \).
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Context:
若存在实数 $r$, 使得圆周 $x^{2}+y^{2}=r^{2}$ 上恰好有 $n$ 个整点, 则 $n$ 可以等于多少？请给出所有可能的n值之和。
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Context:
Two equilateral triangles with perimeters of 12 and 15 are positioned such that their sides are respectively parallel. Find the perimeter of the resulting hexagon.
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Context:
In a plane, a set of points is given such that no three points are collinear. Each point is colored either red, white, or green. There are 213 lines connecting points of different colors and 112 lines connecting points of the same color. Among the triangles formed by these points, 540 triangles have vertices of all different colors, and 612 triangles have exactly two red vertices. Determine the number of triangles with exactly two white vertices.
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Context:
William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\circ$ decline for $1$ kilometer. If William travels at $8$ kilometers per hour during uphill $20^\circ$ sections, $16$ kilometers per hour during flat sections, and $20$ kilometers per hour during downhill $20^\circ$ sections, find the closest integer to the number of minutes it takes William to get to school and back.
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Context:
Find an element in $\mathbb{Z}[\sqrt{-17}]$ that can be expressed both as a product of two irreducible elements and as a product of three irreducible elements.
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Context:
If the mutually prime integers \(A\), \(B\), and \(C\) satisfy \( A \log_{200} 5 + B \log_{200} 2 = C \), then calculate \( A + B + C \). Express your answer as a single integer.
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Context:
$\bigcirc \bigcirc \div \square=14 \cdots 2$, how many ways are there to fill the square?
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Context:
Evaluate the limit: $$\lim_{x\to0}\frac{x\sin(\sin x) - \sin^2x}{x^6}$$
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Context:
Given the function $f(x) = \cos(a\sin x) - \sin(b\cos x)$, derive the range of $a^2+b^2$ such that $f(x)$ has no zeros. Express your answer as an interval.
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Context:
Twenty pairs of integers are formed using each of the integers \( 1, 2, 3, \ldots, 40 \) once. The positive difference between the integers in each pair is 1 or 3. If the resulting differences are added together, what is the greatest possible sum?
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Context:
Analyze the following relationships to determine the number of distinct, unambiguous functions of y with respect to x:

1. x - y = 0
2. y^2 = x
3. |y| = 2x
4. y^2 = x^2
5. y = 3 - x
6. y = 2x^2 - 1
7. y = 3/x

Express your answer as a whole number representing the count of such functions.
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Context:
If $(1-2)^{9}=a_{9}x^{9}+a_{8}x^{8}+\ldots+a_{1}x+a_{0}$, then the sum of $a_1+a_2+\ldots+a$ is \_\_\_\_\_\_.
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Context:
A structure with a design life of 50 years is planned for a site where high intensity earthquakes may occur with a return period of 100 years. The structure is designed to have a 0.98 probability of not suffering damage within its design life. Assuming that the occurrence of high intensity earthquakes at the site follows a Bernoulli sequence and that damage effects between earthquakes are statistically independent, what is the probability of damage to the structure under a single earthquake?
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Context:
Find the smallest two-digit number \( N \) such that the sum of digits of \( 10^N - N \) is divisible by 170.
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Context:
Given a quadratic equation $ax^2+bx+c=0$ (where $a>0$) has no real roots, find the solution set for $ax^2+bx+c<0$.
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Context:
Suppose $x, y \in \mathbb{Z}$ satisfy the equation:
\[
y^4 + 4y^3 + 28y + 8x^3 + 6y^2 + 32x + 1 = (x^2 - y^2)(x^2 + y^2 + 24).
\]
Find the sum of all possible values of $|xy|$.
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Context:
For positive integers \( a \) and \( N \) ,let
 \( r\left( {a,N}\right)  \in  \{ 0,1,\ldots ,N - 1\} \) denote the remainder of \( a \)
 when divided by \( N \) . Determine the number of positive
 integers \( n \leq  {1000000} \) for which
 \( r\left( {n,{1000}}\right)  > r\left( {n,{1001}}\right) \) .
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Context:
Determine the number of distinct ways to completely cover a $10 \times 10$ checkerboard using exactly 25 T-shaped tetrominoes.
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Context:
1 kilogram of soybeans can be made into 3 kilograms of tofu. To produce 1 kilogram of soybean oil, 6 kilograms of soybeans are required. Tofu costs 3 yuan per kilogram, and soybean oil costs 15 yuan per kilogram. A batch of soybeans weighs 460 kilograms, and the total sales revenue from the tofu and soybean oil produced from this batch is 1800 yuan. How many kilograms of this batch of soybeans were made into soybean oil?
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Context:
Given the initial position of a submarine at \((-230, 75)\) and a convoy at \((0, 0)\), with the submarine having a maximum speed of 30 km/h and the convoy traveling southwest at 18.5 km/h, determine the direction the submarine should travel to intercept the convoy most quickly. Assume the units are in kilometers for distance and kilometers per hour for speed.
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Context:
求$1 1 ^ { 1 2 ^ { 1 3 } }$的十位数字（其中，$1 1 ^ { 1 2 ^ { 1 3 } }$表示11的1213次幂）.
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Context:
A line passing through the focus \( F \) of the parabola \( y^2 = 2px \) intersects the parabola at points \( A \) and \( B \). The projections of points \( A \) and \( B \) onto the directrix of the parabola are \( A_1 \) and \( B_1 \), respectively. Find the angle \( \angle A_1 F B_1 \).
