Context:
Find the centroid of the part of a spherical shell centered at the origin with radius 30 that lies in the first octant. The shell is defined by the equation \( z = \sqrt{30^2 - x^2 - y^2} \). Assume symmetry in the first octant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a real 2x2 matrix $B$, determine the conditions on the eigenvalues of $B$ that ensure the convergence of the geometric series used in the expression for $A^{-1}$.
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Context:
In a graph \( G \) with \( m \) vertices of a certain degree, if \( G \) is isomorphic to its complement, determine the number of vertices of the complementary degree in \( G \).
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Context:
Determine the values of \( \alpha \in \mathbb{R} \) for which the integral \( \int_{\mathbb{R}^2} \frac{1}{(1+x^2+xy+y^2)^\alpha} \, dx \, dy \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the conditions on the fixed point (a, b) such that the mapping from the group of all polynomials with real coefficients under addition to itself, which assigns to each polynomial its antiderivative passing through (a, b), is a homomorphism.
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Context:
Find the smallest positive value of \( x \) such that \( \cos(2 \pi x) \), \( \cos(4 \pi x) \), \( \cos(8 \pi x) \), \( \cos(16 \pi x) \), and \( \cos(32 \pi x) \) are all nonpositive.
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Context:
Three lines from the family of lines given by \( C: x \cos t + (y + 1) \sin t = 2 \) enclose an equilateral triangle \( D \). What is the area of the region \( D \)?
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Context:
Determine the extension degree of the constructible subfield $\Omega^{\text{constr}}$ of the splitting field $\Omega^f_{\mathbb{Q}}$ for the polynomial $f=x^9-2x^7+3x^2-6$, given that it is a subfield of the largest real subfield of $\Omega^f_{\mathbb{Q}}$. Provide your answer as an integer.
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Context:
Given that with a probability of \( \frac{1}{3} \), exactly one of eight identical-looking envelopes contains a bill, and with a probability of \( \frac{2}{3} \), all eight envelopes are empty, you open one envelope. Let \( A \) be the event that there is a bill in this envelope, and \( B \) be the event that the other seven envelopes have already been opened and turned out to be empty. Calculate \( P(A \mid B) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest possible value of the expression

$$
\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor,
$$

where \(a, b, c\) and \(d\) vary over the set of positive integers.

(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:
Determine the number of solutions in natural numbers \((x, y)\) for the system of equations: \[\begin{cases} \text{GCD}(x, y) = 20! \\ \text{LCM}(x, y) = 30! \end{cases}\] where \(n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n\).
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Context:
How many six-digit natural numbers exist in which the digits are arranged in ascending order?
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Context:
An ant starts at the point \((1,0)\). Each minute, it walks from its current position to one of the four adjacent lattice points until it reaches a point \((x, y)\) with \(|x|+|y| \geq 2\). What is the probability that the ant ends at the point \((1,1)\)?
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Context:
Let \(B = (B_t)_{t \ge 0}\) be a standard Brownian motion starting at zero. Consider the non-negative stochastic process \(X_t\) that solves the stochastic differential equation:

