Context:
Find $A^2_{}$, where $A^{}_{}$ is the sum of the absolute values of all roots of the following equation: $x = \sqrt{19} + \frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{x}}}}}}}}}$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, we have $AB=36$, $BC=48$, and $CA=60$. The incircle of $\triangle ABC$ is centered at $I$ and touches $AB$, $AC$, and $BC$ at points $M$, $N$, and $D$, respectively. Ray $AI$ meets $BC$ at $K$. The radical axis of the circumcircles of triangles $\triangle MAN$ and $\triangle KID$ intersects lines $AB$ and $AC$ at points $L_1$ and $L_2$, respectively. If $L_1L_2 = x$, compute $x^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of prime factors in the factorization of $12!$, where $12! = 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知数列 $\left\{x_{n}\right\}$ 满足:
$x_{1}=a, x_{2}=b, x_{n+2}=3 x_{n+1}+2 x_{n}$.
若存在整数 $k \geqslant 3$, 使得 $x_{k}=2019$, 则这样的有序正整数对 $(a, b)$ 的个数为 $\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an $11 \times 11$ square, the central cell is colored black. Maxim found the largest rectangular grid area that is entirely within the square and does not contain the black cell. How many cells are in that rectangle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the geometric sequence {a_n}, the product of the first three terms is 27 (i.e., a_1 * a_2 * a_3 = 27), and the sum of the second and fourth terms is 30 (i.e., a_2 + a_4 = 30). Find the common ratio q. Express your answer as a numerical value, including the ± symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. Find the minimum possible value of $BE+DE+CD$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using $1 \times 1, 2 \times 2, 3 \times 3$ tiles to completely cover a $23 \times 23$ floor (without overlapping or leaving gaps), what is the minimum number of $1 \times 1$ tiles needed?
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Context:
A circle with diameter $\overline{PQ}$ of length 10 is internally tangent at $P$ to a circle of radius 20. Square $ABCD$ is constructed with $A$ and $B$ on the larger circle, $\overline{CD}$ tangent at $Q$ to the smaller circle, and the smaller circle outside $ABCD$. The length of $\overline{AB}$ can be written in the form $m + \sqrt{n}$, where $m$ and $n$ are integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute Tor$_1(\mathbb{Q} /\mathbb{Z}, \mathbb{Q} /\mathbb{Z})$. Provide your answer as a group.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Origami. Mitya plans to fold a square sheet of paper \(ABCD\). Mitya calls a fold beautiful if side \(AB\) intersects side \(CD\) and the four resulting right-angled triangles are equal.

Before this, Vanya chooses a random point \(F\) on the sheet. Find the probability that Mitya can make a beautiful fold passing through point \(F\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of non-trivial homomorphisms from the symmetric group $S_5$ to the alternating group $A_6$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ways to express $7^{13}$ as a product of three natural numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n \rightarrow \infty} n \left((n^5 +5n^4)^{1/5} - (n^2 +2n)^{1/2}\right) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a closed form for the following sum:

$$\sum_{k=-\infty}^{\infty} \frac{(-1)^k}{x+(2k+1)}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the sequence defined by the nested expression \(\sqrt{1! + \sqrt{2! + \sqrt{3! + \sqrt{4! + \sqrt{5! + \sqrt{6! + \ldots}}}}}}\) converges to a finite value. If it does, find the limit.
Please reason step by step, and put your final answer within \boxed{}.


Context:
给定正整数 $a 、 b(a \leqslant b)$. 数列 $\left\{f_{n}\right\}$ 满足: $f_{1}=a, f_{2}=b, f_{n+2}=f_{n+1}+f_{n}(n=1,2, \cdots)$. 若对任意的正整数 $n$, 都有 $\left(\sum_{k=1}^{n} f_{k}\right)^{2} \leqslant \lambda f_{n} f_{n+1}$, 求实数 $\lambda$ 的最小值。原始答案的形式为 $m + \sqrt{n}$，请给出 $m + n$ 的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum number of parts the plane can be divided into by $n$ rectangles with parallel sides?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a $3 \times 3$ table, numbers are placed such that each number is 4 times smaller than the number in the adjacent cell to the right and 3 times smaller than the number in the adjacent cell above. The sum of all the numbers in the table is 546. Find the number in the central cell.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In \(\triangle ABC\), \(D\) and \(E\) are the midpoints of the sides \(AB\) and \(AC\) respectively. \(CD\) and \(BE\) intersect at \(P\) with \(\angle BPC = 90^\circ\). Suppose \(BD = 1829\) and \(CE = 1298\). Find \(BC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $a=2^{\frac{4}{3}}$, $b=4^{\frac{2}{5}}$, and $c=25^{\frac{1}{3}}$, determine the relationship between $a$, $b$, and $c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
3. On a circle with center $M$ lie points $A, B$, and $C$. The reflection of $M$ in the line $A B$ lies inside triangle $A B C$ and is the intersection of the angle bisectors of angle $A$ and angle $B$. (The angle bisector of an angle is the line that divides the angle into two equal angles.)

Determine the size of angle $C$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many different ways are there to define a binary operation on a set of $n$ elements, without considering isomorphic structures?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Create a five-digit number using the digits 1, 2, 3, 4, and 5 without any repetition, with the condition that 5 cannot be in the hundred's place, and neither 2 nor 4 can be in the unit's or ten-thousand's place. Calculate the total number of such five-digit numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are \(10!\) permutations \( s_{0}, s_{1}, \ldots, s_{9} \) of the numbers \(0, 1, \ldots, 9\). How many of these permutations satisfy \( s_{k} \geq k - 2 \) for \( k = 0, 1, \ldots, 9 \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
$a$, $b$, $c$ are the three sides of triangle $ABC$, let vectors $\overrightarrow{P}=(a+c,b)$, $\overrightarrow{q}=(b-a,c-a)$. If $\overrightarrow{P} \parallel \overrightarrow{q}$, then the size of angle $C$ is \_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the acute angle of a rhombus, given that the volumes of the solids obtained by rotating the rhombus around its longer diagonal and around its side are in the ratio \(1 : 2\sqrt{5}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sphere with center $O$, there are four points on the surface of the sphere: $S$, $A$, $B$, and $C$, where points $O$, $A$, $B$, and $C$ are coplanar, and $\triangle ABC$ is an equilateral triangle with side length $2$. The plane $SAB$ is perpendicular to the plane $ABC$. Find the maximum volume of the tetrahedron $S-ABC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a regular tetrahedron, how many different edge-length cubes exist such that each vertex of the cube lies on a face of the tetrahedron?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f(x) = x^2 + px + q \). Given that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([2, 4]\), find \( f(f(\ldots f_{2017}\left(\frac{5-\sqrt{11}}{2}\right)) \ldots) \). Round your answer to the nearest hundredth if necessary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of $y$ for which the sum of the coefficients of the odd power terms of $x$ in the expansion of $(x+y)(x+1)^4$ is 32. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given positive integers \(a\), \(b\) (neither a multiple of 5), consider the sequence defined by \(a_1 = 5\) and \(a_{n+1} = a_{a_n} + b\). Determine the largest number of prime numbers that can appear in this sequence before the first composite number is encountered.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a multiplication error involving two positive integers $a$ and $b$, Ron mistakenly reversed the digits of the three-digit number $a$. The erroneous product obtained was $396$. Determine the correct value of the product $ab$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{x \to 0^+}\frac{4e^{x^2}-4\cos(x^3)-2\sin(x^4)-8\sqrt{1+x^2}-x^4+8}{1-\sqrt[3]{1+13x^6}} \) without using L'Hôpital's rule.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many days have passed from March 19, 1990, to March 23, 1996, inclusive?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer  $n$  for which  $315^2-n^2$  evenly divides  $315^3-n^3$ .

*Proposed by Kyle Lee*
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Context:
In the equation $\frac{1}{(\;\;\;)} + \frac{4}{(\;\;\;)} + \frac{9}{(\;\;\;\;)} = 1$, fill in the three brackets in the denominators with a positive integer, respectively, such that the equation holds true. The minimum value of the sum of these three positive integers is $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设$S = \{ 1 , 2 , \cdots , 9 8 \}$，求最小正整数n，使得S的任意一个n元子集中都可以选出10个数，无论怎样将10个数均分成两组，总有一组中存在一个数与另外4个数互质，而另一组中总有一个数与另外4个数不互质． （1998年CMO试题）
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Context:
11.6. Given a tetrahedron \(ABCD\), all of whose faces are similar right triangles with acute angles at vertices \(A\) and \(B\). The edge \(AB\) is equal to 1. Find the length of the shortest edge of the tetrahedron.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设集合 T=\{1,2,\cdots,8\}. 对 T 的非空子集 A，用 S(A) 表示 A 中所有元素的和. 问：T 有多少个非空子集 A，使得 S(A) 是 3 的倍数，但不是 5 的倍数？
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Context:
A square is cut along a diagonal and reassembled to form a parallelogram \( PQRS \). If \( PR=90 \mathrm{~mm} \), what is the area of the original square, in \( \mathrm{mm}^{2} \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
From the set \( M = \{1, 2, \cdots, 36\} \), remove \( n \) numbers such that the sum of any two remaining elements is not a factor of 2015. Find the minimum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The monotonic increasing interval of the function $y=3^{-|x-2|}$ is.
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Context:
Maryam has a fair tetrahedral die, with the four faces of the die labeled 1 through 4.  At each step, she rolls the die and records which number is on the bottom face.  She stops when the current number is greater than or equal to the previous number. (In particular, she takes at least two steps.)  What is the expected number (average number) of steps that she takes?
Please reason step by step, and put your final answer within \boxed{}.


Context:
$已知a>0，函数f(x)=ax-xe^{x}。$
$若存在a，使得f(x) \leq a + b对任意x \in \mathbf{R}成立，求实数b的取值范围.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the graph of the power function $y=f(x)$ passes through the point $\left( \frac{1}{2}, 8 \right)$, find $f(-2) = \_\_\_\_\_\_$.
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Context:
已知f(x)=|1-2x|,,x∈[0,,1]，那么方程.