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Context:
Calculate the flux of the vector field \(\mathbf{a} = y^2 \mathbf{j} + z \mathbf{k}\) through the surface defined by \(z = x^2 + y^2\), bounded by the plane \(z=2\). Assume the normal vector is outward with respect to the region enclosed by the paraboloid.
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Context:
Evaluate the limit $\lim_{x\to 0} \dfrac{\ln(1+ax+x^2)}{\sqrt{x+b}-1}$ given that $a,b\in\mathbb{R}$ and $a+b=2$.
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Context:
Calculate the area of the region defined by the inequalities: \(0 \leq x \leq \sqrt{y}\), \(0 \leq y \leq \sqrt{x}\), and \(x+y \leq \frac{3}{4}\).
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Context:
Consider the graph in 3-space of $0=xyz(x+y)(y+z)(z+x)(x-y)(y-z)(z-x)$. This graph divides 3-space into $N$ connected regions. What is $N$?
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Context:
7. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Find the number of elements in the set $\left\{n \left\lvert\, n=\left[\frac{k^{2}}{2005}\right]\right., 1 \leqslant k \leqslant 2004, k \in \mathbf{N}\right\}$.
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Context:
Given $A=\left\{ x\mid 2x^{2}-3x+1\leqslant 0 \right\}$ and $B=\left\{ x^{2}-(2a+1)x+a(a+1)\leqslant 0 \right\}$, if $(C_{R}B\cap A=\varnothing)$, find the range of values for the real number $a$.
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Context:
A semicircle has diameter $\overline{AD}$ with $AD = 30$. Points $B$ and $C$ lie on $\overline{AD}$, and points $E$ and $F$ lie on the arc of the semicircle. The two right triangles $\vartriangle BCF$ and $\vartriangle CDE$ are congruent. The area of $\vartriangle BCF$ is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m + n$.
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Context:
Find the minimum of $\sum\limits_{k=0}^{40} \left(x+\frac{k}{2}\right)^2$ where $x$ is a real numbers
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Context:
Arrange the numbers 1, 2, 3, 4, 5, 6 in a row, denoting the $i$-th number as $a_i$ ($i=1,2,\ldots,6$). If $a_1 \neq 1$, $a_3 \neq 3$, $a_5 \neq 5$, and $a_1 < a_3 < a_5$, then the number of different arrangements is ___ (answer with a number).
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Context:
There were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Determine the volume, in cubic light years, of the set of all possible locations for a base such that the sum of squares of the distances from the base to each of the stations does not exceed 15 square light years.
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Context:
What is the largest number of integers that can be selected from the natural numbers from 1 to 3000 such that the difference between any two of them is different from 1, 4, and 5?
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Context:
At a certain university, 70% of the professors are women, and 70% of the professors are tenured. If a certain percentage of the professors are women, tenured, or both, and 50% of the men are tenured, what percent of the professors are women, tenured, or both?
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Context:
Determine the cardinality of the set of all infinite binary sequences that do not contain the subsequence 101.
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Context:
John uses the bathroom every certain minutes. During a 2.5-hour movie, he uses the bathroom 3 times. How many minutes are there between each bathroom visit?
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Context:
A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that $1$ belongs to $\mathcal{S}$ and that $2002$ is the largest element of $\mathcal{S},$ what is the greatest number of elements that $\mathcal{S}$ can have?
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Context:
Find the number of integer solutions \((x, y)\) to the equation \(23107\ | \ 5x^2 + 35y^2 + 2023\) such that \(0 \leq x, y \leq 23106.\)
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Context:
A rectangle measures 6 meters by 10 meters. Drawn on each side of the rectangle is a semicircle that has the endpoints of its diameter on the vertices of the rectangle. What percent larger is the area of the large semicircles than the area of the small semicircles? Express your answer to the nearest whole number.
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Context:
The new dormitory manager received 10 keys to open 10 rooms. He knows that each key can only open one of the rooms, but he does not know which key opens which room. Now, to open all the locked rooms, he needs to try at most     times.
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Context:
Given a triangular prism \( P-ABC \) where the three face angles at the top vertex are all \( 60^\circ \), i.e., \( \angle APB = \angle BPC = \angle CPA = 60^\circ \), and the areas of the three lateral faces are \( \frac{\sqrt{3}}{2}, 2, \text{and}\, 1 \), find the volume of the triangular prism.
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Context:
Kelvin the frog is hopping on the coordinate plane $\mathbb{R}^{2}$. He starts at the origin, and every second, he hops one unit to the right, left, up, or down, such that he always remains in the first quadrant $\{(x, y): x \geq 0, y \geq 0\}$. In how many ways can Kelvin make his first 14 jumps such that his 14th jump lands at the origin?
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Context:
The angles in a particular triangle are in arithmetic progression, and the side lengths are $4,5,x$. The sum of the possible values of $x$ equals $a+\sqrt{b}+\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is the value of $a+b+c$?
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Context:
Given a graph with N vertices (3 ≤ N ≤ 1000) and M edges, determine the minimum number of additional vertices that need to be added to ensure that every vertex is either included in the new set of vertices or is adjacent to a vertex in the new set.
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Context:
There are 4 pieces of part $A$ weighing 5 tons each, 6 pieces of part $B$ weighing 4 tons each, 11 pieces of part $C$ weighing 3 tons each, and 7 pieces of part $D$ weighing 1 ton each. If all the parts are to be transported at once, what is the minimum number of trucks, each with a capacity of 6 tons, required?
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Context:
Professor Rackbrain recently asked his young friends to find all five-digit perfect squares for which the sum of the numbers formed by the first two digits and the last two digits equals a perfect cube. For example, if we take the square of the number 141, which is equal to 19881, and add 81 to 19, we get 100 – a number that unfortunately is not a perfect cube.