$$dX_t = \frac{1}{X_t} dt + dB_t$$

Calculate \(E[\sigma]\), where \(\sigma = \inf \{ t \ge 0 : X_t = 1 \}\).
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Context:
Evaluate the infinite series: $$\sum_{k=1}^\infty\frac{k(k+1)(k+2)}{3!}\left(\frac{1}{3}\right)^k$$
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Context:
Given a series of $n$ geomagnetic readings modeled as independent observations from a random variable uniformly distributed on the interval $(0, \theta)$, where $\theta > 1$, and $m$ out of $n$ readings are less than $1$, find the maximum likelihood estimate (MLE) of $\theta$. Assume the remaining $n-m$ readings are greater than $1$.
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Context:
Use Green's Theorem to compute the area enclosed by the parametric curve given by \( x = 2\cos(t) \) and \( y = \sin(2t) \) for \( 0 \le t \le 2\pi \).
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Context:
Evaluate the limit: $$\lim_{n \to \infty} \cos^2(\pi \sqrt[\leftroot{-2} \uproot{1} 3]{n^3+n^2+2n})$$
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Context:
Determine the number of solutions \((x, y, z)\) to the system of equations \(\begin{cases}x+2y+4z=9\\ 4yz+2xz+xy=13\\ xyz=3\end{cases}\) such that at least two of \(x, y, z\) are integers.
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Context:
A bag contains 100 ropes. In each step, two rope ends are randomly selected, tied together, and returned to the bag. This process continues until no free ends remain. What is the expected number of loops formed at the end of this process?
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Context:
Using each of the nine digits exactly once, form prime numbers (numbers that are divisible only by 1 and themselves) such that their sum is minimized.
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Context:
In a regular 2019-gon, numbers are placed at the vertices such that the sum of the numbers in any nine consecutive vertices is 300. It is known that the number at the 19th vertex is 19, and the number at the 20th vertex is 20. What number is at the 2019th vertex?
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Context:
Among the standard products of a certain factory, on average 15% are of the second grade. What is the probability that the percentage of second-grade products among 1000 standard products of this factory differs from 15% by less than 2% in absolute value?
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Context:
Evaluate the integral \( \oint_{|z|=2} \frac{\sin z}{z^3} \, dz \) using contour integration methods.
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Context:
Let \( x \neq y \), and suppose the two sequences \( x, a_{1}, a_{2}, a_{3}, y \) and \( b_{1}, x, b_{2}, b_{3}, y, b_{1} \) are both arithmetic sequences. Determine the value of \( \frac{b_{4}-b_{3}}{a_{2}-a_{1}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In quadrilateral \(ABCD\), the side lengths \(AB\) and \(BC\) are equal to 1, \(\angle B = 100^\circ\), and \(\angle D = 130^\circ\). Find \(BD\).
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Context:
Evaluate the complex integral \( \int_{\gamma} {\frac{z}{\overline z}}dz \) where \( \gamma \) is the edge of the region \( \{1 < |z| < 2 \text{ and } \Im (z) > 0\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Approximate the integral \( \int_0^1 e^{x^2} \, dx \) using the Maclaurin series expansion of \( e^{x^2} \). Ensure that the approximation error is less than 0.0005 from the actual value.
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Context:
A regular triangular prism \( ABC A_1 B_1 C_1 \) is inscribed in a sphere. The base of the prism is \(ABC \) and the lateral edges are \(AA_1, BB_1, CC_1\). Segment \(CD\) is the diameter of this sphere, and points \(K\) and \(L\) are the midpoints of edge \(AA_1\) and \(AB\) respectively. Find the volume of the prism if \(DL = \sqrt{2}\) and \(DK = \sqrt{3}\).
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Context:
Given $x \gt 0$, $y \gt 0$, and $\ln \left(xy\right)^{y}=e^{x}$, find the minimum value of $x^{2}y-\ln x-x$.
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Context:
Calculate the value of \((\boldsymbol{x'}\cdot\nabla)^2\boldsymbol{x}\) where \(\boldsymbol{x'},\boldsymbol{x}\in \mathbb{R}^3\).
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Context:
What is the shortest distance from the origin to the circle defined by \( x^2 - 30x + y^2 - 8y + 325 = 0 \)?
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Context:
There is a cone circumscribed around a sphere with a radius of 1, and the height of the cone is twice the diameter of the sphere. Find the ratio of the total surface area of the cone to the surface area of the sphere.
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Context:
Let \( S \) be the smallest uncountable ordinal, and assume both the Continuum Hypothesis and the Axiom of Choice. Determine the cardinality of the set of all functions from \( S \) to the continuum that map all but a countable number of elements of \( S \) to zero.
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Context:
Let $ABCD$ be a convex quadrilateral with $AC = 20$, $BC = 12$, and $BD = 17$. If $\angle CAB = 80^\circ$ and $\angle DBA = 70^\circ$, then find the area of $ABCD$.
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Context:
Calculate \( \frac{\partial^2 g}{\partial v \, \partial u}(2,2) \) for \( g(u,v)=vf(v^2, 2uv) \) given that \( f(4,8)=2 \), \( \nabla f(4,8)=(5,1) \), \( \frac{\partial^2f}{\partial x^2}(4,8)= \frac{\partial^2f}{\partial y^2}(4,8)=2 \) and \( \frac{\partial^2f}{\partial x \, \partial y}(4,8)=-2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an ellipse $$\frac {x^{2}}{a^{2}}$$ + $$\frac {y^{2}}{b^{2}}$$ = 1 with its right focus F, a line passing through the origin O intersects the ellipse C at points A and B. If |AF| = 2, |BF| = 4, and the eccentricity of the ellipse C is $$\frac {\sqrt {7}}{3}$$, calculate the area of △AFB.
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Context:
In the sequence $\{a_n\}$, $a_1 = 1$, $a_2 = 2$, $a_{n+2}$ is equal to the remainder of $a_n + a_{n+1}$ divided by 3. Find the sum of the first 89 terms of $\{a_n\}$.
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Context:
Evaluate the triple integral \( \int_{-2}^{1} \int_{-2}^{1} \int_{-2}^{1} \frac{x^2}{x^2+y^2+z^2} \,dx \,dy \,dz \).
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Context:
What is the maximum number of even entries possible in a $3 \times 3$ matrix $A$ with integer entries, given that $\det(A) = 1$?
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Context:
For the function \( f(x,y) = \begin{cases} (x^3+y^5-5xy)(x^2+y^2)^{-\beta}, &(x,y) \neq (0,0)\\ 0, & (x,y) = (0,0). \end{cases} \) with \( \beta > 0 \), determine the values of \( \beta \) for which the function is continuous at the origin.
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Context:
Carolyn wants to embroider her new jeans. When embroidering flowers, she can sew 4 stitches/minute, but when embroidering unicorns, she can sew 5 stitches/minute, and when embroidering Godzilla, she can sew 3 stitches/minute. A flower takes 60 stitches to embroider, a unicorn takes 180 stitches, and Godzilla takes 800 stitches. If Carolyn wants to embroider Godzilla crushing 3 unicorns and 50 flowers, and she takes a 5-minute break after every 30 minutes of embroidering, how many total minutes does she need to spend (including breaks) to complete her embroidering project?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a skew-symmetric matrix $M_{ij}$ constructed from $n$ random points in three dimensions $(x_i, y_i, z_i)$, determine the number of purely complex eigenvalues of the matrix.
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Context:
Find the value of the principal square root of the sum of a complex number and its square, given that the square of the complex number is equal to its principal square root but not equal to the number itself.
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Context:
Determine the range of a uniform distribution from which N random numbers are drawn such that, on average, half of the unique numbers drawn are smaller than M/2. Assume that only unique numbers smaller than M/2 are counted.
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Context:
Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer is $S$ is [i]also[/i] removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises of $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S ?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral \( \int_{S(a,\frac{|a|}{2})} \frac{1}{|x|} \, dx \), where \( S(x,r) = \{ x \mid x_1^2 + x_2^2 + x_3^2 = r^2 \} \) and \( 0 \neq a \in \mathbb{R}^3 \).
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Context:
Determine the first-order differential equation for which the solution is given by \( y = \left(\frac{y'}{2}\right)^2 \).
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Context:
Given two 3x3 matrices A and B, where det(A) = 0 and every element of B is 1, and det(A + B) = 1, find the determinant of the matrix A + 2014B.
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Context:
直接从下列 $z$ 变换看出它们所对应的序列。$X(z)=-2 z^{-2}+2 z+1 \quad(0<|z|<\infty)$
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Context:
Calculate the degree of the map \( S: \mathbb{R}^{m+n} \rightarrow \mathbb{R}^{m+n} \) defined by \( S(x, y) = (y, x) \) using the oriented bases of the tangent spaces.
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Context:
5. The unit square in the first quadrant of the coordinate plane ( $0 \leq x, y \leq 1$ ) is divided into squares with side length $2 \cdot 10^{-4}$. How many nodes of this partition (inside the unit square) lie on the parabola $y=x^{2}$?
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Context:
Given a triangle \(ABC\) with a perimeter of 1. A circle is inscribed in angle \(BAC\), lying outside the triangle \(ABC\) and touching side \(BC\) (and the extensions of sides \(AB\) and \(AC\)). Find the length of segment \(AM\), where \(M\) is the point where the circle touches line \(AC\).
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Context:
By what common fraction does $0.\overline{06}$ exceed $0.06$?
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Context:
Determine the number of distinguishable ways to paint a cube using four distinct colors, such that each color appears on at least one face of the cube. Two cubes are considered distinguishable if they cannot be rotated to look identical.
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Context:
Find a $C^∞$ map $f: X \to Y$ where $X \subseteq \mathbb{R}^m$ and $Y \subseteq \mathbb{R}^n$, such that $f$ is one-to-one and onto, its inverse is continuous, but $f$ is not a diffeomorphism.
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Context:
A crude approximation is that the Earth travels in a circular orbit about the Sun at constant speed, at a distance of $150,000,000\text{ km}$ from the Sun. What is the closest value for the acceleration of the Earth in this orbit? Provide your answer as a number without including the unit, and multiply the original answer by 1000.
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Context:
Determine the dimension of the moduli space of complex structures on \( \mathbb{R}^{4} \) with the standard complex structure at infinity.
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Context:
已知椭圆$\Gamma ：\frac{{{x}^{2}}}{{{a2}}}+\frac{{{y}^{2}}}{{{b2}}}=1$与双曲线$\Omega ：\frac{{{x}^{2}}}{{{m2}}}-\frac{{{y}^{2}}}{{{n2}}}=1$共焦点，F$_{1}$、F$_{2}$分别为左、右焦点，曲线$\Gamma $与$\Omega $在第一象限交点为$P$，且离心率之积为1.若$\sin \angle {{F1}}P{{F}_{2}}=2\sin \angle P{{F1}}{{F}_{2}}$，则该双曲线的离心率为____________.
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Context:
Find the lowest common multiple of 10, 14, and a number. The lowest common multiple of these numbers is 140. What is the third number?
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Context:
In the figure, \( AB \) is the diameter of a circle, \( CD \) is a chord parallel to \( AB \), and \( AC \) intersects \( BD \) at \( E \). Given that \(\angle AED = \alpha\), calculate the ratio of the areas of \( \triangle CDE \) and \( \triangle ABE \) in terms of \(\alpha\). Express your answer as a function of \(\alpha\).
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Context:
Evaluate the limit \( \lim_{x \to 0^-} \left(1-(1-\exp(tx^{-\frac{1}{v}}))^v\right)^{1/x} \), where \( t \in (-\infty, 0) \) and \( v \in (0, 1) \).
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Context:
Determine the order of the smallest non-commutative ring without unity.
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Context:
Determine the coefficient of the term $x^3y^4z^5$ in the expansion of $(x + y + z)^8(x + y + z + 1)^8$.
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Context:
Evaluate the integral \( \int_{\mathcal{C}(0,2)} \frac{\sin(1/z)}{z+1}\,dz \) using the residue theorem.
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Context:
A circle is tangent to side \(BC\) of triangle \(ABC\) at its midpoint \(M\), passes through point \(A\), and intersects segments \(AB\) and \(AC\) at points \(D\) and \(E\) respectively. Find angle \(A\), given that \(BC=12\), \(AD=3.5\), and \(EC=\frac{9}{\sqrt{5}}\).
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Context:
Determine the number of distinct cyclic subgroups of order 10 in the group \( \mathbb{Z}_{30} \oplus \mathbb{Z}_{120} \). Identify the generators of these subgroups by locating elements with order 10, ensuring to account for duplicates that generate the same subgroup.
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Context:
Given the function \( f(x) = a \sin x + b \sqrt[3]{x} + c \ln \left(x + \sqrt{x^{2} + 1}\right) + 1003 \), where \( a \), \( b \), and \( c \) are real numbers, and the condition \( f\left(\lg^{2} 10\right) = 1 \), find the value of \( f(\lg \lg 3) \).
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Context:
Determine the limit in the sense of distributions of the sequence of functions \( f_n = n^2\chi_{[-1/n,0)} - n^2\chi_{(0,1/n]} \), where \( \chi \) denotes the characteristic function.
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Context:
Distinct points \( P,Q,R,S \) lie on the circle \( {x}^{2} + {y}^{2} = {25} \) and have integer coordinates. The distances \( {PQ} \) and \( {RS} \) are
 irrational numbers. What is the greatest possible value of the ratio \( \frac{PQ}{RS} \) ?

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Context:
In a series of three training matches, each lasting 90 minutes, coach F of country F evaluates seven players: $A_{1}, A_{2}, \ldots, A_{7}$. At any given moment, exactly one player is on the field. The total playing time for each of $A_{1}, A_{2}, A_{3}, A_{4}$ must be divisible by 7, and the total playing time for each of $A_{3}, A_{5}, A_{6}, A_{7}$ must be divisible by 13. How many different configurations of total playing times are possible for the players?
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Context:
Let  $n{}$  be a natural number. The playing field for a "Master Sudoku" is composed of the  $n(n+1)/2$  cells located on or below the main diagonal of an  $n\times n$  square. A teacher secretly selects  $n{}$  cells of the playing field and tells his student