$f ( f ( f ( x ) ) ) = \frac { 1 } { 2 } x$的解的个数是 .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $\triangle ABC$  with  $AB=AC$  and  $BC=14$  be inscribed in a circle  $\omega$ . Let  $D$  be the point on ray  $BC$  such that  $CD=6$ . Let the intersection of  $AD$  and  $\omega$  be  $E$ . Given that  $AE=7$ , find  $AC^2$ .

*Proposed by Ephram Chun and Euhan Kim*
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Context:
A box contains blue, red, and green flags. The total number of flags in the box is an even number. A group of children are asked to pick up two flags each. If all the flags are used up in the process such that 50% of the children have blue flags and red flags, 30% have red flags and green flags, 10% have blue flags and green flags, and 10% have flags of all three colors, what percentage of children have flags of different colors?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A bicycle wheel has a certain diameter. It makes 624.4536030972898 complete revolutions in 1 km. What is the diameter of the wheel?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the new college entrance examination reform plan, it is stipulated that the scores of each candidate are set according to the mode of $3$ (Chinese, Mathematics, English) $+2$ (Physics, History) $1+4$ (Chemistry, Biology, Geography, Politics) $2$. Then the probability that two students, A and B, have exactly two subjects in common in their elective subjects is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
All the complex roots of $(z + 1)^5 = 32z^5,$ when plotted in the complex plane, lie on a circle.  Find the radius of this circle.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are originally 5 different books placed side by side on a bookshelf. Now, if we want to insert 3 different books in three steps, one book at a time, into the gaps formed by the existing books, calculate the total number of different insertion methods. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the total number of primes $p<100$ such that $\lfloor (2+\sqrt{5})^p \rfloor-2^{p+1}$ is divisible by $p$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square $O P Q R$ is inscribed in $\triangle A B C$. The areas of $\triangle A O R$, $\triangle B O P$, and $\triangle C R Q$ are given as $S_{1}=1$, $S_{2}=3$, and $S_{3}=1$, respectively. Calculate the side length of the square $O P Q R$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $α,β∈(0,π)$, $sin(α-β)=\frac{5}{6}$, $\frac{tanα}{tanβ}=-\frac{1}{4}$, find $\alpha +\beta$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A function $f: \mathbb{R}\to \mathbb{R}$ is essentially increasing if $f(s)\leq f(t)$ holds whenever $s\leq t$ are real numbers such that $f(s)\neq 0$ and $f(t)\neq 0$.

Find the smallest integer $k$ such that for any 2022 real numbers $x_1,x_2,\ldots , x_{2022},$ there exist $k$ essentially increasing functions $f_1,\ldots, f_k$ such that\[f_1(n) + f_2(n) + \cdots + f_k(n) = x_n\qquad \text{for every } n= 1,2,\ldots 2022.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
An electronic clock displays time from 00:00 to 23:59, each moment consisting of four digits. Calculate the probability that the sum of the four digits at any given moment is 23. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A computer screen shows a \( 98 \times 98 \) chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white and white becomes black). Determine the minimum number of mouse-clicks needed to make the chessboard all one color.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many lines in a three dimensional rectangular coordinate system pass through four distinct points of the form $(i, j, k)$, where $i$, $j$, and $k$ are positive integers not exceeding four?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A Turing Machine M has an alphabet with n letters and k states. If M is restricted to only move its head to the right, how many iterations are needed in the worst case to determine if M will loop?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the units (i.e., rightmost) digit of $\left[\frac{10^{20000}}{10^{100}+3}\right]$? Here $[x]$ is the greatest integer $\leqslant x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the universal set $U=\{1,2,3,4,5,6,7,8,9,10\}$, and the set $A=\{a_1, a_2, a_3\}$, find the number of sets $A$ that satisfy $a_3 \geq a_2 + 1 \geq a_1 + 4$. (Answer with a number)
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Context:
Someone in 1860 placed 100,000 frt at 5% interest for the purpose of building and maintaining an orphanage for 100 orphans at some point. When can the construction and opening of the orphanage take place if the construction and equipment costs are 100,000 frt, the staff of the orphanage costs 3,960 frt per year, and the maintenance of one orphan costs 200 frt?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a trapezoid such that $AB \parallel CD$ and let $P = AC \cap BD$, with $AB = 21$, $CD = 7$, $AD = 13$, and $[ABCD] = 168$. Let the line parallel to $AB$ through $P$ intersect the circumcircle of $\triangle BCP$ at $X$. The circumcircles of $\triangle BCP$ and $\triangle APD$ intersect at $P$ and $Y$. Let $XY \cap BC = Z$. If $\angle ADC$ is obtuse, then $BZ = \frac{a}{b}$, where $a$ and $b$ are coprime positive integers. Compute $a + b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $a_{i} \in \{0,1,2,3,4\}$ for every $0 \le i \le 9$, if $6\sum_{i=0}^{9}a_{i} 5^{i} \equiv 1 \pmod{5^{10}}$, find the value of $a_{9}$.
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Context:
Let the set $S = \{P_1, P_2, \dots, P_{12}\}$ consist of the twelve vertices of a regular $12$-gon. A subset $Q$ of $S$ is called "communal" if there is a circle such that all points of $Q$ are inside the circle, and all points of $S$ not in $Q$ are outside of the circle. How many communal subsets are there? (Note that the empty set is a communal subset.)
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Context:
Squares $ABCD$ and $AEFG$, each with side length $12$, overlap such that $\triangle AED$ is an equilateral triangle, as shown in the diagram. The area of the region that lies within both squares, which is shaded in the diagram, is given by $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Determine the value of $m + n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Square $PQRS$ has a side length of $2$ units. Points $T$ and $U$ are on sides $PQ$ and $QR$, respectively, with $PT = QU$. When the square is folded along the lines $ST$ and $SU$, sides $PS$ and $RS$ coincide and lie along diagonal $RQ$. Exprress the length of segment $PT$ in the form $\sqrt{k} - m$ units. What is the integer value of $k+m$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $f(x) = \frac{1}{3^x + \sqrt{3}}$, find the value of $f(-12) + f(-11) + f(-10) + ... + f(0) + ... + f(11) + f(12) + f(13)$.
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Context:
已知凸六边形 $P$ 的所有顶点在一个单位正方形的边上，且所有内角相等. 求 $P$ 的最短边长度的最大可能值。请给出m + n的值，其中原始答案格式为1-\frac{\sqrt{m}}{n}。
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Context:
What is the expected number of trials until we get both outcomes (a success and a failure) in an experiment with a probability of success of 0.81 and a probability of failure of 0.19?
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Context:
A thin diverging lens with an optical power of $D_{p} = -6$ diopters is illuminated by a beam of light with a diameter $d_{1} = 10$ cm. On a screen positioned parallel to the lens, a light spot with a diameter $d_{2} = 20$ cm is observed. After replacing the thin diverging lens with a thin converging lens, the size of the spot on the screen remains unchanged. Determine the optical power $D_{c}$ of the converging lens.
Please reason step by step, and put your final answer within \boxed{}.


Context:
During the steelmaking process, materials with specific chemical elements are added to ensure the steel meets certain required standards. Suppose for a particular use, the amount of a certain element to be added to each ton of steel should be between 500g and 1000g. If we use the 0.618 method to arrange the experiment, then the amount of the element to be added on the second experimental trial could be g.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a, b$ be positive reals with $a>b>\frac{1}{2} a$. Place two squares of side lengths $a, b$ next to each other, such that the larger square has lower left corner at $(0,0)$ and the smaller square has lower left corner at $(a, 0)$. Draw the line passing through $(0, a)$ and $(a+b, 0)$. The region in the two squares lying above the line has area 2013. If $(a, b)$ is the unique pair maximizing $a+b$, compute $\frac{a}{b}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that Liz had no money initially, and her friends gave her one-sixth, one-fifth, and one-fourth of their respective amounts, find the fractional part of the group's total money that Liz has.
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Context:
Evaluate the integral \( \int_0^{\pi} \lfloor \pi^2 \cos^3 x \rfloor \sin x \, dx \), where \( \lfloor x \rfloor \) denotes the floor function.
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Context:
We are given a matrix $A \in M_n (F)$ such that all its entries are either $1$ or $0$. What is the maximum number of $1$'s that can be in matrix $A$ so that it is still invertible?
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Context:
S is the set of all ( $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$ ,  $f$ ) where  $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$ ,  $f$  are integers such that  $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$ . Find the largest  $k$  which divides abcdef for all members of  $S$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Create a cube \( C_{1} \) with edge length 1. Take the centers of the faces and connect them to form an octahedron \( O_{1} \). Take the centers of the octahedron's faces and connect them to form a new cube \( C_{2} \). Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A fair coin is flipped until the number of heads exceeds the number of tails. What is the expected number of flips required?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $2^{\aleph_0} = \aleph_3$, find the value of the expression ${\aleph_2}^{\aleph_0} + {\aleph_0}^{\aleph_2}$.
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Context:
Let $d_1$ and $d_2$ be parallel lines in the plane. We are marking $11$ black points on $d_1$, and $16$ white points on $d_2$. We are drawing the segments connecting black points with white points. What is the maximum number of points of intersection of these segments that lies between the parallel lines (excluding the intersection points on the lines)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which real numbers $a$, there exist pairwise different real numbers $x, y, z$ satisfying
$$\frac{x^3+a}{y+z}=\frac{y^3+a}{x+z}=\frac{z^3+a}{x+y}= -3.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{t\rightarrow 0}\left[-t^{-4} + t^{-5}\left(1+\frac{t^2}{3}\right)\tan^{-1}t\right] \]
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Context:
Tickets for a basketball tournament were $6 for students and $9 for nonstudents. 250 more student tickets were sold than nonstudents, and 850 tickets of each type were sold. What was the total sales amount for the basketball tournament?
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Context:
Origami. Mitya plans to fold a square sheet of paper \(ABCD\). Mitya calls a fold beautiful if side \(AB\) intersects side \(CD\) and the four resulting right triangles are equal.