How many such solutions exist?
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Context:
Suppose that $x$ and $y$ are complex numbers such that $x + y = 1$ and $x^{20} + y^{20} = 20$. Find the sum of all possible values of $x^2 + y^2$.
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Context:
2. 103 Use 94 bricks, each of size $4 \times 10 \times 19$, to stack into a tower 94 bricks high, where each brick can be oriented to provide a height of 4, 10, or 19. If all 94 bricks are used, how many different heights can the tower be (in inches)?
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Context:
Let $\Gamma$ be the maximum possible value of $a+3b+9c$ among all triples $(a,b,c)$ of positive real numbers such that
\[ \log_{30}(a+b+c) = \log_{8}(3a) = \log_{27} (3b) = \log_{125} (3c) .\]
If $\Gamma = \frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers, then find $p+q$.
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Context:
Find the closed form solution for the sequence $$S_n = \frac{x}{1-x^2}+\frac{x^2}{1-x^4}+ \ldots + \frac{x^{2^{n-1}}}{1-x^{2^{n}}}$$ given that $S_n$ follows the recurrence relation $S_n = \frac{S_{n-1}}{1-x^{2^n}} + \frac{x^{2^n}}{1-x^{2^{n+1}}}$.
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Context:
An Ultraman is fighting a group of monsters. It is known that Ultraman has one head and two legs. Initially, each monster has two heads and five legs. During the battle, some monsters split, with each splitting monster creating two new monsters, each with one head and six legs (they cannot split again). At a certain moment, there are 21 heads and 73 legs on the battlefield. How many monsters are there at this moment?
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Context:
Three points are placed independently and at random in a unit square. What is the expected value of the area of the triangle formed by these three points?
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Context:
The whole arc of a circle with radius $R$ is divided into four large and four small parts, which alternate one after another. The large part is twice as long as the small part. Determine the area of the octagon whose vertices are the division points of the arc of the circle.
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Context:
The sum of two natural numbers and their greatest common divisor is equal to their least common multiple. Determine the ratio of the two numbers.
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Context:
Rebecca drinks soda daily, and her consumption varies between 1/4 and 1/2 of a bottle each day. She recently bought a combination of two different sizes of soda bottles: five 6-packs of 500mL sodas and two 12-packs of 1L sodas from a store.

If Rebecca consumes an equal amount of the two different sized sodas and her average daily consumption fluctuates within the given range, approximately how many bottles of soda will she have left after five weeks?
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Context:
What is the maximum number of diagonals of a regular  $12$ -gon which can be selected such that no two of the chosen diagonals are perpendicular?

Note: sides are not diagonals and diagonals which intersect outside the  $12$ -gon at right angles are still considered perpendicular.

*2018 CCA Math Bonanza Tiebreaker Round #1*
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Context:
Determine the maximum number of fleas that can be placed on a $10 \times 10$ chessboard such that each flea jumps to an adjacent square every minute in one of four directions (up, down, left, right) and reverses direction when it cannot continue. During one hour, no two fleas should ever occupy the same square.
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Context:
Eleven girls are standing around a circle. A ball is thrown clockwise around the circle. The first girl, Ami, starts with the ball, skips the next three girls and throws to the fifth girl, who then skips the next three girls and throws the ball to the ninth girl. If the throwing pattern continues, including Ami's initial throw, how many total throws are necessary for the ball to return to Ami?
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Context:
Find the total number of five-digit numbers \(abcde\) that satisfy the following conditions: \( a < b, \, b > c > d, \, d < e, \, a > d, \, b > e. \)
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Context:
A rectangular table of size \( x \) cm \( \times 80 \) cm is covered with identical sheets of paper measuring 5 cm \( \times 8 \) cm. The first sheet is placed at the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet touches the top-right corner. What is the length \( x \) in centimeters?
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Context:
There are 2n people seated around a circular table, and m cookies are distributed among them. The cookies may be passed around under the following rules:
- Each person may only pass cookies to his or her neighbours.
- Each time someone passes a cookie, he or she must also eat a cookie.

Let A be one of these people. Find the least m such that no matter how m cookies are distributed to begin with, there is a strategy to pass cookies so that A receives at least one cookie.
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Context:
George went to a movie theater to see a movie. He paid $16 for the ticket and decided to buy some snacks for himself and a friend. The nachos were half the price of the ticket, and he decided to buy 2 servings. The small popcorn was 25% cheaper than the nachos, and he bought 3 bags. The soda was 75% of the popcorn price, and they got 4 sodas in total. Additionally, they purchased a combo deal that included 2 boxes of candy at a fixed price of $7. There was also a 10% discount on the total food bill, except for the combo deal, and a 5% sales tax applied to the entire purchase. How much did George pay in total for the visit to the movie theater?
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Context:
Yasmin's bank account had an initial balance of X dollars. On Monday, she withdrew half of her balance, and on Tuesday, she deposited $30 into her account. On Wednesday, she deposited a $50 birthday check from her grandmother. This check was worth a quarter of her new balance after the check was added. Finally, on Thursday, Yasmin withdrew $20 from her account. After all these transactions, her final balance was $160. Find the initial balance (X) of Yasmin's account in dollars before any transactions took place.
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Context:
A function $f\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ is linear in each of the $x_{i}$ and $f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\frac{1}{x_{1} x_{2} \cdots x_{n}}$ when $x_{i} \in\{3,4\}$ for all $i$. In terms of $n$, what is $f(5,5, \ldots, 5)$?