- the number of selected cells on each row, and
- that there is one selected cell on each column.
The teacher's selected cells form a Master Sudoku if his student can determine them with the given information. How many Master Sudokus are there?
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Context:
Find the last three digits of the product of all odd numbers from 1 to 1997.
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Context:
One hundred musicians are organizing a festival with several concerts. During each concert, some musicians perform on stage while the others are in the audience. What is the minimum number of concerts required so that each musician has the opportunity to listen to every other musician perform on stage?
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Context:
Let \( t \to y(t) \) be a real-valued smooth function on any open interval containing \([0,1]\). Suppose that \( y(t) \) satisfies the differential equation:
\[ y''(t) + w(t)y(t) = \lambda y(t) \]
where \( \lambda \) is a real constant and \( w(t) \) is a given function. Assume that \( y(t) > 0 \) for all \( t \in [0,1] \) and that \( y'(0) = y'(1) = 0 \). Given that \( \int_{0}^{1}\left(\frac{y'}{y}\right)^{2}\,dt = 10 \) and \( \int_{0}^{1}w(t)\,dt = 20 \), find the value of \( \lambda \).
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Context:
Determine the completion of the space of polynomials \( \mathcal{P} \) with respect to the norm \( \| a_0 + a_1 x + \dots + a_nx^n\| = |a_0| + |a_1| + \dots + |a_n| \).
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Context:
For  $a>0$ , denote by  $S(a)$  the area of the part bounded by the parabolas  $y=\frac 12x^2-3a$  and  $y=-\frac 12x^2+2ax-a^3-a^2$ .
Find the maximum area of  $S(a)$ .
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Context:
Find all pairs of natural numbers \((x, y)\) such that \(5^x + 3^y\) is a perfect square.
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Context:
顶点为 $P$ 的圆锥的轴截面是等腰直角三角形， $A$ 是底面圆周上的点， $B$ 是底面圆内的点， $O$ 为底面圆圆心， $A B \perp O B$ ，垂足为 $B, O H \perp P B$ ，垂足为 $H$ ，且 $P A=4, C$ 为 $P A$ 的中点，求当三棱锥 $O-H P C$ 的体积最大时， $O B$ 的长。答案的形式为 \frac{k \sqrt{m}}{n}，请给出k + m + n的值。
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Context:
Evaluate the limit of the infinite product involving prime numbers: $$ \prod_{p \in \mathbb{P}} \frac{p^4+1}{p^4-1} $$ where $\mathbb{P}$ denotes the set of all prime numbers.
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Context:
Find a function \( f(x) \) such that \( f''(x) = f(x) \), \( f(0) = 0 \), and \( f(x) \to -\infty \) as \( x \to -\infty \).
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Context:
Find the Maclaurin series expansion of \( y = \frac{1+x^2+x^4}{1-x^2+x^4} \) up to the term in \( x^6 \).
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Context:
Let $n$ be a two-digit integer represented as $n = 10a + b$, where $a$ and $b$ are the tens and units digits respectively. If $n$ is $k$ times the sum of its digits, find the constant $x$ such that the number formed by interchanging the digits of $n$ is $x$ times the sum of its digits. Express your answer as an algebraic expression in terms of $k$.
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Context:
A parabola is given by the equation \(x^2 + 44x = y + 88\). Let \(P\) be a variable point on the parabola and \(Q\) be a fixed point on the plane not lying on the parabola. Determine the angle between the tangent at \(P\) and the line segment \(PQ\) when the square of the distance between \(P\) and \(Q\) is minimized.
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Context:
Determine the Euler characteristic of the manifold of fixed rank matrices, \( \mathcal{M}_r \), which is the set of \( n \times m \) matrices over \( \mathbb{R} \) or \( \mathbb{C} \) with rank \( r \).
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Context:
The geometric series \(a+ar+ar^2+\ldots\) has a sum of \(7\), and the terms involving odd powers of \(r\) have a sum of \(3\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
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Context:
Find the sum of all values of $x$ for which the power series \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{x^{2n+1}}{2n+1} \) converges.
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Context:
Find max number $n$ of numbers of three digits such that :
1. Each has digit sum $9$
2. No one contains digit $0$
3. Each $2$ have different unit digits
4. Each $2$ have different decimal digits
5. Each $2$ have different hundreds digits
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \((X_i)_{i=1}^{20}\sim N_6(\mu,\Sigma)\). Determine the distribution of the quadratic form \((X_1-\mu)^T\Sigma^{-1}(X_1-\mu)\).
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Context:
There are $2n$ complex numbers that satisfy both $z^{28} - z^{8} - 1 = 0$ and $|z| = 1$. These numbers have the form $z_{m} = \cos\theta_{m} + i\sin\theta_{m}$, where $0\leq\theta_{1} < \theta_{2} < \dots < \theta_{2n} < 360$ and angles are measured in degrees. Find the value of $\theta_{2} + \theta_{4} + \dots + \theta_{2n}$.
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Context:
Given a quadratic inequality in $x$, $ax^2+2x+b>0$, whose solution set is $\{x|x\neq c\}$, find the range of $$\frac {a^{2}+b^{2}+7}{a+c}$$ (where $a+c\neq 0$).
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Context:
A and B alternately throw a pair of dice until one of them wins. A wins if he rolls a total of 7, and B wins if he rolls a total of 5. A starts the game. What is the probability that one of the dice shows a face of 3 on the last throw of the game?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Question 74, Given $a \geq b \geq c \geq d>0, a^{2}+b^{2}+c^{2}+d^{2}=\frac{(a+b+c+d)^{2}}{3}$, find the maximum value of $\frac{a+c}{b+d}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of convergence for the error between the exact solution and the cubic Hermite spline approximation, given that the numerical solution used to construct the spline has an order of convergence $p$ and the function $f$ in the initial value problem is only $C^2$ continuous.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Decrypt the RSA message given that the public exponent is \( e = 3 \) and the message \( m < N^{1/3} \), where \( c \equiv m^3 \pmod{N} \). Find the plaintext message \( m \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x) = e^x(\sin x - \cos x)$ for $0 \le x \le 2011\pi$, find the sum of all maximum values of the function $f(x)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the line integral over the smooth Jordan curve \( \gamma \) in \( \mathbb{R}^2 \setminus \{(0,0)\} \) from \((1,0)\) to \((1,0)\), which winds once around the origin in the clockwise direction:
\[ \int_{\gamma}\frac{y}{x^2+y^2}dx-\frac{x}{x^2+y^2}dy \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a circle with radius \(a\) and a diameter \(AB\), chords are drawn perpendicular to \(AB\), intercepting equal arcs along the circumference of the circle. Find the limit of the average length of these chords as the number of chords tends to infinity.
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Context:
Find all positive integers \(n\) such that \(n^3 + n + 1\) is a perfect square.
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Context:
Determine the value of $k$ for which the quadratic form associated with the bilinear form, given by the matrix \( A = \begin{pmatrix} 0 & 0 & k \\ 0 & k & 0 \\ k & 0 & 0 \end{pmatrix} \), has a signature of \((1, 2)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a hyperbola with the equation $x^2 - y^2 = 9$. A line passing through the left focus $F_1$ of the hyperbola intersects the left branch of the hyperbola at points $P$ and $Q$. Let $F_2$ be the right focus of the hyperbola. If the length of segment $PQ$ is 7, then calculate the perimeter of $\triangle F_2PQ$.
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Context:
Given $p$: $|x-3|\leqslant 2$ and $q$: $(x-m+1)(x-m-1)\leqslant 0$, determine the range of the real number $m$ if $\neg p$ is a necessary but not sufficient condition for $\neg q$.
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Context:
Given a family of functions $f_b(x)$ such that $f_b(0)=b$ and $f_b(x)=2^af_b(x-a)$, find an expression for $f_b(2x)$ in terms of $f_b(x)$. Provide your answer as a function of $f_b(x)$.
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Context:
已知点集 $A=\left\{(x, y) \left\lvert\,(x-3)^{2}+(y-4)^{2} \leq\left(\frac{5}{2}\right)^{2}\right.\right\}, B=\left\{(x, y) \left\lvert\,(x-4)^{2}+(y-5)^{2}>\left(\frac{5}{2}\right)^{2}\right.\right\}$, 则点集 $A \cap B$ 中的整点 （即横、纵坐标均为整数的点）的个数为 $\qquad$.
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Context:
Lilly, Fiona, Jack, and Emily are working together to clean 3 rooms. For the first room, a quarter of the time is spent cleaning by Lilly and Fiona combined, while Jack spent 1/3 of the time cleaning, and Emily was responsible for the rest of the cleaning. In the second room, Jack and Emily each handle 25% of the cleaning, leaving Lilly and Fiona to split the remaining 50%. In the third room, Emily spends 40% of the time cleaning, Lilly and Jack both contribute 20% each, and Fiona takes care of the remaining 20%. It takes them a total of 12 hours to clean all 3 rooms. How long, in minutes, was Emily cleaning in the second room?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a random walk where $X_1, X_2, \ldots$ are independent and identically distributed random variables with a Bernoulli$(\frac{1}{2})$ distribution, meaning $P(X_i = 0) = P(X_i = 1) = \frac{1}{2}$. Define $S_0 = 0$ and $S_n = \sum_{i=1}^n X_i$. Let $\tau_{1000} = \inf\{ n \ge 1 : S_n = 1000 \}$. Find the expected value $\mathbb{E}[\tau_{1000}]$. Provide your answer as a single numerical value or state if it is infinite.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three points \( A \), \( B \), and \( C \) are randomly selected on the unit circle. Find the probability that the side lengths of triangle \( \triangle ABC \) do not exceed \( \sqrt{3} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知$\textit{a}$,$\textit{b}$,$\textit{c}$,$\textit{d}$∈(0,$\textit{+∞}$),且$\textit{S}= frac{a}{a + b + c} + frac{b}{b + c + d} + frac{c}{c + d + a} + frac{d}{a + b + d}, 则S的取值范围是$__________$\textit{.}$
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Context:
Bill has the expression $1+2+3+\cdots+8.$ He replaces two different addition symbols with multiplication symbols uniformly at random. The value that he obtains on average can be expressed as a common fraction $\\tfrac{m}{n}.$ Find $m+n.$