Before this, Vanya selects a random point \(F\) on the sheet. Find the probability that Mitya can make a beautiful fold that passes through point \(F\).
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Context:
You roll a fair 12-sided die repeatedly. The probability that all the prime numbers show up at least once before seeing any of the other numbers can be expressed as a fraction \( \frac{p}{q} \) in lowest terms. What is \( p+q \)?
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Context:
If \(b > 1\), \(\sin x > 0\), \(\cos x > 0\), and \(\log_b \sin x = a\), find the value of \(\log_b \cos x\). The original answer is in the form \(\frac{m}{n} \log_b(1-b^{p})\). Please provide the value of \(m + n + p\).
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Context:
$x \in \mathbf{R}$, 用 $[x]$ 表示不超过 $x$ 的最大整数, 并用 $\{x\}=x-[x]$ 表示小数部分, 已知: $a_{1}=\sqrt{2}$, $a_{n+1}=\left[a_{n}\right]+\frac{1}{\left\{a_{n}\right\}}$, 求 $\sum\limits_{k=1}^{2024} a_{k}$. 原始答案为 $a(\sqrt{b}+c)$ 的形式，请给出a+b+c的值。
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Context:
Given a lawn with a rectangular shape of $120$ feet by $200$ feet, a mower with a $32$-inch swath width, and a $6$-inch overlap between each cut, and a walking speed of $4000$ feet per hour, calculate the time it will take John to mow the entire lawn.
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Context:
Four disks with disjoint interiors are mutually tangent. Three of them are equal in size and the fourth one is smaller. Find the ratio of the radius of the smaller disk to one of the larger disks.
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Context:
Eight unmarked envelopes contain the numbers: 1, 2, 4, 8, 16, 32, 64, 128. Eve chooses a few envelopes randomly. Alie takes the rest. Both sum up their numbers. Eve's sum is 31 more than Alie's. Find the number of envelopes that Eve took. Express your answer as a whole number.
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Context:
10. A sphere touches all the edges of the pyramid $S A B C$, and the lateral edges $S A, S B$, and $S C$ at points $A^{\prime}, B^{\prime}$, and $C^{\prime}$. Find the volume of the pyramid $S A^{\prime} B^{\prime} C^{\prime}$, if $A B=B C=S B=5$ and $A C=4$.
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Context:
James decides to cut down some trees. In the first 2 days, he cuts down 20 trees each day. For the next 3 days, his 2 brothers help him cut down trees. They cut 20% fewer trees per day than James. Then, their cousin joins them for the following 4 days; the cousin manages to cut down 15% more trees per day than James. Finally, they hire a professional woodcutter who has a daily productivity of 30 trees. They all work together for the next 5 days. How many total trees were cut down during this entire period?
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Context:
Calculate the number of distinct (unordered) hands that can be dealt to a player from a deck of 52 cards when choosing 10 cards. Express your answer as a single digit, which represents the digit A in the number 158A00A4AA0.
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Context:
设 S=\int_{-\infty}^{+\infty}\df{\dd x}{1+x^2+x^4}，求 [100S^2].
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Context:
A triangle is divided into 1000 triangles. What is the maximum number of distinct points that can be vertices of these triangles?
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Context:
The height of a rhombus, drawn from the vertex of its obtuse angle, divides the side of the rhombus in the ratio $1:3$ as measured from the vertex of its acute angle. What fraction of the rhombus's area is the area of the circle inscribed in it?
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Context:
If the price of a book is first decreased by 25%, then increased by 20%, after which it is decreased by 15% and finally increased by 30%, then the net change in the price will be:
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Context:
A train of a certain length is running with a speed of 55 km/hr. In 10.45077684107852 seconds, it will pass a man who is running at 7 km/hr in the direction opposite to that in which the train is going. What is the length of the train?
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Context:
Determine the number of subgroups of order $q$ in a finite abelian $q$-group, where $q$ is a prime number.
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Context:
Find the number of ordered integer pairs \((a, b)\) such that the equation \(x^{2} + a x + b = 167y\) has integer solutions \((x, y)\), where \(1 \leq a, b \leq 2004\).
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Context:
Given the function $f(x) = x^{n-1}\ln(x)$, determine the expression for its $n$-th derivative $f^{(n)}(x)$.
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Context:
In $\triangle ABC$, the angles opposite to sides $a$, $b$, $c$ are $A$, $B$, $C$, respectively. If $\sin^{2}A+\sin^{2}B-\sin^{2}C=\sin A\sin B$, find the measure of angle $C$.
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Context:
If  $x = cos 1^o cos 2^o cos 3^o...cos 89^o$  and  $y = cos 2^o cos 6^o cos 10^o...cos 86^o$ , then what is the integer nearest to  $\frac27 \log_2 \frac{y}{x}$  ?
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Context:
Consider the equation: 250,957 + x^3 = 18,432,100. What is the value of x?
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Context:
Among the positive integers less than $10^{4}$, how many positive integers $n$ are there such that $2^{n} - n^{2}$ is divisible by 7?
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Context:
In a square $A B C D$ with side length 2, $E$ is the midpoint of $A B$. The square is folded along segments $E C$ and $E D$ such that segments $E A$ and $E B$ coincide (point $B$ coincides with point $A$), forming the tetrahedron $C D E A$. Find the volume of this tetrahedron.
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Context:
Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$ , of play blocks which satisfies the conditions:
$(a)$ If $16$ , $15$ , or $14$ students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and $(b)$ There are three integers $0 < x < y < z < 14$ such that when $x$ , $y$ , or $z$ students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.
Find the sum of the distinct prime divisors of the least possible value of $N$ satisfying the above conditions.
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Context:
In a rhombus $P Q R S$ with $P Q=Q R=R S=S P=S Q=6$ and $P T=R T=14$, what is the length of $S T$?
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Context:
Find the number of permutations of $\{1,2,...,n\}$ like $\{a_1,...,a_n\}$ st for each $1 \leq i \leq n$:
$$a_i | 2i$$
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Context:
Determine the values of \( \alpha \) for which the integral \( \int_{1}^{\infty}\left(\sum_{j = 1}^{10}x^{j}\right)^{-(n +1)/10}\,\frac{-n + 1}{10}\,x^{\alpha}\,\mathrm{d}x \) is convergent, given that \( n > 0 \) and \( \alpha \) is a real number.
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Context:
Let $S$ be the set $\{1,2,3,...,19\}$. For $a,b \in S$, define $a \succ b$ to mean that either $0 < a - b \le 9$ or $b - a > 9$. How many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x \succ y$, $y \succ z$, and $z \succ x$?
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Context:
Points \( C \) and \( D \) have the same \( y \)-coordinate of 25, but different \( x \)-coordinates. What is the difference between the slope and the \( y \)-intercept of the line containing both points?
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Context:
Three players are playing table tennis, where the player who loses a game gives up their place to a player who was not participating. It turns out that the first player played 10 games and the second player played 21 games. How many games did the third player play?
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Context:
A circle can be inscribed in the quadrilateral \(ABCD\). Let \(K\) be the intersection point of its diagonals. It is known that \(AB > BC > KC\), \(BK = 4 + \sqrt{2}\), and the perimeter and area of triangle \(BKC\) are 14 and 7, respectively.

Find \(DC\).
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Context:
Consider a circle $O$ with an inscribed triangle $\triangle ABC$. Let the midpoints of arcs $\overparen{AB}$, $\overparen{BC}$, and $\overparen{CA}$ be $C_1$, $A_1$, and $B_1$ respectively, forming an inscribed triangle $\triangle A_1B_1C_1$. Then, take the midpoints of arcs $\overparen{A_1B_1}$, $\overparen{B_1C_1}$, and $\overparen{C_1A_1}$ as $C_2$, $A_2$, and $B_2$ respectively, forming another inscribed triangle $\triangle A_2B_2C_2$. Determine the behavior of the angles of $\triangle A_nB_nC_n$ as $n$ approaches infinity.
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Context:
Evaluate $$\lim_{n \to \infty} \sum_{k=1}^n \left[\left(\cos\left(\frac{2\pi(k+1)}n \right) +\cos\left(\frac{2\pi k}n\right) \right) \left(\sin\left(\frac{2\pi(k+1)}n \right) -\sin\left(\frac{2\pi k}n\right) \right) \right]$$
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Context:
Evaluate the integral $$\iint_R\sqrt \frac{x+y}{x-2y}dA$$ where $R$ is the region bounded by $ y - \frac{x}{2} = 0, y = 0,x+y = 1.$
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Context:
A circle passing through the vertices $A$, $B$, and $C$ of the parallelogram $ABCD$ is tangent to line $AD$ and intersects line $CD$ at points $C$ and $M$. Find the length of segment $AD$, if $BM = 9$ and $DM = 8$.
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Context:
Find the smallest number in which all digits are different and the sum of all digits equals 32.
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Context:
Given sets $P = \{1, 5, 10\}$ and $S = \{1, 3, a^2+1\}$, if the union of $S$ and $P$ is $S \cup P = \{1, 3, 5, 10\}$, find the value of the real number $a$.
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Context:
To investigate a non-luminous black planet in distant space, Xiao Feitian drives a high-speed spaceship equipped with a powerful light, traveling straight towards the black planet at a speed of 100,000 km/s. When Xiao Feitian had just been traveling for 100 seconds, the spaceship instruments received light reflected back from the black planet. If the speed of light is 300,000 km/s, what is the distance from Xiao Feitian's starting point to the black planet in 10,000 kilometers?
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Context:
Determine the initial value to be used when applying the method of characteristics to solve the partial differential equation: \[ u_t + (1-x)u_x = 0, \quad x \in [0,1], \ t>0 \] with the initial condition \( u(x,0) = g(x), \ x \in [0,1] \).
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Context:
Let $n \geq 3$ be an integer. Rowan and Colin play a game on an $n \times n$ grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid and Colin is allowed to permute the columns. A grid coloring is [i]orderly[/i] if: [list] [*]no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and [*]no matter how Colin permutes the columns of the coloring, Rowan can then permute the rows to restore the original grid coloring. [/list] In terms of $n$, how many orderly colorings are there?
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Context:
7

An engineer arrives at the train station at 8 o'clock in the morning every day. At exactly 8 o'clock, a car arrives at the station and takes the engineer to the factory. One day, the engineer arrived at the station at 7 o'clock and started walking towards the car. Meeting the car, he got in and arrived at the factory 20 minutes earlier than usual. How long did the engineer walk? The speeds of the car and the engineer are constant.