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Context:
3. For any four-element subset $\mathrm{P}$ of the set $\{1,2,3, \ldots, 12\}$, denote

$$
\mathrm{Q}=\{3 x: x \in \mathrm{P}\} \quad \text { and } \quad \mathrm{R}=\{4 x: x \in \mathrm{P}\} \text {. }
$$

Determine the number of such sets $P$ for which the numbers from $P, Q, R$ give all possible non-zero remainders when divided by 13.

(Jaromír Šimša)
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Context:
Teams A and B play a basketball match. The rule is that if one team wins 4 games, that team wins the match and the game ends (best of seven). The probability of teams A and B winning each game is 1/2. Let ξ be the number of games required. Calculate Eξ.
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Context:
How many sequences \( a_{1}, a_{2}, \ldots, a_{8} \) of zeroes and ones have \( a_{1}a_{2} + a_{2}a_{3} + \cdots + a_{7}a_{8} = 5 \)?
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Context:
Given the parabola equation y^2 = 8x and a line passing through the focus F(2,0) with a slope angle of 60 degrees, calculate the length of the chord intercepted by the parabola. Express your answer as a single fraction.
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Context:
Mac needs to fill 2012 barrels with apple cider. He starts with 0 energy. Each minute, he can either rest to gain 1 energy or use his current energy. If he has \( n \) energy, he can expend \( k \) energy \((0 \leq k \leq n)\) to fill up to \( n(k+1) \) barrels. What is the minimal number of minutes required for Mac to fill all the barrels?
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Context:
George Arfken's book: Mathematical Methods for Physicists has the following problem in a chapter on contour integration: $\displaystyle \int_0^{\infty} \dfrac{(\log x)^2}{x^2 + 1} dx$. Their suggestion is to make the substitution $x \rightarrow z=e^t$. What is the value of the integral?
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Context:
Let $C$ be a circumference of radius $a$ in the plane $2x+2y+z=4$, centered at the point (1,2,-2). Given the vector field $F(x,y,z)=(y-x,z-x,x-y)$, determine the value of $a$ such that the line integral $\oint_C F \cdot dr = -\frac{8\pi}{3}$. Use Stokes' Theorem to solve the problem.
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Context:
Consider the recursive function defined by \( C_n = C_{n-1} + iC_{n-2} \), with initial conditions \( C_1 = 1 \) and \( C_2 = 1 \). If \( C_{10} \) is expressed in the form \( a + bi \), find the value of \( b \).
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Context:
In three 90-minute training matches, seven players \( \mathrm{A}_{1}, \mathrm{A}_{2}, \ldots, \mathrm{A}_{7} \) from country \( F \) are observed. At any moment, exactly one player is on the field. Players \( \mathrm{A}_{1}, \mathrm{A}_{2}, \mathrm{A}_{3}, \mathrm{A}_{4} \) have total playing times divisible by 7, and players \( \mathrm{A}_{5}, \mathrm{A}_{6}, \mathrm{A}_{7} \) have total playing times divisible by 13. With unlimited substitutions, how many different total playing time combinations are possible for each player?
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Context:
Let  $P$  be a  $10$ -degree monic polynomial with roots  $r_1, r_2, . . . , r_{10} \ne $  and let  $Q$  be a  $45$ -degree monic polynomial with roots  $\frac{1}{r_i}+\frac{1}{r_j}-\frac{1}{r_ir_j}$  where  $i < j$  and  $i, j \in \{1, ... , 10\}$ . If  $P(0) = Q(1) = 2$ , then  $\log_2 (|P(1)|)$  can be written as  $a/b$  for relatively prime integers  $a, b$ . Find  $a + b$ .
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Context:
Let \( n \) be an integer and
$$
m = (n-1001)(n-2001)(n-2002)(n-3001)(n-3002)(n-3003).
$$

Given that \( m \) is positive, find the minimum number of digits of \( m \).
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Context:
In a trapezoid, the diagonals intersect at a right angle, and one of them is equal to the midsegment. Determine the angle that this diagonal forms with the bases of the trapezoid.
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Context:
Given that $||\vec{a}+\vec{b}||=||\vec{a}-\vec{b}||=2||\vec{a}||$, find the angle between the vectors $(\vec{a}-\vec{b})$ and $\vec{b}$. Express your answer in radians, as a fraction of $\pi$.
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Context:
The volume of the solid generated by rotating the circle $x^2 + (y + 1)^2 = 3$ around the line $y = kx - 1$ for one complete revolution is what?
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Context:
A checkered paper square of size \( 8 \times 8 \) was folded several times along the lines of the grid to form a small \( 1 \times 1 \) square. This small square was then cut along a segment connecting the midpoints of two opposite sides of the square. Into how many parts could the original square break apart by this cut?
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Context:
In the ring $R=\mathbb{Z}_5[x]$, consider the element $b=(x-4)(x-3)^2$. Let $A$ be the set of elements $a$ in $R$ such that the equation $at=b$ has a solution $t\in R$. Determine the number of elements in the set $A$. \(\text{Express your answer as an integer.}\)
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Context:
A trader starts with 500 bags of rice in stock. On the first day, she sells 25% of her stock and restocks twice the number of sold bags. On the second day, she sells 30% of her new stock and restocks 50% more rice bags than she sold on that day. By the end of the third day, the total number of bags she has left is 20% less than her stock after the second day restocking. How many bags of rice did she sell on the third day?
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Context:
What is the units digit when $5^4$ is multiplied out?
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Context:
Given a vector \( \vec{x} \in \mathbb{R}^n \) (or \( \mathbb{C}^n \)), determine the maximum possible rank of a matrix \( A \in \mathbb{R}^{m \times n} \) (or \( \mathbb{C}^{m \times n} \)) such that \( A\vec{x} = \vec{0} \).