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Context:
Two circles (centres $d$ apart) have radii $15,95$. The external tangents to the circles cut at $60$ degrees, find $d$.
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Context:
Find a number $n \geq 9$ such that for any $n$ numbers, not necessarily distinct, $a_1, a_2, \ldots, a_n$, there exists 9 numbers $a_{i_1}, a_{i_2}, \ldots, a_{i_9}$ ($1 \leq i_1 < i_2 < \ldots < i_9 \leq n$) and $b_i \in \{4, 7\}$, $i = 1, 2, \ldots, 9$ such that $b_1 a_{i_1} + b_2 a_{i_2} + \ldots + b_9 a_{i_9}$ is a multiple of 9.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, $a$, $b$, and $c$ respectively are the sides opposite to angles $A$, $B$, and $C$. If $\frac{\cos C}{\cos B}=\frac{2a-c}{b}$, then $B=$ _________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the ring \( S=\begin{Bmatrix}\begin{pmatrix}a&b\\0&a\end{pmatrix}: a,b\in \mathbb{Z}\end{Bmatrix} \). Determine the number of endomorphisms of \( S \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the product of all the solutions to the equation $$\log_{7x}2023 \cdot \log_{289x} 2023 = \log_{2023x} 2023?$$ Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The area of polygon $ABCDEF$ is 52 with $AB=8$, $BC=9$ and $FA=5$. What is $DE+EF$? [asy]
pair a=(0,9), b=(8,9), c=(8,0), d=(4,0), e=(4,4), f=(0,4);
draw(a--b--c--d--e--f--cycle);
draw(shift(0,-.25)*a--shift(.25,-.25)*a--shift(.25,0)*a);
draw(shift(-.25,0)*b--shift(-.25,-.25)*b--shift(0,-.25)*b);
draw(shift(-.25,0)*c--shift(-.25,.25)*c--shift(0,.25)*c);
draw(shift(.25,0)*d--shift(.25,.25)*d--shift(0,.25)*d);
draw(shift(.25,0)*f--shift(.25,.25)*f--shift(0,.25)*f);
label("$A$", a, NW);
label("$B$", b, NE);
label("$C$", c, SE);
label("$D$", d, SW);
label("$E$", e, SW);
label("$F$", f, SW);
label("5", (0,6.5), W);
label("8", (4,9), N);
label("9", (8, 4.5), E);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of ways to split the numbers from $1$ to $12$ into $4$ non-intersecting sets of size $3$ such that each set has sum divisible by $3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The numbers $1, 2,. . . , 10000, $ were placed in the cells of a $100 \times  100$ square, each once; in this case, numbers differing by $1$ are written in cells adjacent to each side. After that we calculated distances between the centers of every two cells whose numbers differ by exactly $5000$. Let $S$ be the minimum of these distances What is the largest value $S$ can take?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all natural numbers \( n \) such that \( w = \frac{2^{n+1}(1+i)}{(1-i)^{3n+5}} \) is a purely real number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum dimension of a subspace M of the space of 6x6 skew-symmetric matrices over the real numbers, where all non-zero elements of M have rank 2.
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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 $\left(\sum_{k=1}^{n} f_{k}\right)^{2} \leqslant \lambda \cdot f_{n} f_{n+1}$, 求实数 $\lambda$ 的最小值. 原始答案为 $m + \sqrt{n}$ 的形式，请给出 m + n 的值.
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Context:
Find the smallest constant \( c \) such that for any function \( f \in C^4([0,1]) \) and any \( x, y \in [0,1] \), the following inequality holds:
\[ \left\vert\frac{f(x)-f(y)}{x-y}-\frac{f'(x)+f'(y)}{2}\right\vert\le c\vert x-y\vert ^2\max_{\xi\in[0,1]}\vert f'''(\xi)\vert \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the definite integral \( I = \int_{0}^{\infty} (x^2 - 3x + 1) e^{-x} \ln^3(x) \, dx \) using the series expansion \( e^{-x} = \sum_{n=0}^{\infty} \frac{(-x)^n}{n!} \).
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Context:
Let $r_1, r_2, \ldots, r_{20}$ be the roots of the polynomial $x^{20} - 7x^3 + 1$. If \[\frac{1}{r_1^2+1} + \frac{1}{r_2^2+1} + \cdots + \frac{1}{r_{20}^2+1}\] can be expressed as $\frac{m}{n}$ where $m$ and $n$ are positive coprime integers, find $m+n$. \(\text{Express your answer as a single integer.}\)
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Context:
When $m=$______, the equation $\frac{5x}{x-2}+1=\frac{m}{x-2}$ will have a repeated root.
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Context:
If the function $f(x)= \frac{x^{3}}{3}- \frac{a}{2}x^{2}+x+1$ has an extreme point in the interval $\left( \frac{1}{2},3\right)$, then the range of the real number $a$ is ______.
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Context:
Identify a group $G$ that contains elements $a$ and $b$ such that $a^2 = e$, $b^2 = e$, and the order of the element $ab$ is infinite.
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Context:
X and Y are jointly normal random variables with a common mean of 0, a common variance of 1, and a covariance of 1/2. Calculate the probability that X + Y is less than or equal to \( \sqrt{3} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A survey was sent to some customers, 7 of whom responded. Then the survey was redesigned and sent to another 63 customers, 9 of whom responded. The response rate increased by approximately 5% from the original survey to the redesigned survey. How many customers were sent the original survey?
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Context:
Determine the quotient of \( \mathbb{C}^* \) by the inverse action, where the map \( f: \mathbb{C}^* \longrightarrow \mathbb{C}^* \) is defined as \( z \mapsto z^{-1} \). What is the resulting quotient space?
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Context:
Let \( c \) be a complex number, and consider the equation that holds for every complex number \( z \):
\[(z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct)\]
where \( r \), \( s \), and \( t \) are distinct complex numbers. Determine the number of distinct possible values of \( c \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $\alpha$, $\beta$, and $\gamma$ are the roots of the polynomial $x^3 - x^2 + px - 1 = 0$, and that $(\alpha^3 + 1)(\beta^3 + 1)(\gamma^3 + 1) = 2019$, find the product of all possible values of $p$. Note that $p$ may be a complex number.
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Context:
Find the maximal number of red points in a critical coloring of the plane, where each point has coordinates in the set $S=\{0,1,\ldots,99\}$, such that for each $i,j\in S$, at least one of the four points $(i,j),(i + 1,j),(i,j + 1)$, and $(i + 1, j + 1)$ $(99 + 1\equiv0)$ is red. The coloring should lose its critical property if any red point is recolored to white.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a 1000 by 1000 grid, an s-tile is defined as a path of squares that makes two turns in opposite directions and has a length of at least 4. What is the maximum area that can be covered by s-tiles in this grid?
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Context:
(5) In a simple graph with eight vertices, what is the maximum number of edges in a graph that does not contain a quadrilateral? (A simple graph is one where no vertex is connected to itself, and any two vertices are connected by at most one edge.)
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Context:
Define the sequences $c_n$ and $d_n$ as follows: $c_1 = 6$, $c_{n+1} = (c_n + 1)!$, $d_1 = 6$, $d_{n+1} = 6^{d_n + 1}$. Determine which is larger, $c_{d_2}$ or $d_{c_2}$.
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Context:
In a language with only three letters - $A, B$, and $C$, a word is defined as a sequence of exactly 100 letters, with exactly 40 consonants and 60 letters being $A$. Determine the maximum number of such words that can be selected such that any two selected words have at least one position where they both have consonants, but different consonants.
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Context:
The sequence ${a_n}$ satisfies the equation $a_{n+1} + (-1)^n a_n = 2n - 1$. Determine the sum of the first 80 terms of the sequence.
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Context:
Evaluate the double integral \( J = \int\limits_0^1 \int\limits_0^1 \frac{x-y}{(x+y)^3} \, dy \, dx .\)
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Context:
Given a positive integer \(k\), let \(S_{k}\) be the set of all visible lattice points \((x, y)\) such that \(x^{2}+y^{2}=k^{2}\). A lattice point is visible if the line segment connecting the point and the origin does not pass through another lattice point. Let \(D\) denote the set of all positive divisors of \(2021 \cdot 2025\). Compute the sum \(\sum_{d \in D}|S_{d}|\), where \(|A|\) denotes the number of elements in the set \(A\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a random variable $X$ with the cumulative distribution function (CDF):
\[ F(x) = \begin{cases} 1 - \left(\frac{2000}{2000 + x}\right), & x \ge 0 \\ 0, & \text{otherwise} \end{cases} \]
Let $Y = \max(0, X - 500)$. Calculate the 60th percentile of the positive values of $Y$, ignoring values of $Y$ where $0 \le X \le 500$. Provide your answer as a single number.
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Context:
If triangle $PQR$ has sides of length $PQ = 8,$ $PR = 7,$ and $QR = 5,$ then calculate
\[\frac{\cos \frac{P - Q}{2}}{\sin \frac{R}{2}} - \frac{\sin \frac{P - Q}{2}}{\cos \frac{R}{2}}.\]
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Context:
Evaluate the limit: \[ \lim _{x\to \infty }\frac{\left(x!\right)^3\left(3\left(x+1\right)\right)!}{\left(\left(x+1\right)!\right)^3\left(3x\right)!} \]
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Context:
A collection of stained glass windows consists of 9 squares arranged in a 3x3 grid. Each square can be either red or blue, with k of them being red. The set is designed such that any possible window configuration can be achieved by rotating or flipping one unique window from the set. Given that the total number of red squares in the set exceeds 100, determine the value of k. Use Burnside's lemma, considering the 8 isometries of a square: the identity, 3 rotations (90°, 180°, 270°), and 4 reflections (vertical, horizontal, and two diagonals).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $n \in \mathbb{N}^*$, and $n$ is an odd number, then the remainder obtained when dividing $6^{n}+C_{n}^{1} \cdot 6^{n-1}+C_{n}^{2} \cdot 6^{n-2}+...+C_{n}^{n-1} \cdot 6$ by $8$ is $\_\_\_\_\_\_\_\_.$
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Context:
How many ways can $8$ mutually non-attacking rooks be placed on a $9 \times 9$ chessboard so that all $8$ rooks are on squares of the same color?