#
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Context:
$$
\begin{array}{l}
\text { Find }  \lim_{n \to \infty}  a_{n} \text { where }\left(a_{n}\right)_{n \geq 1} \text { is defined by }\\
a_{n}=\frac{1}{\sqrt{n^{2}+8 n-1}}+\frac{1}{\sqrt{n^{2}+16 n-1}}+\frac{1}{\sqrt{n^{2}+24 n-1}}+\ldots+\frac{1}{\sqrt{9 n^{2}-1}}
\end{array}
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Context:
Determine the number of group structures that make the unit circle $S^1$ a topological group.
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Context:
Determine the smallest integer $n$ such that the Simpson's Rule approximation for the integral \( \int_0^1 9e^{x^2} \, dx \) is accurate to within 0.0001.
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Context:
Pat Peano has plenty of 0's, 1's, 3's, 4's, 5's, 6's, 7's, 8's and 9's, but he has only twenty-two 2's.  How far can he number the pages of his scrapbook with these digits?
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Context:
Determine the number of terms, $n$, required in the series expansion of $\pi = 4\arctan(1/2) + 4\arctan(1/3)$ such that the error is less than $10^{-10}$. Provide your answer as an integer.
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Context:
An $10 \times 25$ rectangle is divided into two congruent polygons, and these polygons are rearranged to form a rectangle again. Determine the length of the smaller side of the resulting rectangle.
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Context:
On the extensions of the sides of triangle $ABC$, points $A_{1}$, $B_{1}$, and $C_{1}$ are taken such that $\overrightarrow{A B_{1}} = 2 \overrightarrow{A B}$, $\overrightarrow{B C_{1}} = 2 \overrightarrow{B C}$, and $\overrightarrow{C A_{1}} = 2 \overrightarrow{A C}$. If the area of triangle $ABC$ is $S$, find the area of triangle $A_{1} B_{1} C_{1}$. Express your answer in terms of $S$. \( \boxed{} \)
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Context:
Max repeatedly throws a fair coin in a hurricane. For each throw, there is a $4 \%$ chance that the coin gets blown away. He records the number of heads $H$ and the number of tails $T$ before the coin is lost. (If the coin is blown away on a toss, no result is recorded for that toss.) What is the expected value of $|H-T|$?
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Context:
Given the equation $(t+1)\cos x-t\sin x=t+2$ has a real root in $(0,\pi)$, the maximum value of the real number $t$ is \_\_\_\_\_.
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Context:
A sports team in a certain class consists of 18 soccer players, 12 basketball players, and 6 table tennis players (each person participates in only one sport). Now, a sample of size $n$ is drawn from these athletes. If both systematic sampling and stratified sampling methods are used, no individuals need to be excluded; when the sample size is $n+1$, if systematic sampling is used, 1 individual needs to be excluded. Then, the sample size $n$ is \_\_\_\_\_.
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Context:
An obtuse triangle with positive area has side lengths 10, 17, and \(x\). If \(x\) is an integer, what is the sum of all possible values of \(x\)?
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Context:
A spherical soap bubble lands on a horizontal wet surface and forms a hemisphere. The volume of the hemisphere is known to be $36\pi$ cm³. Find the radius of the original bubble.
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Context:
Let $D$ be a point on side $[AB]$ of triangle $ABC$ with $|AB| = |AC|$, such that $[CD]$ is an angle bisector and $m(\widehat{ABC}) = 40^\circ$. Let $F$ be a point on the extension of $[AB]$ beyond $B$ such that $|BC| = |AF|$. Let $E$ be the midpoint of $[CF]$. If $G$ is the intersection of lines $ED$ and $AC$, what is the value of $x$ if $m(\widehat{FBG}) = x^\circ$?
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Context:
A unit cube has vertices \(P_1, P_2, P_3, P_4, P_1', P_2', P_3'\), and \(P_4'\). Vertices \(P_2, P_3\), and \(P_4\) are adjacent to \(P_1\), and for \(1\leq i\leq 4\), vertices \(P_i\) and \(P_i'\) are opposite to each other. A regular octahedron has one vertex in each of the segments \(P_1P_2, P_1P_3, P_1P_4, P_1'P_2', P_1'P_3'\), and \(P_1'P_4'\). Find the octahedron's side length. The original answer is in the form \(\frac{a\sqrt{b}}{c}\), please give the value of \(a + b + c\).
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Context:
3. Find the minimum value of the expression

$$
A=\left(5\left(\cos x_{1}+\ldots+\cos x_{n}\right)+\sin x_{1}+\ldots+\sin x_{n}\right) \cdot\left(\cos x_{1}+\ldots+\cos x_{n}-5\left(\sin x_{1}+\ldots+\sin x_{n}\right)\right)
$$
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Context:
Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. Calculate the percentage of the blend that is pear juice. Express your answer as a percentage (%).
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Context:
Given $m\in R$, the distance from point $P(x,y)$ where the line $x+my=0$ intersects with the line $mx-y-2m+4=0$ to the line $l:(x-1)\cos θ+(y-2)\sin θ=3$ is maximized at $\_\_\_\_\_\_$.
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Context:
A buyer took goods worth 10 rubles from a seller and gave 25 rubles. The seller did not have change, so he exchanged money with a neighbor. After they finished the transaction and the buyer left, the neighbor discovered that the 25 rubles were counterfeit. The seller returned 25 rubles to the neighbor and started thinking. What loss did the seller incur?
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Context:
Sequence \(a_{1}, a_{2}, \cdots\) is defined as follows: \(a_{n}=2^{n}+3^{n}+6^{n}-1\), for \(n=1,2,3, \cdots\). Find all positive integers that are relatively prime to every term of this sequence.
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Context:
In triangle $ABC,$ $AB = 20$ and $BC = 15.$  Find the largest possible value of $\tan A.$
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Context:
The integers from 1 to $2008^2$ are written on each square of a $2008 \times 2008$ board. For every row and column, the difference between the maximum and minimum numbers is computed. Let $S$ be the sum of these 4016 numbers. Find the greatest possible value of $S$.
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Context:
How many distinct sets of $5$ distinct positive integers $A$ satisfy the property that for any positive integer $x \leq 29$, a subset of $A$ sums to $x$?
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Context:
Alyssa spent half of her allowance at the movies and 20% on snacks. She saved 1/4 of her allowance for a future purchase. She washed the family car and earned 8 dollars and also mowed the lawn for 5 dollars. If she ended up with 12 dollars after spending on movies and snacks, and saving for the future purchase, what is her weekly allowance?
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Context:
Determine the radius of convergence for the series: $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \left(\left|\frac{x-1}{x+1}\right|-1\right)^n$.
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Context:
It is known that in triangle $ABC$, $AB = a$, and $\angle C = \alpha$. Find the radius of the circle passing through vertices $A$, $B$, and the center of the circle inscribed in triangle $ABC$.
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Context:
11. From the three-digit numbers $100, 101, 102, \ldots, 699, 700$, if $n$ different numbers are taken, such that there are always three numbers among them with the same sum of digits. Then the minimum value of $n$ is $\qquad$
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Context:
在RtΔABC中，$\angle C = 9 0 ^ { \circ } , A C = 1 1$，点D在边AC上且AD=10,∠BDC=3∠BAC..若将 ΔABC的周长写成$\frac { 1 } { 2 } ( m + \sqrt { n } )$ 的形式（m，n为整数），则m+n= ·
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Context:
Given that $x$ and $y$ are independent standard Gaussian random variables, calculate the conditional probability $P(x > 0 \mid x > y)$. Provide your answer as a numerical value.
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Context:
Find $B^2$, where $B$ is the sum of the absolute values of all roots of the equation:
\[x = \sqrt{26} + \frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{x}}}}}}}}}.\]
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Context:
Determine the domain of the function \( \frac{1}{\sqrt{g(x)}} \), given that \( g: \mathbb{R} \rightarrow \mathbb{R} \) satisfies the functional equation \( g(x-y) = g(x) + g(y) \).
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Context:
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. If $\frac{S_3}{S_6} = \frac{1}{3}$, then $\frac{S_9}{S_6} = \_\_\_\_\_\_$.
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Context:
Given an arithmetic sequence $\{a_n\}$ that satisfies $a_3 + a_5 = 14$ and $a_2a_6 = 33$, find the value of $a_1a_7$.
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Context:
Let $p \geq 3$ be a prime number. Each side of a triangle is divided into $p$ equal parts, and each of the division points is connected to the opposite vertex. Calculate the maximum number of pairwise disjoint regions into which the triangle is divided.
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Context:
A hectoliter barrel is full of wine; 1 liter of liquid flows out in 24 hours, which is replaced by the inflow of 1 liter of water. After how much time will the contents of the barrel be half wine and half water?
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Context:
Each of the 2009 distinct points in the plane is colored either blue or red. On every blue-centered unit circle, there are exactly two red points. Determine the greatest possible number of blue points.
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Context:
Find the expected value of the following expression: $$E[min(-B_s,-B_t)^2]$$ where $$ s,t \in \mathbb{R}^{+}$$ and $B_s$ is a Brownian motion.
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Context:
Calculate $$\int_{0}^{\infty} \frac{x-\sin(x)}{x^3(1+x^2)}$$
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Context:
Given integers \(0 \le a_1 < a_2 < \dots < a_c\), determine a bound on the least modulus \(m\) such that for all \(i \neq j\), \(a_i \not\equiv a_j \pmod{m}\).
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Context:
Construct a twice-derivable odd function $g(x)$ such that $g(-2)=g(1)=g(4)=1$. Then find the minimum number of solutions to the equation $(g'(x))^2+g''(x)=0$.
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Context:
Let $V = \mathbb{C}^{n \times n}$. Define the linear transformation $T: V \to V$ by $T(u) = (u,v)w$ for fixed $v, w \in V$. Determine the rank of the transformation $T$. \( \operatorname{rank} T = ? \)
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Context:
A quadrilateral has consecutive sides of lengths $70$, $90$, $130$, and $110$. It is both inscribed in a circle and has a circle inscribed within it. The point of tangency of the inscribed circle to the side of length $130$ divides that side into segments of lengths $x$ and $y$. Find the value of $|x-y|$.
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Context:
Let $A B C$ be an equilateral triangle of side length 15 . Let $A_{b}$ and $B_{a}$ be points on side $A B, A_{c}$ and $C_{a}$ be points on side $A C$, and $B_{c}$ and $C_{b}$ be points on side $B C$ such that $\triangle A A_{b} A_{c}, \triangle B B_{c} B_{a}$, and $\triangle C C_{a} C_{b}$ are equilateral triangles with side lengths 3, 4 , and 5 , respectively. Compute the radius of the circle tangent to segments $\overline{A_{b} A_{c}}, \overline{B_{a} B_{c}}$, and $\overline{C_{a} C_{b}}$.
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Context:
Find the mass percentage of O in Dinitrogen monoxide. What is the mass percentage?
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Context:
In the trapezoid \(ABCD\), point \(K\) is the midpoint of base \(AB\), and \(M\) is the midpoint of base \(CD\). Find the area of the trapezoid, given that \(DK\) is the angle bisector of \(\angle D\), \(BM\) is the angle bisector of \(\angle B\), the largest angle at the lower base is \(60^\circ\), and the perimeter is 30.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two trains start toward each other on parallel tracks at the same time from towns some miles apart. One train travels 6 mph faster than the other train. The rate of each train is 48 mph when they meet in 5 hours. How far apart are the towns?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A gas station has two output pipes A and B, and one input pipe C. When the tank is empty, opening A (to fuel cars) and C (to fill the tank) simultaneously will fill the tank in 2 hours; if B and C are opened simultaneously instead, it will take 4 hours to fill the tank. When the tank is full, opening A and B simultaneously for 3 hours, and then opening C for 5 hours (with A and B still open), the tank is full again. How many hours will it take to fill the tank when it is empty if only C is opened?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of subsets $S$ of $\{1,2, \ldots 6\}$ satisfying the following conditions: - $S$ is non-empty. - No subset of $S$ has the property that the sum of its elements is 10.
Please reason step by step, and put your final answer within \boxed{}.