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Context:
A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically displays the result. Starting with the display "1," what is the fewest number of keystrokes you would need to reach "200"? Express your answer as a single integer.
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Context:
Evaluate the flux of the vector field \( \vec{F} = -9\hat{j} - 3\hat{k} \) across the surface \( z = y \) bounded by the sphere \( x^2 + y^2 + z^2 = 16 \).
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Context:
A train that is 456 meters long is running at a speed of 43 km/hour. In what time will it pass a bridge that is 254 meters long?
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Context:
If $g(x)$ is a cubic polynomial with roots $p, q, r$, and its derivative $g'(x)$ divides $g(2x)$, find the value of $C$ such that $Cpq = 16r^2$. Given $g(x) = A(x-p)(x-q)(x-r)$ and $g(2x) = A(2x-p)(2x-q)(2x-r)$, determine the value of $C$.
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Context:
Given that a vowel (v) is followed by a consonant (c) 75% of the time and a consonant is followed by a vowel 65% of the time, determine the most likely sequence of vowels and consonants when three successive letters are chosen.
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Context:
For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1, \sqrt{2k})$. Evaluate
\[
\sum_{k=1}^\infty (-1)^{k-1} \frac{A(k)}{k}.
\]
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Context:
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube?
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Context:
Given the circle \( M: 2x^{2} + 2y^{2} - 8x - 8y - 1 = 0 \), and the line \( l: x + y - 9 = 0 \), let \( A \) be a point on \( l \). Construct \(\triangle ABC\) such that \(\angle BAC = \frac{\pi}{4}\), with side \( AB \) passing through the center of the circle \( M \) and points \( B \) and \( C \) lying on the circle \( M \). What is the range of the y-coordinate of point \( A \)?
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Context:
If the equation $\sin ^{2}x+2\sin x-1+m=0$ with respect to $x$ has a solution, then the range of the real number $m$ is \_\_\_\_\_\_.
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Context:
In a hexagon, how many different triangles whose vertices are vertices of the hexagon can be formed by drawing diagonals?
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Context:
If $3(\sin ^{5}\theta +\cos ^{5}2\theta ) \gt 5(\sin ^{3}\theta +\cos ^{3}2\theta ),\theta \in \left[0,2\pi \right)$, then the range of $\theta$ is ______.
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Context:
在 $\triangle A B C$ 中, 已知 $\overrightarrow{A B} \cdot \overrightarrow{A C}=2 \overrightarrow{B C} \cdot \overrightarrow{B A}=3 \overrightarrow{C A} \cdot \overrightarrow{C B}$, 则 $\triangle A B C$ 最大角的正弦值为。原始答案为$\frac{m\sqrt{n}}{k}$的格式，其中$m,n,k$均为正整数，请给出$m + n+k$的值。
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Context:
Let \( S \) be the part of the sphere \( x^2+y^2+z^2=25 \) which lies above the plane \( z=4 \). Orient \( S \) by upward pointing normal vectors. Evaluate \( \int\!\!\!\int_{S}\text{curl}F\,dS \) where \( F=yze^xi+(x+ze^x)j+yzk \). (Hint: Consider a different surface.)
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Context:
A pack of 20 cards is shuffled using the "interlacing" method, where the original order 1, 2, 3, ..., 20 becomes 1, 11, 2, 12, ..., 10, 20 after one shuffle. Determine how many times this shuffle must be repeated for the cards to return to their original order.
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Context:
Example 6.21. On average, $85 \%$ of the items coming off the conveyor are of the first grade. How many items need to be taken so that with a probability of 0.997, the deviation of the frequency of first-grade items from 0.85 in absolute value does not exceed 0.01?
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Context:
A subset $M$ of $\{1, 2, . . . , 2006\}$ has the property that for any three elements $x, y, z$ of $M$ with $x < y < z$, $x+ y$ does not divide $z$. Determine the largest possible size of $M$.
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Context:
Determine the angle of inclination between two lines that lie in the second bisector plane, which is located in the second and fourth quadrants and forms equal angles with the two image planes.
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Context:
Let $S = \{1, 2, \ldots, 8\}$. How many ways are there to select two disjoint subsets of $S$?
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Context:
A set S contains triangles whose sides have integer lengths less than 7, and no two elements of S are congruent or similar. Calculate the largest number of elements that S can have.
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Context:
Consider the stochastic process defined by:
\[ \tilde{W}(t) := W(t) + \int_{0}^{t} e^{W(s)}\mathbb{1}_{\{|W(s)\le 1\}}\,ds \quad \forall \, t \in [0,T] \]
where \( W(t) \) is a Wiener process. For the stopping time \( \sigma := \inf\{t \ge 0 : W(t) = 1 - \int_{0}^{t} e^{W(s)}\mathbb{1}_{\{|W(s)\le 1\}}\,ds \} \), compute the expectation:
\[ \mathbb{E}_{\mathcal{Q}}\left[e^{-\sigma/2} \right] \]
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Context:
A cylinder with a radius of $20 \text{ cm}$ touches the ground along the line $e$. Perpendicular to the line $e$, a $50 \text{ cm}$ long rod is placed resting against the cylinder, such that the end of the rod on the ground is $40 \text{ cm}$ away from the line $e$. How high is the other end of the rod?
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Context:
Huahua is writing letters to Yuanyuan with a pen. When she finishes the 3rd pen refill, she is working on the 4th letter; when she finishes the 5th letter, the 4th pen refill is not yet used up. If Huahua uses the same amount of ink for each letter, how many pen refills does she need to write 16 letters?