(Two rooks are said to be attacking each other if they are placed in the same row or column of the board.)
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Context:
Let $a_1, a_2, \ldots, a_{100}$ be positive integers, satisfying
$$
\frac{a_1^2+a_2^2+\ldots+a_{100}^2} {a_1+a_2+\ldots+a_{100}}=100.
$$
What is the maximal value of $a_1$?
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Context:
Find the number of ways to color all natural numbers from 1 to 20 in blue and red so that both colors are used and the product of all red numbers is relatively prime to the product of all blue numbers.
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Context:
There exists complex numbers  $z=x+yi$  such that the point  $(x, y)$  lies on the ellipse with equation  $\frac{x^2}9+\frac{y^2}{16}=1$ . If  $\frac{z-1-i}{z-i}$  is real, compute  $z$ .

*(Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 3)*
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Context:
Determine the angle between the hour and minute hands of a clock showing 1:10, assuming both hands move at constant speeds.
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Context:
For any positive integer $n$, let $x_n = \mathrm{C}_{2n}^{n}$. Find all positive integers $h$ (where $h > 1$) such that the sequence $\left\{ x_n \right\}$, when taken modulo $h$, becomes a periodic sequence starting from a certain term.
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Context:
In a group of people participating in some curriculum, 25 of them practice yoga, 18 study cooking, 10 study weaving, 4 of them study cooking only, some of them study both the cooking and yoga, 4 of them participate all curriculums. If 5 people study both cooking and weaving, how many people study both cooking and yoga?
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Context:
Five glass bottles can be recycled to make a new bottle. Additionally, for every 20 new bottles created, a bonus bottle can be made from residual materials. Starting with 625 glass bottles, how many total new bottles can eventually be made from recycling and bonuses? (Keep counting recycled and bonus bottles until no further bottles can be manufactured. Do not include the original 625 bottles in your count.)
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Context:
Find the largest integer \( m \) such that \( a^{12} \equiv 1 \pmod{m} \) for all integers \( a \) that are relatively prime to \( m \).
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Context:
A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A = (-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths?
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Context:
Evaluate \( \lim_{n \rightarrow \infty} \int_0^1 \frac{nx}{1 + n^2x^4} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A man travels between three towns forming a triangle by walking, riding, and driving. He can walk, ride, and drive a mile in \(a\), \(b\), and \(c\) minutes respectively. If he starts from town B, he takes \(a + c - b\) hours; if he starts from town C, he takes \(b + a - c\) hours; and if he starts from town A, he takes \(c + b - a\) hours. Determine the total length of the circuit in miles.
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Context:
Determine the condition under which the eigenvalue sets of the matrices \(I - BAA^T\) and \(I - A^TBA\) are equal, where \(A \in \mathbb{R}^{m \times n}\) and \(B \in \mathbb{R}^{m \times m}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A man leaves point P at 6 a.m. and reaches point Q at 10 a.m. Another man leaves point Q at 8 a.m. and reaches point P at 12 noon. At what time do they meet?
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Context:
Given a yearly 3x3 transition matrix with eigenvalues \( \lambda = 1, \frac{1}{2}, 0 \), and corresponding eigenvectors \( x_1 = (60, 20, 30) \) and \( x_2 = (-60, -30, -90) \), the initial state vector is \( v_0 = 500x_1 + 200x_2 + 100x_3 \), where \( x_3 \) is the eigenvector for \( \lambda = 0 \). Determine the number of people who will be sick with the disease 2 years later.
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Context:
设 A,B 是集合 \{1,2,\cdots,20\} 的两个子集，A\cap B=\emptyset，且 n\in A 时，2n+2\in B. 记 M(A) 为 A 的元素之和，则 M(A) 的最大值是__________.
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Context:
A side of beef lost 35 percent of its weight in processing. If the side of beef weighed a certain amount after processing, and it weighed 861.54 pounds before processing, how many pounds did it weigh after processing?
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Context:
Find $1 - 0.\overline{123}$.
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Context:
4. The numbers $x$ and $y$ are such that the equalities $\cos y + \cos x = \sin 3x$ and $\sin 2y - \sin 2x = \cos 4x - \cos 2x$ hold. What is the smallest value that the sum $\sin y + \sin x$ can take?
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Context:
Determine the value of \( c \) that leads to the fastest convergence of the sequence \( D_n(c) = \frac {4^n}{\binom {2n}{n} \sqrt{4n + c}} \) to \( \frac{\sqrt{\pi}}{2} \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Provide an example of a bounded sequence that is not convergent, but where the differences of consecutive terms converge to zero.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ying lives on Strangeland, a tiny planet with 4 little cities that are each 100 miles apart from each other. One day, Ying begins driving from her home city of Viavesta to the city of Havennew, which takes her about an hour. When she gets to Havennew, she decides she wants to go straight to another city on Strangeland, so she randomly chooses one of the other three cities (possibly Viavesta), and starts driving there. Ying drives like this for most of the day, making 8 total trips between cities on Strangeland, choosing randomly where to drive to next from each stop. She then stops at her final city of destination, digs a hole, and buries her car.

Let $p$ be the probability Ying buried her car in Viavesta and let q be the probability she buried it in Havennew. Find the value of $p + q$.
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Context:
Let \( f: \mathbb{R} \to \mathbb{R} \) be a function that is three times differentiable and satisfies the equation \( f(x^2)f''(x) = f'(x)f'(x^2) \) for all \( x \in \mathbb{R} \). Given that \( f(1) = 1 \) and \( f'''(1) = 8 \), find the value of \( f'(1) + f''(1) \).
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Context:
A point \( P \) inside an equilateral triangle \( ABC \) is at distances \( PA = 6 \), \( PB = 8 \), and \( PC = 10 \) from the vertices. Find the closest integer to the area of \( \triangle ABC \).
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Context:
Given a function $f: \mathbb{N}^+ \rightarrow \mathbb{N}^+$ that is strictly increasing and satisfies $f(f(n)) = 3n$ for all positive integers $n$, find $f(100)$.
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Context:
Determine the distance from the origin to the set \( A \) in \( \mathbb{R}^{2d} \) where \( d > 1 \), defined as:
\[
A = \left\{ \begin{pmatrix} x_1 & y_1 & \ldots & x_d & y_d \end{pmatrix}': x_i^2 + y_i^2 > c, \ i = 1, \ldots, d \right\},
\]
with \( c \) being a positive constant. Calculate \( d(0, A) \), the distance from the origin to the set \( A \).
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Context:
Calculate the flux of the vector field \( \bar{F} = \langle x, y, x \rangle \) across the plane defined by \( 1 - (x + y + z) = 0 \) in the first octant, using the unit normal vector \( \langle \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \rangle \).
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Context:
7. From the sides and diagonals of a regular 12-sided polygon, three different segments are randomly selected. The probability that their lengths are the side lengths of a triangle is $\qquad$
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Context:
Let \(\kappa\) and \(\lambda\) be infinite cardinals. Consider a random function \(\phi: \kappa \times \lambda \rightarrow \{0,1\}\) constructed by flipping a fair coin for each element of the domain. Determine the probability \(P(\kappa, \lambda)\) that there exists an \(\alpha < \kappa\) such that \(\forall x, \phi(\alpha, x) = 0\).
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Context:
Find the total number of different integer values the function $$f(x)=[x]+[2 x]+\left[\frac{5 x}{3}\right]+[3 x]+[4 x]$$ takes for real numbers $x$ with $0 \leq x \leq 100$. Note: $[t]$ is the largest integer that does not exceed $t$.
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Context:
An article costing $C$ dollars is sold for $100 at a loss of $x$ percent of the selling price. It is then resold at a profit of $x$ percent of the new selling price $S'$. If the difference between $S'$ and $C$ is $1\frac {1}{9}$ dollars, then $x$ is:
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Context:
## Task B-3.6.

A rectangle is given with side lengths $a$ and $b$, where $a = \frac{4}{7} b$. Determine the probability that a randomly selected point within the given rectangle is closer to the shorter side and is less than $\frac{a}{8}$ away from it.
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Context:
设在 $5 \times 5$ 的方格表的第 $i$ 行第 $j$ 列所填的数为 $a_{i j}\left(a_{i j} \in\{0,1\}\right), a_{i j}=a_{j i}(1 \leqslant i、j \leqslant 5)$ .则表中共有五个 1 的填表方法总数为 $\qquad$ （用具体数字作答).
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Context:
Determine the number of roots of the polynomial \( f(x) = x^{134} + x^{127} + x^7 + 1 \) over the field \( \mathbb{F}_{2^{1463}} \).
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Context:
Let $X(t)$ be the price of JetCo stock at time $t$ years from the present. Assume that $X(t)$ follows a geometric Brownian motion with zero drift and volatility $\sigma = 0.4/yr$. If the current price of JetCo stock is 8.00 USD, calculate the probability that the price will be at least 8.40 USD six months from now.
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Context:
In a game, 15 players each choose 6 cards from separate decks of 40 cards. Calculate the expected number of cards that at least two players will share.
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Context:
A stick of length 2 meters is made of uniformly dense material. A point is chosen randomly on the stick, and the stick is broken at that point. The left portion of the stick is discarded. This process is repeated indefinitely on the remaining portion of the stick. What is the probability that one of the discarded left parts has a length greater than 1 meter?
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Context:
In a regular quadrilateral pyramid $S A B C D$, where each edge is equal to 2, a section is constructed by a plane parallel to the diagonal of the base $A C$ and the lateral edge $S B$ of the pyramid, intersecting the edge $A B$. Find the perimeter of the polygon obtained in this section, if the lower base of the section is equal to $\sqrt{2}$.
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Context:
On the leg \( BC \) of the right triangle \( ABC \), a circle is constructed with \( BC \) as the diameter. The circle intersects the hypotenuse at point \( D \) such that \( AD : BD = 1 : 3 \). The altitude from vertex \( C \) (the right angle) to the hypotenuse is 3. Find the leg \( BC \).
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Context:
Given a $7 \times 7$ matrix $A$ such that $2A^2 - A^4 = I$, where $I$ is the identity matrix, and $A$ has two distinct eigenvalues, each with a geometric multiplicity of 3, determine the total number of non-zero entries in the Jordan canonical form of $A$.
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Context:
设 V 为关于 x 的次数小于或等于 10 的实多项式构成的集合, 且 a为实数。定义 $V$ 到 V 的线性映射 F 为 $F(f)=\left(x^{2} f\right)^{\prime \prime}+f^{\prime}-a f$ 。记 I 为使得 F 不是满射的 a 的所有可能值之和，则 $[I]=$
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Context:
In how many ways can 5 married couples be seated around a circular table such that man 1 is seated next to his wife, and man 3 is not seated next to his wife?
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Context:
What number must we add to $3-5i$ to get $2+7i$? Here, $i^2 = -1.$
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Context:
Example 19 (Problem from the 2nd American Mathematical Invitational Competition) Given
$$
\left\{\begin{array}{l}
\frac{x^{2}}{2^{2}-1^{2}}+\frac{y^{2}}{2^{2}-3^{2}}+\frac{z^{2}}{2^{2}-5^{2}}+\frac{w^{2}}{2^{2}-7^{2}}=1, \\
\frac{x^{2}}{4^{2}-1^{2}}+\frac{y^{2}}{4^{2}-3^{2}}+\frac{z^{2}}{4^{2}-5^{2}}+\frac{w^{2}}{4^{2}-7^{2}}=1, \\
\frac{x^{2}}{6^{2}-1^{2}}+\frac{y^{2}}{6^{2}-3^{2}}+\frac{z^{2}}{6^{2}-5^{2}}+\frac{w^{2}}{6^{2}-7^{2}}=1, \\
\frac{x^{2}}{8^{2}-1^{2}}+\frac{y^{2}}{8^{2}-3^{2}}+\frac{z^{2}}{8^{2}-5^{2}}+\frac{w^{2}}{8^{2}-7^{2}}=1,
\end{array}\right.
$$