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:
The sides of triangle $DEF$ are in the ratio $3:4:5$. Segment $EG$ is the angle bisector drawn to the shortest side, dividing it into segments $DG$ and $GE$. If the length of side $DE$ (the base) is 12 inches, what is the length, in inches, of the longer segment of side $EF$ once the bisector is drawn from $E$ to $EF$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
\(a\) and \(b\) are positive reals and \(a > b\). Let \(C\) be the plane curve given by \(r = a - b \cos \theta\). For what values of \(\frac{b}{a}\) is \(C\) convex?
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Context:
In a right circular cone ($S-ABC$), $SA =2$, the midpoints of $SC$ and $BC$ are $M$ and $N$ respectively, and $MN \perp AM$. Determine the surface area of the sphere that circumscribes the right circular cone ($S-ABC$).
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Context:
Given a regular pentagon \(ABCDE\). Point \(K\) is marked on side \(AE\), and point \(L\) is on side \(CD\). It is known that \(\angle LAE + \angle KCD = 108^\circ\), and \(AK: KE = 3:7\). Find the ratio \(CL: AB\).

A regular pentagon is a pentagon in which all sides are equal and all angles are equal.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two cubical dice each have removable numbers $1$ through $6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is $7$? Express your answer in the form \(\frac{k}{m}\), where k and m are coprime, and find the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function f(x) = x^2 - x + sqrt(2x^4 - 6x^2 + 8x + 16), find the minimum value of f(x). Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $d_1, d_2, \ldots , d_{k}$  be the distinct positive integer divisors of  $6^8$ . Find the number of ordered pairs  $(i, j)$  such that  $d_i - d_j$  is divisible by  $11$ .
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Context:
Let  $ABCD$  be a quadrilateral with  $\overline{AB}\parallel\overline{CD}$ ,  $AB=16$ ,  $CD=12$ , and  $BC<AD$ . A circle with diameter  $12$  is inside of  $ABCD$  and tangent to all four sides. Find  $BC$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of zeros (counting multiplicities) of the polynomial \( g(z) = z^{5} + 3z^{3} - 2z^2 + 5z - 4 \) within the annulus \( 1 < |z| < 2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( n \) is an even number greater than 4, what is the probability of randomly selecting 4 vertices from a regular \( n \)-gon to form a rectangle?
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Context:
Calculate \( \limsup\limits_{j\to\infty} |a_j|^{1/j} \), where
\[ a_j = \sum_{j=1}^\infty \frac{(1+1/j)^{2j}}{e^j} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
How would you evaluate the following sum using combinatorial techniques or algebraic manipulation? $$\sum_{k=1}^{n} k\binom{2n}{n+k}$$
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Context:
A watch was sold at a loss of 10%. If it was sold for Rs. 140 more, there would have been a gain of 3%. What is the cost price of the watch?
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Context:
A ship sails at a speed of 30 nautical miles per hour. At point A, it is determined that an oil well P is at a bearing of South 60° East. The ship then sails north for 40 minutes to reach point B, where the oil well P is determined to be at a bearing of South 30° East. The ship changes its course to North 60° East and sails for another 80 minutes to reach point C. Calculate the distance between point P and point C.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\phi$ be the Euler totient function. Let $\phi^k (n) = (\underbrace{\phi \circ  ... \circ  \phi}_{k})(n)$ be $\phi$ composed with itself $k$ times. Define $\theta (n) = \min \{k \in \mathbb{N} \mid \phi^k (n)=1 \}$.

For example:
- $\phi^1 (13) = \phi(13) = 12$
- $\phi^2 (13) = \phi (\phi (13)) = 4$
- $\phi^3 (13) = \phi(\phi(\phi(13))) = 2$
- $\phi^4 (13) = \phi(\phi(\phi(\phi(13)))) = 1$

Thus, $\theta (13) = 4$.

Let $f(r) = \theta (13^r)$. Determine $f(2012)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, points $A$, $B$, and $C$ are located such that $AB = 35$ units, $BC = 40$ units, and $CA = 45$ units. A point $X$ lies on side $AB$ such that $CX$ bisects $\angle ACB$. Given that $BX = 21$ units, find the length of segment $AX$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a set of N numbers and two values, K and X, you are allowed to pick exactly K different numbers from the set. For each picked number, perform the operation: number XOR X. This operation can be performed any number of times. Find the maximum sum of the numbers in the set after performing these operations. For example, consider set A = {10, 15, 20, 13, 2, 1, 44}, K = 4, and X = 14. What is the maximum sum of the set after the operations?
Please reason step by step, and put your final answer within \boxed{}.