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Context:
Find the value of the following limit:
$$\lim_{n\to\infty} \frac {\cos 1 \cdot \arccos \frac{1}{n}+\cos\frac {1}{2} \cdot \arccos \frac{1}{(n-1)}+ \cdots +\cos \frac{1}{n} \cdot \arccos{1}}{n}$$
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Context:
Two types of shapes formed by unit squares (each with an area of 3) are to be placed into an $8 \times 14$ large rectangular grid, with the condition that no two shapes share a common point. What is the maximum number of these shapes that can be placed into the $8 \times 14$ rectangular grid?
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Context:
Consider a function \( f \) that assigns a number \( f(\boldsymbol{v}) \) to each vector \( \boldsymbol{v} \) in space, originating from a common point \( O \). The function satisfies the condition that for any vectors \( \boldsymbol{u}, \boldsymbol{v} \) and any scalars \( \alpha, \beta \), the value \( f(\alpha \boldsymbol{u} + \beta \boldsymbol{v}) \) does not exceed at least one of the values \( f(\boldsymbol{u}) \) or \( f(\boldsymbol{v}) \). Determine the maximum number of distinct values that the function \( f \) can take.
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Context:
The function \( f(x) = \frac{x^{2}}{8} + x \cos x + \cos (2x) \) (for \( x \in \mathbf{R} \)) has a minimum value of ___
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Context:
Line segments connecting an internal point of a convex, unequal-sided $n$-gon with its vertices divide the $n$-gon into $n$ equal triangles.

For what smallest value of $n$ is this possible?
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Context:
A company sells a brand of cars in two locations, A and B, with profits (in ten thousand yuan) of $L_1 = -x^2 + 21x$ and $L_2 = 2x$, respectively, where $x$ is the sales volume (in units). If the company sells a total of 15 units in both locations, calculate the maximum profit it can achieve. Express your answer in ten thousand yuan.
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Context:
Calculate the derivative \( \frac{d|a|^2}{da} \), where \( a \) is a complex number.
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Context:
Two trains of length 90 m and 95 m are a certain distance apart. They start moving towards each other on parallel tracks, at speeds 64 kmph and 92 kmph. The trains meet after 5.768769267689355 seconds. What is the initial distance between the two trains?
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Context:
A car has four tires, each with an exponential lifetime averaging 5000 miles, and one spare tire with an exponential lifetime averaging 1000 miles. When a primary tire fails, the spare is used. Calculate the expected number of miles the car can travel before any of the remaining three primary tires or the spare fails.
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Context:
Find the 257th derivative of $e^{-t} \sin t$.
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Context:
Given \( a, b \in \mathbf{R}^{*} \), find the minimum value of \( y = \frac{a}{\sin^{3/2} \theta} + \frac{b}{\cos^{3/2} \theta} \) for \( \theta \in \left(0, \frac{\pi}{2}\right) \).
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Context:
The prduct of a few primes is ten times as much as the sum of the primes. What are these (not necessarily distinct) primes?
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Context:
If the line $3x+4y-3=0$ is parallel to the line $6x+my+14=0$, then the distance between them is.
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Context:
For positive integers $m$ and $n$, the decimal representation for the fraction $\frac{m}{n}$ begins with $0.711$ followed by other digits. Find the least possible value for $n$.
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Context:
We have that $\lceil x \rceil - \lfloor x \rfloor = 0.$ Then, what is $\lceil x \rceil - x$?
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Context:
7. In the Cartesian coordinate system $x O y$, $F$ is the focus of the parabola $\Gamma: y^{2}=2 p x(p>0)$. Point $B$ is on the $x$-axis and to the right of point $F$. Point $A$ is on $\Gamma$, and $|A F|=$ $|B F|$. The second intersection points of lines $A F$ and $A B$ with $\Gamma$ are $M$ and $N$, respectively. If $\angle A M N=90^{\circ}$, then the slope of line $A F$ is $\qquad$
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Context:
Find the coefficient of \(x^4\) in the expansion of \(\frac{(x-2)^2}{(x^2-1)(x^2-4)}\).
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Context:
In the Cartesian coordinate system, the coordinates of the point $A(2,3)$ with respect to the $x$-axis are ______.
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Context:
14. For each integer $n \geqslant 2$, determine the maximum value of the sum $a_{0}+a_{1}+a_{2}+\cdots+a_{n}$ satisfying the conditions $a_{0}=1, a_{i} \geqslant a_{i+1}+a_{i+2}, i=0$. $1, \cdots, n-2$ (where $a_{0}, a_{1}, \cdots a_{n}$ are non-negative).
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Context:
One dimension of a cube is tripled, another is decreased by `a/2`, and the third dimension remains unchanged. The volume gap between the new solid and the original cube is equal to `2a^2`. Calculate the volume of the original cube.
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Context:
Compute the integral of the differential form \( \omega = -x_1^2 x_2 \,dx_1 + x_1 x_2 \log(1+e^{2x_2})\,dx_2 \) along the unit circle \( \Gamma = \{x \in \mathbb{R}^2 \mid x_1^2 + x_2^2 = 1\} \) in the trigonometric direction. Use Green's theorem if necessary. What is the value of the integral?
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Context:
Evaluate the triple integral \( \iiint_V (3x^{2}+2y+z) \, dxdydz \) over the region defined by the inequalities:
\[
\begin{array}{c}
|x-y| \le 1 \\
|y-z| \le 1 \\
|x+z| \le 1
\end{array}
\]
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Context:
令 f(x)=[x]+[2x]+[3x]+[4x]+[5x]，当 x 取遍 [0,50] 中的实数时，求 f(x) 能取到的整数值的个数.