Find the value of $x^{2}+y^{2}+z^{2}+w^{2}$.
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Context:
In the convex quadrilateral ABCD, it is known that ∠DAB = ∠BCD = 90°, and ∠CDA : ∠ABC = 2 : 1, AD : CB = 1 : √3. What is the measure of ∠BDA?
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Context:
A nonempty word is called pronounceable if it alternates between vowels (A, E, I, O, U) and consonants (all other letters) and contains at least one vowel. How many pronounceable words can be formed using the letters P, U, M, A, C at most once each? Words of length shorter than $5$ are allowed.
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Context:
Given that a point $P$ is taken within a right circular cone $(M-ABC)$ such that the volume of the cone $(P-ABC)$ is not greater than one third of the volume of the cone $(M-ABC)$, what is the probability?
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Context:
Find the value of $k$ such that the curves $y=k\sqrt{x^2-1}$ and $y=\ln x$ are tangential.
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Context:
If the $60$th percentile of the data $0$, $1$, $2$, $3$, $4$, $5$, $7$, $8$, $9$, $10$ is $n$, find the constant term in the expansion of $(\frac{1}{x}-2x)^n$.
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Context:
Let $\{x_n\}_{n=1}^{\infty}$ be a convergent sequence in a normed space $(S, \|\cdot\|)$. Find the limit of the sequence \[\left\{\frac{x_1 + 2x_2 + \cdots + nx_n}{n^2}\right\}_{n=1}^{\infty}.\]
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Context:
Referees from China, Japan, and Korea, two from each country, will serve as judges for badminton courts 1, 2, and 3 at the Asian Games. It is stipulated that each court will have two referees, and referees from the same country cannot be assigned to the same court. The number of different allocation methods is ____. (Fill in the number)
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Context:
Petya was running down an escalator, counting the steps. Exactly halfway down, he tripped and tumbled the rest of the way (he tumbles 3 times faster than he runs). How many steps are on the escalator if Petya counted 20 steps with his feet (before falling) and 30 steps with his sides (after falling)?
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Context:
In a class consisting of 31 people, each group of 4 students contains at least one student who is friends with the other 3 students. In this class, what is the minimum number of students who are friends with all the other students except themselves?
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Context:
Find the determinant of an $n \times n$ matrix $A$ where each element $a_{i,j}$ is defined as $\gcd(i,j)$. What is the determinant of $A$?
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Context:
Let $J$ , $E$, $R$, and $Y$ be four positive integers chosen independently and uniformly at random from the set of factors of $1428$. What is the probability that $JERRY = 1428$? Express your answer in the form $\frac{a}{b\cdot 2^n}$ where $n$ is a nonnegative integer, $a $and $b$ are odd, and gcd $(a,b) = 1$.
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Context:
Given that the function $y = (m^2 + 2m - 2)x^{\frac{1}{m-1}}$ is a power function, find the value of $m$.
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Context:
Given a right circular cone $(P-ABC)$ with lateral edges $(PA)$, $(PB)$, $(PC)$ being pairwise perpendicular, and base edge $AB = \sqrt{2}$, find the surface area of the circumscribed sphere of the right circular cone $(P-ABC)$.
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Context:
In how many ways can a soccer game between two teams end with a total of 12 goals scored, such that the scores of the two teams never differ by more than 2 goals?
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Context:
Evaluate the integral \( \int_{0}^{\infty} xe^{-ikx} \, dx \) where \( k \) is a real number.
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Context:
Evaluate the integral:

$$\frac{\exp\left(-\frac{N}{2}\right)}{2\pi N} \iint \exp\left(-\left( \frac {x^2+2xy+2Ny+2y^2}{2N}\right)\right) \,dx\,dy$$

where $N > 0$.
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Context:
A traveler token starts on the center tile of a hexagonal grid with 19 tiles. Each turn, a die is rolled to determine the neighboring tile the traveler moves to, with all six directions being equally likely. The game ends on the turn the traveler leaves the board. What is the expected number of turns the game will last?
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Context:
Evaluate the limit of the integral \( \int_0^{\pi/2}\dfrac {dx} {\sqrt{\sin^2(x)+\epsilon \cos^2(x)}} \) as \( \epsilon \to 0^{+} \).
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Context:
In a tournament with 8 players where each player plays every other player exactly once, the winner of a game earns 1 point, the loser earns 0 points, and each player earns 0.5 points if the game is a draw. At the end of the tournament, each player has a different score, and the player who finishes in second place has a score equal to the combined scores of the four players who finish in the bottom four positions. Determine the result of the game between the player who finishes in third place and the player who finishes in seventh place.
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Context:
The price of a shirt is increased by 20%, then reduced by 15%, followed by an increase of 10%, and finally, a decrease of 5%. Calculate the final price of the shirt.
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Context:
Determine the conditions on the parameter $m$ such that every nontrivial solution of the differential equation \( y'' + \frac{m}{x}y = 0 \) has an infinite number of positive zeros.
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Context:
设正四棱锥 $P-A B C D$ 的底面边长与高相等, 点 $G$ 为侧面 $\triangle P B C$ 的重心,则直线 $A G$ 与底面 $A B C D$ 所成角的正弦值的原始答案为 \(\frac{\sqrt{m}}{n}\) 的形式，请给出 m + n 的值。
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Context:
Determine the values of \(a \in \mathbb{R}\) for which the integral \(\int_{0}^{\frac{\pi}{2}} \log(\cos(x)) \cdot (\tan(x))^a \, dx\) converges.
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Context:
Evaluate the improper integral: \[ \int_0^\infty \frac{1}{1+x^2 \sin^2 x} \, dx. \]
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Context:
Let $h(x) = 2x - 3$ and $j(x) = \frac{x}{4}$. Calculate the value of \[h(j^{-1}(h^{-1}(h^{-1}(j(h(12)))))).\]
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Context:
Given that point M is a moving point on the parabola $y^2=8x$, F is the focus of the parabola, and point A is on the circle $C: (x-3)^2+(y+1)^2=1$, then the minimum value of $|AM|+|MF|$ is.
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Context:
What is the minimum number of cells that need to be coloured in a 5x5 square grid so that every 1x4 or 4x1 rectangle in the grid has at least one coloured cell? Express your answer as a single integer.
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Context:
Given three complex numbers $z$, $w$, and $u$ such that $|z|=1$, $|w|=2$, $|u|=3$, and $|z+w-u|=|u+z-w|$, determine the greatest possible value of $|z-u|$. Provide your answer as a single number.
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Context:
Determine whether there exists a natural number $k$ such that for every even integer $n \ge k$, there exist positive integers $a, b, x, y$ with $\gcd(a, b) = 1$ and $a, b > 1$, satisfying $n = a^x + b^y$, where at least one of $x, y$ is greater than $1$. If such a $k$ exists, provide its value.
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Context:
The very hungry caterpillar lives on the number line. For each non-zero integer $i$, a fruit sits on the point with coordinate $i$. The caterpillar moves back and forth; whenever he reaches a point with food, he eats the food, increasing his weight by one pound, and turns around. The caterpillar moves at a speed of $2^{-w}$ units per day, where $w$ is his weight. If the caterpillar starts off at the origin, weighing zero pounds, and initially moves in the positive $x$ direction, after how many days will he weigh 10 pounds?
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Context:
Find the smallest natural number \( n \) such that among any \( n \) irrational numbers, there will always be 3 numbers for which the sum of each pair remains irrational.
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Context:
Given the vector field $F(x,y,z)=(z^2-x,-xy,3z)$ and the surface $S$ of the solid delimited by the equations $z=4-y^2, x=0, x=3$ and $z=0$, with the normal vector exterior, evaluate $\iint_S F\cdot n \,ds$ without using The Divergence Theorem.
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Context:
In an $8 \times 8$ chessboard, color the squares using 8 colors such that each square is colored in one color, neighboring squares sharing an edge have different colors, and each row contains all 8 colors. How many different coloring methods are there?
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Context:
Find all positive integer solutions (x, y) to the equation \(x^4 + 4x^3 + 6x^2 + 4x + 16 = y^2\).
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Context:
Determine the smallest positive integer \( q \) with the following property: for every integer \( m \) with \( 1 \leqslant m \leqslant 1006 \), there exists an integer \( n \) such that