Context:
9. (20 points) Inside an acute triangle $A B C$, a point $M$ is marked. The lines $A M, B M$, $C M$ intersect the sides of the triangle at points $A_{1}, B_{1}$ and $C_{1}$ respectively. It is known that $M A_{1}=M B_{1}=M C_{1}=3$ and $A M+B M+C M=43$. Find $A M \cdot B M \cdot C M$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a \( k \times k \) chessboard, a set \( S \) of 25 cells that are in a \( 5 \times 5 \) square is chosen uniformly at random. The probability that there are more black squares than white squares in \( S \) is 48%. Find \( k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the value of $n$ such that
\[(1 + \tan 1^\circ)(1 + \tan 2^\circ)(1 + \tan 3^\circ) \dotsm (1 + \tan 30^\circ) = 2^n.\]
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Context:
Find a generator of the cyclic Galois group \( \operatorname{Gal}(\mathbb{Q}[\xi_{n}]/\mathbb{Q}) \), where the elements of the group are automorphisms of the form \( \sigma_{\ell}: \mathbb{Q}[\xi_{n}] \rightarrow \mathbb{Q}[\xi_{n}] \) defined by \( z = a_{0} + a_{1}\xi_{n} + \ldots + a_{k}\xi_{n}^{k} \rightarrow z(\ell) = a_{0} + a_{1}\xi_{n}^{\ell} + \ldots + a_{k}\xi_{n}^{\ell k} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A homogeneous fair six-sided die has numbers 1, 2, 3, 4, 5, and 6 on its faces. The die is rolled four times, resulting in numbers \( a, b, c, d \) in that order. What is the probability that there is at least one number equal to 4 in the sequence \( a, a+b, a+b+c, a+b+c+d \)? Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the smallest base-10 integer that can be represented as $CC_6$ and $DD_8$, where $C$ and $D$ are valid digits in their respective bases?
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Context:
Given that the general term of the sequence $\{a_{n}\}$ is ${a}_{n}=97-3n(n∈{N}^{*})$, find the value of $n$ for which the sum of the first $n$ terms of the sequence $\{{a}_{n}{a}_{n+1}{a}_{n+2}\}(n∈{N}^{*})$ reaches its maximum value.
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Context:
A decagon is inscribed in a rectangle such that the vertices of the decagon divide each side of the rectangle into five equal segments. The perimeter of the rectangle is 160 centimeters, and the ratio of the length to the width of the rectangle is 3:2. What is the number of square centimeters in the area of the decagon?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the limit:
\[ \lim_{n\rightarrow\infty}\left[\frac{f(x)-n}{(f(x))^2-\frac{n^3(n+2)}{4}}\right] \]
where \( f:(0,\infty)\rightarrow\mathbb{N} \) is defined as
\[ f(x)=\left[\frac{x^2+x+1}{x^2+1}\right]+\left[\frac{4x^2+x+2}{2x^2+1}\right]+\left[\frac{9x^2+x+3}{3x^2+1}\right]+\cdots+\left[\frac{(nx)^2+x+n}{nx^2+1}\right] \]
and \([.]\) denotes the greatest integer function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
9. Let $P(x)=x^{2020}+x+2$, which has 2020 distinct roots. Let $Q(x)$ be the monic polynomial of degree $\binom{2020}{2}$ whose roots are the pairwise products of the roots of $P(x)$. Let $\alpha$ satisfy $P(\alpha)=4$. Compute the sum of all possible values of $Q\left(\alpha^{2}\right)^{2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a regular triangular prism \( S-ABC \) with height \( SO = 3 \) and base side length of 6, a perpendicular is drawn from point \( A \) to the opposite side face \( SBC \) with foot \( O' \). On \( AO' \), a point \( P \) is selected such that \( AP : PO' = 8 \). Find the area of the section passing through point \( P \) and parallel to the base.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given positive real numbers $u$, $v$, and $w$ that satisfy $u^2 + v^2 + w^2 = 8$, find the minimum value of $\frac{u^4}{9} + \frac{v^4}{16} + \frac{w^4}{25}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Triangle $XYZ$ is a right, isosceles triangle. Angle $X$ measures 45 degrees. What is the number of degrees in the measure of angle $Y$?
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Context:
Determine the number of pairs of integers, \((a, b)\), with \(1 \leq a \leq 100\) so that the line with equation \(b=ax-4y\) passes through point \((r, 0)\), where \(r\) is a real number with \(0 \leq r \leq 3\), and passes through point \((s, 4)\), where \(s\) is a real number with \(2 \leq s \leq 4\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $F\_1$ and $F\_2$ are the two foci of the hyperbola $C$: $\frac{x^{2}}{9} - \frac{y^{2}}{m} = 1$, and point $P$ is on $C$ such that $\overrightarrow{PF_{1}} \cdot \overrightarrow{PF_{2}} = 0$. If the directrix of the parabola $y^{2} = 16x$ passes through a focus of the hyperbola $C$, find the value of $|\overrightarrow{PF_{1}}| \cdot |\overrightarrow{PF_{2}}|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$ \lim_{n \rightarrow \infty} \sum_{k=1}^n \left( \sqrt{n^4+k}\ \sin\frac{2k\pi}{n} \right). $$
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Context:
A knight is placed on the lower-left corner of an \( n \times n \) checkerboard. Determine the value of \( n \) such that the minimum number of moves required for the knight to reach the upper-right corner is equal to the minimum number of moves required to reach the lower-right corner.
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Context:
Given that the sum of the coefficients of the last three terms in the expansion of the binomial $({\sqrt{x}+\frac{2}{{\sqrt{x}}}})^n$ is $79$, find the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all values of $m$ so that the function
\[ c(x) = \frac{3x^2 - 4x + m}{7x^2 + m - 6} \]
is defined for all real numbers.
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Context:
Equilateral triangle \(ABC\) has circumcircle \(\Omega\). Points \(D\) and \(E\) are chosen on minor arcs \(AB\) and \(AC\) of \(\Omega\) respectively such that \(BC=DE\). Given that triangle \(ABE\) has area 3 and triangle \(ACD\) has area 4, find the area of triangle \(ABC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the dimension of the variety \( Z(x_1x_2, x_2x_3, x_1x_3) \) in \( \mathbb{A}_k^3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ralph watches TV for 3 hours a day from Monday to Friday. On each of these days, he watches one 1-hour show and four 30-minute shows. However, on Monday, Wednesday, and Friday, Ralph also plays video games for 1 hour before watching TV, and this time replaces one of the 1-hour shows he initially planned to watch.

On Saturday and Sunday, Ralph watches TV for 6 hours a day, where he watches three 1-hour shows and four 45-minute shows. However, on these days, he also takes a 30-minute break to do some chores, which reduces his total TV watching time by 30 minutes on both days.

How many hours does Ralph spend watching TV in one week considering his video gaming time and chores?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $x_1 \dots, x_{42}$ , be real numbers such that  $5x_{i+1}-x_i-3x_ix_{i+1}=1$  for each  $1 \le i \le 42$ , with  $x_1=x_{43}$ . Find all the product of all possible values for  $x_1 + x_2 + \dots + x_{42}$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $x_{1}, \ldots, x_{n}$ denote the $n$th roots of unity, evaluate
$$
\pi\left(x_{i}-x_{j}\right)^{2} \quad(i<j)
$$What is the value when n=3。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the terms of the geometric sequence $\{a_n\}$ are positive, and the common ratio is $q$, if $q^2 = 4$, then $$\frac {a_{3}+a_{4}}{a_{4}+a_{5}}$$ equals \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^\infty \ln(x)e^{-x} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a parabola with its vertex at the origin \( O \) and focus \( F \), \( PQ \) is a chord that passes through \( F \), where \( |OF|=a \) and \( |PQ|=b \). Find the area \( S_{\triangle OPQ} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of sequences \(\left(a_{1}, a_{2}, \cdots, a_{n}\right)\) such that every parking space is occupied, given that there are \(n\) parking spaces and \(n\) drivers. Each driver parks in their preferred space if available, otherwise in the nearest available space, or drives away if no space is available. The values of \(a_{1}, a_{2}, \cdots, a_{n}\) do not need to be distinct.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the Cartesian coordinate system, the coordinates of point $P(-2, \sqrt{5})$ with respect to the $x$-axis are ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a regular triangle ABC with side length $\sqrt{3}$, all three vertices are on the surface of sphere O, and the angle between OA and plane ABC is 60°, then the surface area of sphere O is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A ball is passed among 10 people seated in a circle, with chairs numbered 0 to 9 in a clockwise direction. Initially, the ball is with the person on chair 0. In the first step, the ball is passed 1 turn (\(1^1\)) to chair 1. In the second step, it is passed 4 turns (\(2^2\)) to chair 5. In the third step, it is passed 27 turns (\(3^3\)) to chair 2, and so on. After 2020 steps, on which chair will the ball be? Provide your answer as the chair number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of possible \( \mathbb{Z}[x] \)-module structures on the abelian group \( \mathbb{Z}/5\mathbb{Z} \).
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Context:
How many positive four-digit integers of the form $\_\_35$ are divisible by 35?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Some vertices (the vertices of the unit squares) of a \(6 \times 6\) grid are colored red. We need to ensure that for any sub-grid \(k \times k\) where \(1 \leq k \leq 6\), at least one red point exists on its boundary. Find the minimum number of red points needed to satisfy this condition.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A regular pentagon can have the line segments forming its boundary extended to lines, giving an arrangement of lines that intersect at ten points. How many ways are there to choose five points of these ten so that no three of the points are collinear?
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Context:
Given the quadratic inequality $x^{2}-5x+4 \lt 0$ is a necessary and sufficient condition for the interval $a-1 \lt x \lt a+1$, calculate the range of real number $a$. Express your answer in interval notation.
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Context:
黑板上写着乘积$a _ { 1 } \cdot a _ { 2 }$ $\cdots \cdot a _ { 1 0 0 }$，其中$a _ { 1 } , a _ { 2 } , \cdots ,$100为正整数．如果将其中的一个乘号改为加号（保持其余乘号），发现在所得的99个和数中有32个是偶数．试问，在$a _ { 1 } , a _ { 2 }$,···,$a _ { 1 0 0 }$中至多有多少个偶数？
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit without using L'Hôpital's rule:
\[ \lim_{t\to\frac\pi4}\frac{\int_{\pi/4}^t (\sin t-\sin\left(\frac\pi4\right))dt-\frac{\left(\sin t-\sin\left(\frac\pi4\right)\right)\left(t-\frac\pi4\right)}2}{\frac{\left(\sin t-\sin\left(\frac\pi4\right)\right)\left(t-\frac\pi4\right)}2\left(t-\frac\pi4\right)} \]

Use the identity:
\[ \lim_{t\to0}\frac{\tan t - t}{t^3}=\frac13 \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two trains of different lengths run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. The time they take to cross each other is 26.99784017278618 seconds. If the shorter train is 250 m long, how long is the longer train?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the partial derivative \( \frac{\partial X}{\partial x_0}(t,0) \) for the function \( X: \mathcal{R} \times \mathcal{R} \rightarrow \mathcal{R} \) defined by the differential equation:
\[ \frac{dX}{dt}(t,x_0)=X(t,x_0)+\sin(X^2(t,x_0)), \]
with the initial condition \( X(0,x_0)=x_0. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function f(x) = log2(x) + x - 2, which is an increasing function on the interval (0, +∞), find the solution interval where f(x) = 0. Express your answer as an interval in standard notation.
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Context:
In a box, there are 2 one-cent coins, 4 five-cent coins, and 6 ten-cent coins. Six coins are drawn in sequence without replacement, and each coin has an equal probability of being selected. Calculate the probability that the total value of the drawn coins is at least 50 cents. Express your answer as a fraction.
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Context:
Find the number of critical points of the function $h(x) = e^{3f(x) - 2x + 1} + e^{2f(x) + 3x - 2}$, given that $f(x)$ has a minimum of $-3$ at $x= 0$ and a maximum of $3$ at $x = -2$.
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Context:
The real number \(x\) satisfies the equation \(x + \frac{1}{x} = \sqrt{3}\). Evaluate the expression \(x^{7} - 5x^{5} + x^{2}\).
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Context:
On January 1, a bicycle dealer had an inventory of 200 bicycles. Starting February 1, the number of bicycles in stock began to decrease at unpredictable rates each month. The stock decreased by 4 for February, 6 for March, and 8 for April. In each following month, the decrease rate will be the sum of the previous two months' rates.