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Context:
Given a square $ABCD$ with side length $6$. We draw line segments from the midpoints of each side to the vertices on the opposite side. For example, we draw line segments from the midpoint of side $AB$ to vertices $C$ and $D$. The eight resulting line segments together bound an octagon inside the square. What is the area of this octagon?
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Context:
Find the remainder when the determinant \( \begin{vmatrix} { 2014 }^{ 2014 } & { 2015 }^{ 2015 } & { 2016 }^{ 2016 } \\ { 2017 }^{ 2017 } & { 2018 }^{ 2018 } & { 2019 }^{ 2019 } \\ { 2020 }^{ 2020 } & { 2021 }^{ 2021 } & { 2022 }^{ 2022 } \end{vmatrix} \) is divided by 5.
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Context:
in a quadrilateral $ABCD$, $E$ and $F$ are on $BC$ and $AD$ respectively such that the area of triangles $AED$ and $BCF$ is $\frac{4}{7}$ of the area of $ABCD$. $R$ is the intersection point of digonals of $ABCD$. $\frac{AR}{RC}=\frac{3}{5}$ and $\frac{BR}{RD}=\frac{5}{6}$.
 find $\frac{AF}{FD}$.
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Context:
Evaluate the complex function \( f(x) = x^{2/3} \) at \( x = -1 \).
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Context:
Let $X = \{(x,y) : x^2 + y^2 < 9\}$ and $K = \{(x,y) : 4 \leq x^2 + y^2 \leq 5 \text{ or } 7 \leq x^2 + y^2 \leq 8\}$. Determine the number of relatively compact connected components of $X \setminus K$ within $X$. Provide your answer as a single integer.
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Context:
In unit square $A B C D$, points $E, F, G$ are chosen on side $B C, C D, D A$ respectively such that $A E$ is perpendicular to $E F$ and $E F$ is perpendicular to $F G$. Given that $G A=\frac{404}{1331}$, find all possible values of the length of $B E$.
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Context:
A circle passes through the vertices $K$ and $P$ of triangle $KPM$ and intersects its sides $KM$ and $PM$ at points $F$ and $B$, respectively. Given that $K F : F M = 3 : 1$ and $P B : B M = 6 : 5$, find $K P$ given that $B F = \sqrt{15}$.
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Context:
Find the maximum number of permutations of the set $\{1, 2, 3, \ldots, 2014\}$ such that for every two different numbers $a$ and $b$ in this set, there exists at least one permutation where $b$ comes immediately after $a$.
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Context:
顺次连结圆$x ^ { 2 } + y ^ { 2 } = 9$与双曲线xy=3的交点，得到一个凸四边形.则此凸四边形的面积为
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Context:
Find the number of pairs $(a,b)$ of natural numbers such that:
1. $b$ is a 3-digit number.
2. $a+1$ divides $b-1$.
3. $b$ divides $a^{2} + a + 2$.
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Context:
If for any ${x}_{1},{x}_{2}∈[1,\frac{π}{2}]$, $x_{1} \lt x_{2}$, $\frac{{x}_{2}sin{x}_{1}-{x}_{1}sin{x}_{2}}{{x}_{1}-{x}_{2}}＞a$ always holds, then the maximum value of the real number $a$ is ______.
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Context:
Jill has some unit cubes which are all black, grey, or white. She uses 27 of them to build a 3 × 3 × 3 cube. She wants the surface to be exactly one-third black, one-third grey, and one-third white. The smallest possible number of black cubes she can use is \(X\) and the largest possible number of black cubes she can use is \(Y\). What is the value of \(Y - X\)?
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Context:
A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly $\frac1{1985}$.
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Context:
Determine the set of accumulation points for the sequence \( A = \{ \sqrt{n} - \sqrt{m} : m, n \in \mathbb{N} \} \).
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Context:
Given the six whole numbers 10-15, compute the largest possible value for the sum, S, of the three numbers on each side of the triangle.
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Context:
A circle is circumscribed around an isosceles triangle whose two base angles are equal to $x^{\circ}$. Two points are chosen independently and randomly on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}$. Find the sum of the largest and smallest possible value of $x$.
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Context:
Find the smallest integer \( n_0 \) such that for the sequence \( a_n = \frac{n^2 - 3n + (-1)^n}{3n^2 - 7n + 5} \), the inequality \( |a_n - \frac{1}{3}| < 0.01 \) holds for all \( n > n_0 \).
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Context:
What is the smallest integer $n$ such that $n^2 - 14n + 45 > 0$?
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Context:
若函数$f(x)=-{{x}^{2}}+a+2\ln x\text{ }$在$\left[ \frac{1}{{{e2}}},e \right]$上有两个不同的零点，则实数$a$的取值范围为_____.
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Context:
Find the greatest negative value of the expression \( x - y \) for all pairs of numbers \( (x, y) \) that satisfy the equation \( (1 - \operatorname{ctg} x)(1 + \operatorname{ctg} y) = 2 \).
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Context:
Evaluate the integral \( \int_M (x^2+y^2) \, \mathrm{d}V \) where \( M = \{(x, y) \in \mathbb{R}^2: 1 \leq x^2 - y^2 \leq 9, 2 \leq xy \leq 4, x, y \geq 0\} \).
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Context:
Find the order of the quotient ring \( \mathbb{Z}_{10}[x] / \langle 5x^2 + 2x + 1 \rangle \).