\[
\frac{m}{1007} q < n < \frac{m+1}{1008} q
\]
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Context:
Let $n$ be a positive integer. Let there be $P_{n}$ ways for Pretty Penny to make exactly $n$ dollars out of quarters, dimes, nickels, and pennies. Also, let there be $B_{n}$ ways for Beautiful Bill to make exactly $n$ dollars out of one dollar bills, quarters, dimes, and nickels. As $n$ goes to infinity, the sequence of fractions \frac{P_{n}}{B_{n}}$ approaches a real number $c$. Find $c$.
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Context:
The sequence starts with 12, 13, 15, 17, 111, 113, 117, 119, 123, followed by a number, and then 131. What is the tenth number in the sequence?
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Context:
Define the sequence $\left\{a_{n}\right\}$: $a_{1}=1, a_{2}=2, a_{n+2}=a_{n}+a_{n+1}$ for $n \in \mathbf{N}_{+}$. Calculate the value of $\left[\frac{a_{2}}{a_{1}}\right] \cdot\left\{\frac{a_{3}}{a_{2}}\right\} \cdot\left\{\frac{a_{4}}{a_{3}}\right\} \cdot \cdots \ \left\{\frac{a_{99}}{a_{98}}\right\} \cdot\left[\frac{a_{98}}{a_{2}}\right]$. Express your answer as a single number.
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Context:
A $2008 \times 2009$ rectangle is divided into unit squares. In how many ways can you remove a pair of squares such that the remainder can be covered with $1 \times 2$ dominoes?
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Context:
Let \(\mathcal{P}\) be a parabola, and let \(V_{1}\) and \(F_{1}\) be its vertex and focus, respectively. Let \(A\) and \(B\) be points on \(\mathcal{P}\) so that \(\angle A V_{1} B=90^{\circ}\). Let \(\mathcal{Q}\) be the locus of the midpoint of \(AB\). It turns out that \(\mathcal{Q}\) is also a parabola, and let \(V_{2}\) and \(F_{2}\) denote its vertex and focus, respectively. Determine the ratio \(F_{1} F_{2} / V_{1} V_{2}\).
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Context:
Two people start at the same time and move with the same pace on a 5 x 5 square grid, one starting from the bottom left corner and the other from the top right corner. Calculate the probability that they meet on the grid. Express your answer as a fraction in the form \( \frac{a}{b} \).
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Context:
A sequence is defined as \( {a}_{1} = {2025} \) and for all \( n \geq  2 \) ,
 \( {a}_{n} = \frac{{a}_{n - 1} + 1}{n} \)
 Determine the smallest \( k \) such that \( {a}_{k} < \frac{1}{2025} \) .
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Context:
Find the distribution of the random variable \( Y = X^2 \), where \( X \sim N(0,1) \).
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Context:
Evaluate the expression $(1^1+2^1+2^{2}+\dots)(1^1+3^1+3^{2}+\dots)(1^1+5^1+5^{2}+\dots)\dots$ using the properties of the Riemann zeta function.
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Context:
6 animals of a circus have to be placed inside some cages, one in each cage. If a certain number of cages are too small for 6 of the animals, there are 240 ways of caging the animals. How many cages are too small for the animals?
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Context:
A line divides the length of an arc of a circle in the ratio 1:3. In what ratio does it divide the area of the circle?
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Context:
Find all positive integers $k$ such that there exist positive integers $a$ and $b$ satisfying the equation:
\[
a^2 + 4 = (k^2 - 4)b^2.
\]
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Context:
Form a four-digit number without repeating digits using the numbers 0, 1, 2, 3, 4, 5, 6, where the sum of the digits in the units, tens, and hundreds places is even. How many such four-digit numbers are there? (Answer with a number)
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Context:
Determine the number of terms needed in the series expansion of \( \int_{0}^{1} \frac{e^x-1}{x} \, dx \) to approximate the integral with an error less than or equal to \( 10^{-6} \).
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Context:
Find the area enclosed by the curve $g(x)$, the lines $x=-3$ and $x=5$, and the $x$-axis, where $g(x)$ is the inverse of $f(x)=x^3+3x+1$. Express your answer as a single numerical value.
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Context:
Compute the largest possible number of distinct real solutions for \( x \) to the equation
 \( {x}^{6} + a{x}^{5} + {60}{x}^{4} - {159}{x}^{3} + {240}{x}^{2} + {bx} + c = 0, \)
 where \( a,b \) ,and \( c \) are real numbers.
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Context:
Given \( x \in (0,1) \) and \( \frac{1}{x} \notin \mathbf{Z} \), define
\[
a_{n}=\frac{n x}{(1-x)(1-2 x) \cdots(1-n x)} \quad \text{for} \quad n=1,2, \cdots
\]

A number \( x \) is called a "good number" if and only if it makes the sequence \(\{a_{n}\}\) satisfy
\[
a_{1}+a_{2}+\cdots+a_{10} > -1 \quad \text{and} \quad a_{1} a_{2} \cdots a_{10} > 0
\]

Find the sum of the lengths of all intervals on the number line that represent all such "good numbers".
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Context:
Let \( X_{1}, X_{2}, \ldots, X_{100} \) be a sequence of mutually distinct nonempty subsets of a set \( S \). Any two sets \( X_{i} \) and \( X_{i+1} \) are disjoint and their union is not the whole set \( S \), that is, \( X_{i} \cap X_{i+1}=\emptyset \) and \( X_{i} \cup X_{i+1} \neq S \), for all \( i \in \{1, \ldots, 99\} \). Find the smallest possible number of elements in \( S \).
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Context:
A spaceship travels randomly between three planets: A, B, and C. The journey starts at planet A and ends when the spaceship reaches planet C. The transition probabilities and travel times (in months) between the planets are given in the matrices below. Calculate the expected time (in months) for the spaceship to reach planet C starting from planet A.

**Probability Matrix**

| Origin | Destination A | Destination B | Destination C |
|--------|---------------|---------------|---------------|
| A      | 0.05          | 0.90          | 0.05          |
| B      | 0.50          | 0.50          | 0.00          |
| C      | 0.00          | 0.00          | 1.00          |

**Travel Time Matrix (in months)**

| Origin | Destination A | Destination B | Destination C |
|--------|---------------|---------------|---------------|
| A      | 5             | 2             | 10            |
| B      | 3             | 3             | 0             |
| C      | 0             | 0             | 0             |
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Context:
The sequence $ (a_n)$ is given by $ a_1=1,a_2=0$ and:

$ a_{2k+1}=a_k+a_{k+1}, a_{2k+2}=2a_{k+1}$ for $ k \in \mathbb{N}.$

Find $ a_m$ for $ m=2^{19}+91.$
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Context:
Determine the maximum value of \(|f'''(x)|\) for a function \(f\) that is infinitely differentiable on the real numbers, given that \(f(0) = 1\), \(f'(0) = 1\), \(f''(0) = 2\), and \(f(1) < 5\).
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Context:
Find an element of order 5 in the field \( \mathbb{Z}_3[x]/\langle x^4 + x + 2 \rangle \).
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Context:
Calculate the divergence \( \nabla \cdot \hat{n} \) where \( \hat{n} \) is a unit vector normal to a cylinder of radius \( R \) and infinite length \( L = \infty \).
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Context:
Given the function $f(x)={x}^{3}+a{x}^{2}+bx+{a}^{2},$, which has an extreme value of $10$ at $x=1$, find the value of $a+b$.
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Context:
The diagonal of an isosceles trapezoid is $a$, and the midline is $b$. Find the height of the trapezoid.
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Context:
Unit circle \(\Omega\) has points \(X, Y, Z\) on its circumference so that \(X Y Z\) is an equilateral triangle. Let \(W\) be a point other than \(X\) in the plane such that triangle \(W Y Z\) is also equilateral. Determine the area of the region inside triangle \(W Y Z\) that lies outside circle \(\Omega\).
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Context:
Find the expression for the supremum of the absolute values of the inner products between a unit vector $x$ in a Hilbert space $H$ and vectors $y$ in a subspace $S$ with norm less than or equal to 1.
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Context:
Max likes to collect model trains. He asks for one for every birthday of his, and asks for two each Christmas. Max always gets the gifts he asks for, and asks for these same gifts every year for 5 years. At the end of the 5 years, his parents give him a certain multiple of the number of trains he already has. Max has 45 trains now. What is the ratio of the number of trains his parents gave him at the end of the 5 years to the number of trains he already had?
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Context:
Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The third monkey takes the remaining bananas from the pile, keeps one-twelfth of them, and divides the rest equally between the other two. Given that each monkey receives a whole number of bananas whenever the bananas are divided, and the numbers of bananas the first, second, and third monkeys have at the end of the process are in the ratio $3: 2: 1$, what is the least possible total for the number of bananas?
Please reason step by step, and put your final answer within \boxed{}.