In addition, the dealer experiences higher sales during holidays: 5 extra bicycles will be sold during an event in May, 10 during another event in July, and 15 during one in September. Furthermore, the dealer will receive new shipments of bicycles in June and August, which increases the stock by 50 and 60 bicycles, respectively.

Considering the unpredictable rate of decrease in stock, the additional bicycles sold during holidays, and the new shipments, how many fewer bicycles will be in stock on October 1 than were in stock on January 1?
Please reason step by step, and put your final answer within \boxed{}.


Context:
We will call two numbers almost equal if they are either equal to each other or differ from each other by no more than one unit. A grid rectangle with sides equal to the natural numbers \(a\) and \(b\) is such that it is impossible to cut out a rectangle along the grid lines that has an area almost equal to half of the original rectangle. What is the smallest value that the number \( |a-b| \) can take?
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Context:
Let $n$ be an odd number. In a circular arrangement of $n$ buttons, each colored either black or white, pressing a button changes the colors of its two adjacent buttons but not itself. Initially, each button is black or white with probability $\frac{1}{2}$, independently. What is the probability that all buttons can be turned to the same color through a sequence of moves?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a matchstick of length 1 and an infinite grid of 1x1 squares, what is the probability that the matchstick lands overlapping at least one of the matchsticks in the grid?
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Context:
A standard deck of 52 cards is shuffled and dealt to 4 players (N, S, E, and W), each receiving 13 cards. If players N and S together hold exactly 10 cards of a specified suit, what is the probability that the remaining 3 cards of that suit are all in the hand of either player E or player W?
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Context:
Consider a Markov chain with state space $\mathbb{R}$. Its transition function is given by $P(x,A) = \mu([x − 1/4,x + 1/4] \cap A)$, where $\mu$ is the Lebesgue measure. If the initial distribution is concentrated at the origin, calculate the probability $P(|\omega_{3}| \leq 1/8)$. Provide your answer as a single probability value.
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Context:
Let $a, b, c$ be positive integers such that $0 < a, b, c < 11$. They satisfy the following congruences: \[ 3a + b + c \equiv abc \pmod{11} \] \[ a + 3b + c \equiv 2abc \pmod{11} \] \[ a + b + 3c \equiv 4abc \pmod{11} \] Find the sum of all possible values of $abc$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the name of the equivalence relation defined by $A \simeq B$ if $A = P^t BP$ for some invertible matrix $P$?
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Context:
$21$ bandits live in the city of Warmridge, each of them having some enemies among the others. Initially each bandit has $240$ bullets, and duels with all of his enemies. Every bandit distributes his bullets evenly between his enemies, this means that he takes the same number of bullets to each of his duels, and uses each of his bullets in only one duel. In case the number of his bullets is not divisible by the number of his enemies, he takes as many bullets to each duel as possible, but takes the same number of bullets to every duel, so it is possible that in the end the bandit will have some remaining bullets.
Shooting is banned in the city, therefore a duel consists only of comparing the number of bullets in the guns of the opponents, and the winner is whoever has more bullets. After the duel the sheriff takes the bullets of the winner and as an act of protest the loser shoots all of his bullets into the air. What is the largest possible number of bullets the sheriff can have after all of the duels have ended?

Being someones enemy is mutual. If two opponents have the same number of bullets in their guns during a duel, then the sheriff takes the bullets of the bandit who has the wider hat among them.

Example: If a bandit has $13$ enemies then he takes $18$ bullets with himself to each duel, and they will have $6$ leftover bullets after finishing all their duels.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the definite integral \( \int_{0}^{2} (x^2 + 1) \, d\lfloor x \rfloor \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many $3 \times 3$ matrices $A$ with entries either 0 or 1 exist such that the system $A\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ has exactly two distinct solutions?
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Context:
Let $A$ be an $n \times n$ diagonal matrix with the characteristic polynomial $(x-a)^p(x-b)^q$, where $a$ and $b$ are distinct real numbers. Determine the dimension of the real vector space $V$ consisting of all $n \times n$ matrices $B$ such that $AB = BA$. Provide your answer as a single integer.
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Context:
Determine the smallest integer $m$ such that the wheel graph $W_4$, consisting of a triangle and a central hub, can be embedded in $\mathbb{R}^m$ with every edge having length 1.
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Context:
Find the number of triples(a,b,c)of positive integers such that
 (a) \( {ab} \) is a prime;
 (b) \( {bc} \) is a product of two primes;
 (c) \( {abc} \) is not divisible by square of any prime and
 (d) \( {abc} \leq  {30} \) .
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Context:
Compute the length of the $\mathbb{R}[x]$-module $M=\mathbb{C}[x]/\mathbb{R}[x]$. Provide your answer as a single integer.
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Context:
计算下列实变函数定积分．

$\int _ { 0 } ^ { \infty } \frac { \sin m x } { x ( x ^ { 2 } + a ^ { 2 } ) } d x ( m > 0 , a > 0 )$
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Context:
Determine the radius of convergence for the power series \( \sum_{n=1}^\infty \left[\frac {z(z+n)}{n}\right]^{n} \).
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Context:
Given that f(x) is an even function defined on R, and for x >= 0, f(x) = (x^2 - 2x)e^x, find the number of extreme points of the function f(x). Express your answer as a single integer.
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Context:
In a $5$-regular graph $G$ with $m$ edges and girth at least $6$, how many paths of length $5$ are there?
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Context:
Given that $|\vec{a}|=|\vec{b}|=1$ and $\vec{a} \perp \vec{b}$, find the projection of $2\vec{a}+\vec{b}$ in the direction of $\vec{a}+\vec{b}$.
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Context:
Find the inverse Z-transform of the function \( \frac{1}{z-a} \).
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Context:
Let \( a \) and \( b \) be nonnegative real numbers such that
\[
\sin (ax + b) = \sin 17x
\]
for all integers \( x \). Find the smallest possible value of \( a \).
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Context:
Equilateral triangles $ABC$ and $A_{1}B_{1}C_{1}$ with a side length of 12 are inscribed in a circle $S$ such that point $A$ lies on the arc $B_{1}C_{1}$, and point $B$ lies on the arc $A_{1}B_{1}$. Find $AA_{1}^{2} + BB_{1}^{2} + CC_{1}^{2}$.
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Context:
Find the infimum of the expression \( \inf\limits_{k \ge n} k^{2}\chi_{[0,\frac{1}{k}]} \), where \( \chi \) is the indicator function.
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Context:
What is the difference between the place value of 2 in the numeral 7669?
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Context:
For a real number $x$, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of $x$. The sum of all real numbers $\alpha$ that satisfy the equation $$\alpha^2 + \{\alpha\} = 21$$ can be expressed in the form $$\frac{\sqrt{a} - \sqrt{b}}{c} - d$$ where $a$, $b$, $c$, and $d$ are positive integers, and $a$ and $b$ are not divisible by the square of any prime. Compute $a + b + c + d$.
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Context:
Find the characteristic polynomial of the $n \times n$ matrix $C$, where $C$ has 1s on the superdiagonal and the last entry of the first row, and 0s elsewhere.
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Context:
Suppose that \( T_1 \) is a \( \alpha \times 100\% \) lower confidence limit for \( \theta \) and \( T_2 \) is a \( \alpha \times 100\% \) upper confidence limit for \( \theta \). Further assume that \( P(T_1<T_2)=1 \). Determine a \( (2\alpha-1) \times 100\% \) confidence interval for \( \theta \).
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Context:
1. The sequence is defined by the following conditions: $x_{1}=1, x_{n+1}=\cos \left(\operatorname{arcctg}\left(x_{n}\right)\right)$. Find $x_{4} 000000$
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Context:
Given two matrices $A$ and $B$ of size $n \times m$ with integer entries, determine the minimum number of moves required to make them identical. In each move, you can choose one of the matrices and either increment all elements of a selected row or column by 1. If it is impossible to make the matrices identical, state that it is impossible. For example, for $A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 2 \\ 0 & 0 \end{bmatrix}$, the matrices can be made identical in 2 moves.
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Context:
Given the sets $M=\{a|a=(1,2)+\lambda(3,4), \lambda\in\mathbb{R}\}$ and $N=\{a|a=(-2,-2)+\lambda(4,5), \lambda\in\mathbb{R}\}$, calculate the intersection $M\cap N$. Express your answer as a set of points, including coordinates.
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Context:
The value of log 2 ( log of a certain number ) is 2. What is that number?
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Context:
A reduction in the price of petrol by 20% enables three motorists A, B, and C to buy additional gallons for their budgets of $200, $300, and $400 respectively. Motorist A was able to buy 8 more gallons, motorist B was able to buy 15 more gallons, and motorist C was able to buy 22 more gallons. Find the original price of petrol per gallon.
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Context:
设数列 \{a_n\} 满足 a_0=\df{1}{2}, a_{n+1}=a_n+\df{a_n^2}{2012}, n\geq0. 求所有的整数 k，使得 a_k<1<a_{k+1}.
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Context:
Evaluate the integral \( \int_0^{2\pi} \log |1-re^{i\theta}| \, d\theta \) where \( r \) is a real number such that \( |r| < 1 \).
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Context:
After a storm deposits some gallons of water into the city reservoir, the reservoir is 85% full. If the original contents of the reservoir totaled 220 billion gallons, and the reservoir was approximately 55.00000000000001% full before the storm, how many gallons of water did the storm deposit into the reservoir?
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Context:
there exists a $k_{0} \in \mathbb{N}$ such that for every $k \in \mathbb{N}, k > k_{0}$, there exists a finite number of lines in the plane not parallel to one of them, that divide the plane exactly in k regions. Find $k_{0}$.
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Context:
Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. Determine the value of the sum:
\[
\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}
\]
This sum can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
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Context:
Given $\pi < \alpha + \beta < \frac{4}{3}\pi$ and $-\pi < \alpha - \beta < -\frac{\pi}{3}$, the range of values for $2\alpha$ is \_\_\_\_\_\_.
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Context:
Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to form such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have?
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Context:
If \(a, b, c\) satisfy the equations \(a+b+c=1\), \(a^{2}+b^{2}+c^{2}=2\), and \(a^{3}+b^{3}+c^{3}=3\), what is \(a b c\)?
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Context:
Let $p_{}$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of $5$ heads before one encounters a run of $2$ tails. Given that $p_{}$ can be written in the form $m/n$ where $m_{}$ and $n_{}$ are relatively prime positive integers, find $m+n$.
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Context:
Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence of positive real numbers that satisfies $$\sum_{n=k}^{\infty}\binom{n}{k} a_{n}=\frac{1}{5^{k}}$$ for all positive integers $k$. The value of $a_{1}-a_{2}+a_{3}-a_{4}+\cdots$ can be expressed as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Compute $100a+b$.
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Context:
Nine children, numbered 1 to 9, each have a character pinned on their chest. The characters are: 惠 (Hui), 州 (Zhou), 西 (Xi), 湖 (Hu), 丰 (Feng), 鳄 (E), 平 (Ping), 菱 (Ling), 南 (Nan). They are standing in five circles, with the sum of the numbers of children in each circle being 13. What is the highest possible number for the child with the character "丰" (Feng) on their chest?
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Context:
Given $a > 0$, $b < 0$, and $a + b > 0$, arrange $a$, $b$, $-a$, $-b$ in ascending order.
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Context:
A dishonest dealer claims to sell a product at its cost price. He uses a counterfeit weight which is 20% less than the real weight. Further greed overtook him and he added 15% impurities to the product. What is the net profit percentage of the dealer?
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Context:
The real number $x$ satisfies
\[3x + \frac{1}{2x} = 3.\]Find
\[64x^6 + \frac{1}{729x^6}.\]
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Context:
In the tetrahedron $ABCDE$ with congruent side edges and a volume of one unit, the base is the rectangle $ABCD$. A plane intersects the tetrahedron through edge $BC$, creating a cross-section that intersects side edges $EA$ and $ED$ at points $A_1$ and $D_1$, respectively. The lines $BA_1$ and $CD_1$ intersect at point $M$. If $EM = 2AB$, find the volume of the tetrahedron $EMA_1D_1$. Express your answer as a single number.
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Context:
How many even integers are there between \( \frac{12}{3} \) and \( \frac{50}{2} \)?
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Context:
8.1. For all triples $(x, y, z)$ satisfying the system