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Context:
Let $M$ be the number of ways to write $3050$ in the form $3050 = b_3 \cdot 10^3 + b_2 \cdot 10^2 + b_1 \cdot 10 + b_0$, where the $b_i$'s are integers, and $0 \le b_i \le 99$. Find $M$.
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Context:
Suppose that $X_1, X_2, \ldots, X_n$ are independent and identically distributed random variables with the characteristic function $\chi(t) = e^{-|t|^{1.9}}$. Determine the weak limit of $n^{-5/9} S_n$ as $n$ approaches infinity, where $S_n = \sum_{k=1}^n X_k$. Provide your answer in terms of a probability distribution.
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Context:
Calculate the determinant of the following $n \times n$ matrix, where $n$ is a multiple of $3$:
\[\begin{pmatrix}
0 & 0 & 1 & & & & & & &\\
& & & 0 & 0 & 1 & & & &\\
& & & & & &\ddots\\
& & & & & & & 0 & 0 & 1\\
0 & 1 & 0 & & & & & & &\\
& & & 0 & 1 & 0 & & & &\\
& & & & & &\ddots\\
& & & & & & & 0 & 1 & 0\\
1 & 0 & 0 & & & & & & &\\
& & & 1 & 0 & 0 & & & &\\
& & & & & &\ddots\\
& & & & & & & 1 & 0 & 0
\end{pmatrix}\]
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Context:
Let \( n \) be a non-negative integer that ends with exactly \( k \) zeros in its decimal notation and is greater than \( 10^k \). Given that \( k = k(n) \geq 2 \), determine the number of different ways \( n \) can be expressed as the difference of two squares of non-negative integers, as a function of \( k \).
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Context:
For a graph $G$ with $n \geq 1$ vertices, determine the maximum number of vertices in its block-cutpoint graph $BC(G)$. Provide your answer as a single integer.
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Context:
Factorize the polynomial $x^9 - 1$ in the finite field $\mathbb{F}_3[x]$. Provide the complete factorization into irreducible polynomials.
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Context:
Convex quadrilateral $ABCD$ has $AB = 18$, $\angle A = 60^\circ$, and $\overline{AB} \parallel \overline{CD}$. In some order, the lengths of the four sides form an arithmetic progression, and side $\overline{AB}$ is a side of maximum length. The length of another side is $a$. What is the sum of all possible values of $a$?
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Context:
Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length 3 units and the remaining four sides of length 2 units. The answer is in the form $r+s \sqrt{t}$ with $r, s$, and $t$ being positive integers. Please provide the value of $r + s + t$.
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Context:
Michael is celebrating his fifteenth birthday today. How many Sundays have there been in his lifetime?
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Context:
Let $ABCD$ be a parallelogram with $\angle BAD < 90^\circ.$ A circle tangent to sides $\overline{DA},$ $\overline{AB},$ and $\overline{BC}$ intersects diagonal $\overline{AC}$ at points $P$ and $Q$ with $AP < AQ,$ as shown. Suppose that $AP=3,$ $PQ=9,$ and $QC=16.$ Then the area of $ABCD$ can be expressed in the form $m\sqrt{n},$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n.$
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Context:
We repeatedly toss a coin until we get either three consecutive heads ($HHH$) or the sequence $HTH$ (where $H$ represents heads and $T$ represents tails). What is the probability that $HHH$ occurs before $HTH$?
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Context:
Four identical isosceles triangles $A W B, B X C, C Y D$, and $D Z E$ are arranged with points $A, B, C, D$, and $E$ lying on the same straight line. A new triangle is formed with sides the same lengths as $A X, A Y,$ and $A Z$. If $A Z = A E$, find the largest integer value of $x$ such that the area of this new triangle is less than 2004.
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Context:
Into how many smallest non-overlapping tetrahedrons can a cube be divided?
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Context:
设正实数 a,b 满足 a+2b=1，令 I 为 \df{b^2+a+2}{ab} 的最小值，则 [I]=__________.
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Context:
已知 $f(x)=e^{x}-m x$.
若 $x>0$ 时, 不等式 $(x-2) f(x)+m x^{2}+2>0$ 恒成立, 求实数 $m$ 的取值范围;
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Context:
Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide?
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Context:
在边长为 8 的正方形 $A B C D$ 中, $M$ 是 $B C$ 边的中点, $N$ 是 $D A$ 边上一点, 且 $D N=3 N A$。若对于常数 $m$ ，在正方形的边上恰有 6 个不同的点 $P$ ，使 $\overrightarrow{P M} \cdot \overrightarrow{P N}=m$ ，则实数 $m$ 的取值范围是 \((a, b)\)。请计算a+b的值。
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Context:
Two spheres are inscribed in a dihedral angle in such a way that they touch each other. The radius of one sphere is 1.5 times the radius of the other, and the line connecting the centers of the spheres forms an angle of $30^{\circ}$ with the edge of the dihedral angle. Find the measure of the dihedral angle. In the answer, write down the cosine of this angle, rounding it to two decimal places if necessary.
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Context:
2. 55 In decimal, find the smallest natural number: its square number starts with 19 and ends with 89.
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Context:
10 In the Cartesian coordinate system, given the point set $I=\{(x, y) \mid x, y$ are integers, and $0 \leqslant x \leqslant 5,0 \leqslant y \leqslant 5\}$, then the number of different positioned squares with vertices from the set $I$ is $\qquad$.
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Context:
Given \( A^{2}=0.012345678987654321 \times (1+2+3+\cdots+9+\cdots+3+2+1) \) and \( B^{2}=0.\dot{0}1234567\dot{9} \), find the value of \( 9 \times 10^{9} \times (1 - |A|) B \).
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