$$
\left\{\begin{array}{l}
2 \sin x=\operatorname{tg} y \\
2 \cos y=\operatorname{ctg} z \\
\sin z=\operatorname{tg} x
\end{array}\right.
$$

find the smallest value of the expression $\cos x-\sin z$.
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Context:
记 m=7997\times7999^3-7998\times7996^3，求 \sqrt[3]{m}.
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Context:
John is flipping his favorite bottle, which currently contains $10$ ounces of water. However, his bottle is broken from excessive flipping, so after he performs a flip, one ounce of water leaks out of his bottle. When his bottle contains k ounces of water, he has a $\frac{1}{k+1}$ probability of landing it on its bottom. What is the expected number of number of flips it takes for John’s bottle to land on its bottom ?
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Context:
Suppose that \( f(x) \) is a function defined for every real number \( x \) with \( 0 \leq x \leq 1 \) with the properties that

- \( f(1-x)=1-f(x) \) for all real numbers \( x \) with \( 0 \leq x \leq 1 \),
- \( f\left(\frac{1}{3} x\right)=\frac{1}{2} f(x) \) for all real numbers \( x \) with \( 0 \leq x \leq 1 \), and
- \( f(a) \leq f(b) \) for all real numbers \( 0 \leq a \leq b \leq 1 \).

What is the value of \( f\left(\frac{6}{7}\right) \) ?
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Context:
How many numbers should there be in a lottery for the probability of getting an ambo to be $\frac{5}{473}$, when drawing five numbers?
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Context:
On the New Year's table, there are 4 glasses in a row: the first and third contain orange juice, while the second and fourth are empty. While waiting for guests, Valya absent-mindedly and randomly transfers juice from one glass to another. Each time, she can take a full glass and pour all its contents into one of the two empty glasses.

Find the expected number of transfers needed for the first and third glasses to become empty and the second and fourth glasses to become full.
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Context:
In the card game War, assuming cards are put into the bottom of the pile when taken, calculate the probability that a game will end and not result in an infinite loop.
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Context:
In rectangle $ABCD$, point $E$ is chosen on $AB$ and $F$ is the foot of $E$ onto side $CD$ such that the circumcircle of $\vartriangle ABF$ intersects line segments $AD$ and $BC$ at points $G$ and $H$ respectively. Let $S$ be the intersection of $EF$ and $GH$, and $T$ the intersection of lines $EC$ and $DS$. If $\angle SF T = 15^o$ , compute the measure of $\angle CSD$.
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Context:
In the two regular tetrahedra \(A-OBC\) and \(D-OBC\) with coinciding bases, \(M\) and \(N\) are the centroids of \(\triangle ADC\) and \(\triangle BDC\) respectively. Let \(\overrightarrow{OA}=\boldsymbol{a}, \overrightarrow{OB}=\boldsymbol{b}, \overrightarrow{OC}=\boldsymbol{c}\). If point \(P\) satisfies \(\overrightarrow{OP}=x\boldsymbol{a}+y\boldsymbol{b}+z\boldsymbol{c}\) and \(\overrightarrow{MP}=2\overrightarrow{PN}\), find the value of the expression \(9x+81y+729z\).
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Context:
Given an even function $f(x)$ with the domain $(-1,0)\cup(0,1)$, and $f(\frac{1}{e})=0$. When $0 < x < 1$, $(1-x^{2})\ln x(1-x^{2})f′(x) > 2xf(x)$. Find the range of $x$ for which $f(x) < 0$. Express your answer as an interval or union of intervals.
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Context:
已知三个复数 $z_{1}=(5-a)+(6-4 b) \mathrm{i}, z_{2}=(2+2 a)+(3+b) \mathrm{i}, z_{3}=(3-a)+(1+3 b) \mathrm{i}(a, b \in$ $\mathbf{R})$. 若 $\left|z_{1}\right|+\left|z_{2}\right|+\left|z_{3}\right|$ 取最小值, 则 $3 a+6 b$ 的结果为 \frac{k}{m} 的形式，请给出 $k+m$ 的值。
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Context:
Let $m$ and $n$ be distinct even numbers. Determine the value of the following expression:

$$\sum_{i=0}^{\frac{(m-2)}{2}} [\frac{ni}{m}] + \sum_{i=0}^{\frac{(n-2)}{2}} [\frac{mi}{n}]$$
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Context:
3. In Chernomorsk, there are $n$ residents. All of them formed a party. Then this party split into two non-intersecting factions, each of which declared itself a party. Each subsequent day, each party formed on the previous day, consisting of at least two members, split into two factions. Factions consisting of at least two people immediately declared themselves parties. No other parties were formed. When the process ended, each resident of Chernomorsk paid a membership fee of 1 ruble to each party they belonged to. What is the maximum amount the total fees could have been?
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Context:
Calculate the sum of the series \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{2n+3}\) to within an error of \(\pm 0.02\).
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Context:
The base of an oblique prism is an equilateral triangle with a side length of $a$. One of the lateral faces of the prism is perpendicular to the base plane and is a rhombus with a diagonal of length $b$. Find the volume of the prism.
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Context:
A shop sells two kinds of products $A$ and $B$ at the price $\$ 2100$. Product $A$ makes a profit of $20\%$, while product $B$ makes a loss of $20\%$. Calculate the net profit or loss resulting from this deal.
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Context:
There are three solutions with different percentages of alcohol. If you mix them in a ratio of $1:2:3$, you get a $20\\%$ solution. If you mix them in a ratio of $5: 4: 3,$ you will get a solution with $50\\%$ alcohol content. What percentage of alcohol will the solution contain if equal amounts of the original solutions are mixed?
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Context:
For all $x \in (0, +\infty)$, the inequality $(2x - 2a + \ln \frac{x}{a})(-2x^{2} + ax + 5) \leq 0$ always holds. Determine the range of values for the real number $a$.
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Context:
Rachel and Brian are playing a game on a grid with 1 row of 2011 squares. Initially, there is one white checker in each of the first two squares from the left, and one black checker in the third square from the left. At each stage, Rachel can choose to either run or fight. If Rachel runs, she moves the black checker 1 unit to the right, and Brian moves each of the white checkers one unit to the right. If Rachel chooses to fight, she pushes the checker immediately to the left of the black checker 1 unit to the left; the black checker is moved 1 unit to the right, and Brian places a new white checker in the cell immediately to the left of the black one. The game ends when the black checker reaches the last cell. How many different final configurations are possible?
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Context:
Two trains of equal length are running on parallel lines in the same direction at 52 km/hr and 36 km/hr. The faster train passes the slower train in 36 seconds. A third train of length L meters is running at 60 km/hr on a parallel track in the same direction. Calculate the time it takes for the faster train to overpass both the slower train and the third train one after another.
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Context:
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins or indefinitely. If Nathaniel goes first, determine the probability that he ends up winning.
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Context:
If line $l_1: ax+3y+1=0$ is parallel to line $l_2: 2x+(a+1)y+1=0$, find the value of $a$.
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Context:
Find the mass percentage of O in BaO
Please reason step by step, and put your final answer within \boxed{}.