Context:
An 18 inch by 24 inch painting is mounted in a wooden frame where the width of the wood at the top and bottom of the frame is twice the width of the wood at the sides. If the area of the frame is equal to the area of the painting, calculate the ratio of the shorter side to the longer side of this frame. Express your answer as a simplified ratio in the form a:b.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a monotonic function f(x) defined on (0, +∞), for ∀x ∈ (0, +∞), it holds that f[f(x)-log2(x)] = 3. Find the interval in which the solution of the equation f(x)-f''(x)=2 lies. Express your answer as an interval.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the outer normal of the set of $k \times k$ positive definite symmetric matrices, $S^k_+$, as a subset of the set of $k \times k$ symmetric matrices, $S^k$, with respect to the entry-wise inner product.
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Context:
Let \( A = \{1, 2, 3, 4, 5, 6\} \). Determine the number of distinct bijective functions \( f: A \rightarrow A \) such that \( f(f(f(n))) = n \) for all \( n \in A \).
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Context:
Suppose $A_1, A_2, A_3, \ldots, A_{20}$ is a 20-sided regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon, but the sides are not the sides of the polygon?
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Context:
The hyperbola $C$ has its center at the origin and its foci on the $x$-axis. $C$ intersects the directrix of the parabola $y^{2}=16x$ at points $A$ and $B$, with $|AB|=4 \sqrt {3}$. Determine the length of the real axis of $C$.
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Context:
In a game where each pancake is placed randomly on one of N stacks, calculate the expected number of additional pancakes needed to cover the (k+1)-th pancake on a stack after the k-th pancake has been covered.
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Context:
In a queue of 100 people waiting to enter a hall with 100 numbered seats from 1 to 100, the first person chooses any seat and sits down. Each subsequent person, from the 2nd to the 100th, sits in their corresponding seat number if it is available; otherwise, they choose any unoccupied seat. How many ways can the 100 seats be occupied such that the 100th person sits in seat number 100?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a division sum, the divisor is 10 times the quotient and some times the remainder. The remainder is 46, and the dividend is 5290. What is the ratio of the divisor to the remainder?
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Context:
Calculate the volume of the solid generated by revolving the curve defined by the parametric equations \(x(t) = a(t - \sin(t))\), \(y(t) = a(1 - \cos(t))\), and \(z(t) = a(1 - \cos(t))\sin(\theta)\) around the x-axis, where \(t\) and \(\theta\) range from \([0, 2\pi]\).
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Context:
Two lines, parallel to the bases of a trapezoid, divide each of the lateral sides into three equal parts. The entire trapezoid is divided into three sections by these lines. Find the area of the middle section if the areas of the outer sections are \(S_{1}\) and \(S_{2}\).
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Context:
Let \( K \) be a field and \( a \in K \) such that \( f(x) = x^n - a \) is irreducible. If \( m \in \mathbb{N} \) divides \( n \) and \( \alpha \) is a root of \( f(x) \) in an extension field of \( K \), find the minimal polynomial of \( \alpha^m \).
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Context:
In a 12-hour interval (from 0 hours to 12 hours), how many minutes are there when the value of the hour is greater than the value of the minutes?
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Context:
Determine the asymptotic complexity of the recurrence relation given by $T(1) = 1$ and $T(n) = 3T(\sqrt[3]{n}) + \log_{2}(\log_{2}n)$. Express your answer using big O notation.
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Context:
Compute the area of an ideal triangle in the Poincaré disk model, where the triangle's vertices are on the boundary of the disk. One side of the triangle is a vertical geodesic that intersects a semicircle at the points (0,0) and (1,0). The semicircle is described by the equation \((x-\frac{1}{2})^2 + y^2 = \frac{1}{4}\) with \(y > 0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jude bought three chairs for his house, all at the same price. He also bought a table, four lamps, two sets of plates at $20 for each set, and a painting. The table costs twice as much as one of the chairs, and the lamps each cost half of the table's price. The painting originally costs $100 but was 25% off. After giving the cashier $350, Jude got a change. How much did each of the chairs cost?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of solutions \((\alpha, \beta)\) in the finite field \(F_{q^2}\) such that \(\alpha + \alpha^q + \beta \beta^q = 0\).
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Context:
Identify the smallest non-abelian group that is not isomorphic to a cyclic group, a direct product of cyclic groups, or a semi-direct product of cyclic groups, excluding finite abelian groups and groups of order up to 7.
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Context:
Given that $X$ follows a uniform distribution $Uniform(1, 9.9)$ and, conditionally on $X = x$, $Y$ is normally distributed $Normal(1.4, x^2)$, find the covariance between $X$ and $Y$, denoted as $Cov(X, Y)$. Calculate $E[XY]$ as part of your solution.
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Context:
You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out you must open all the doors and disarm all the mines. In the room is a panel with $3$ buttons, which conveniently contains an instruction manual. The red button arms a mine, the yellow button disarms two mines and closes a door, and the green button opens two doors. Initially $3$ doors are closed and $3$ mines are armed. The manual warns that attempting to disarm two mines or open two doors when only one is armed/closed will reset the system to its initial state. What is the minimum number of buttons you must push to get out?
Please reason step by step, and put your final answer within \boxed{}.


Context:
From a point on a circle, two chords are drawn with lengths of 10 cm and 12 cm. Find the radius of the circle if the distance from the midpoint of the shorter chord to the longer chord is 4 cm.
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Context:
Determine the number of elements of order \( p \) in the group \( \text{GL}(2, \mathbb{Z}_p) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given three complex numbers $(z_1, z_2, z_3)$ lying on the unit circle and related by the equation $ \frac{ z_1^2}{z_2 z_3} + \frac{ z_2^2}{z_1 z_3} + \frac{ z_3^2}{z_1 z_2} +1 =0$, find the sum of all possible values of $|z_1 + z_2 + z_3|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rectangle ABCD is given. On the side AB, \( n \) different points are chosen strictly between A and B. Similarly, \( m \) different points are chosen on the side AD between A and D. Lines are drawn from the points parallel to the sides. How many rectangles are formed in this way?
Please reason step by step, and put your final answer within \boxed{}.


Context:
考虑1,2···，20的排列$( a _ { 1 } a _ { 2 } \cdots a _ { 2 0 } )$，对此排列进行如下操作：对换其中某两个数所处的位置，目标是将此排列变为（1,2，···，20）．设对每一个排列$a = ( a _ { 1 } , a _ { 2 } , \cdots ,$ $a _ { 2 0 } )$达到目标所需要进行操作次数的最小值记为$k _ { a }$，求$k _ { a }$的最大值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $\sqrt{x+3}$ is the simplest quadratic radical, then the possible value of $x$ is ________. (Fill in the correct answer)
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Context:
Observe the following equations: $a + b = 1$, $a^2 + b^2 = 3$, $a^3 + b^3 = 4$, $a^4 + b^4 = 7$, $a^5 + b^5 = 11$. Identify the pattern in the sequence of results on the right side of the equations and calculate $a^{10} + b^{10}$ directly. Provide your answer as a single integer.
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Context:
Calculate the partial derivative of the function \( f(A) = \langle Ax, By \rangle \) with respect to the matrix \( H \), where \( A, B \in L(\mathbb{R}^n, \mathbb{R}^k) \), \( x, y \in \mathbb{R}^n \), and \( H \in L(\mathbb{R}^n, \mathbb{R}^k) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( a_{1}, a_{2}, \cdots, a_{2014} \) be a permutation of the positive integers \( 1, 2, \cdots, 2014 \). Define
\[ S_{k} = a_{1} + a_{2} + \cdots + a_{k} \quad (k=1, 2, \cdots, 2014). \]

What is the maximum number of odd numbers among \( S_{1}, S_{2}, \cdots, S_{2014} \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \( \triangle ABC \), \( BC = 4 \) and \( \angle BAC = 60^\circ \). Let \( I \) be the incentre of \( \triangle ABC \). The circle passing through \( B, I \), and \( C \) meets the perpendicular bisector of \( BC \) at a point \( X \) inside \( \triangle ABC \). Find \( AX \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( \text{Revolution}(x) = x^3 + Ux^2 + Sx + A \), where \( U \), \( S \), and \( A \) are all integers and \( U + S + A + 1 = 1773 \). Given that \( \text{Revolution} \) has exactly two distinct nonzero integer roots \( G \) and \( B \), find the minimum value of \( |GB| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Darnell pays $12 for unlimited texting, calling, and data on his phone each month. An alternative phone plan would charge $1 per 30 texts, $3 per 20 minutes of calls, $5 per 2GB of data usage, and an additional $2 per month for international calls. Darnell sends 60 texts, spends 60 minutes on the phone, uses 3GB of data, and makes 10 minutes of international calls each month. How many dollars less would he pay on the alternative plan?
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Context:
A semicircle with radius 2021 has diameter $AB$ and center $O$. Points $C$ and $D$ lie on the semicircle such that $\angle AOC < \angle AOD = 90^{\circ}$. A circle of radius $r$ is inscribed in the sector bounded by $OA$ and $OC$ and is tangent to the semicircle at $E$. If $CD=CE$, compute $\lfloor r \rfloor$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of $6 \times 6$ Latin squares that do not contain a $3 \times 2$ saturated sub-matrix (SSM).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of terms required in the Taylor series expansion of $\arctan x$ around $0$ to approximate $\pi$ to 10 decimal places when $x = 1$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n \to \infty} \left\{ \frac{n!}{\prod_{k=1}^{n} H_k} \right\}^{\frac{1}{n}} \] where \( H_k = \sum_{i=1}^{k} \frac{1}{i} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f\left( x \right)=\sin \left( \omega x+\varphi \right)\left( \omega > 0,\left| \varphi \right|\leqslant \dfrac{\pi }{2} \right)$, and knowing that $x=-\dfrac{\pi }{4}$ is a zero point of $f\left( x \right)$, $x=\dfrac{\pi }{4}$ is the axis of symmetry of the graph of $y=f\left( x \right)$, and $f\left( x \right)$ is monotonic in $\left( \dfrac{\pi }{18},\dfrac{5\pi }{36} \right)$, find the maximum value of $\omega$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the radius of convergence of the power series \( \sum_{n=0}^\infty n!z^{n!} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
$f(0)=0, f(1) = 1$, and for every $n \geq 1$, $f(3n-1)= f(n)-1, f(3n)= f(n), f(3n+1)=f(n)+1$. So $f(2011)$ is
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find $10110_2\times10100_2\div10_2$. Express your answer in base 2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the paths from \((0,0)\) to \((6,3)\) that only take steps of unit length up and right. Compute the sum of the areas bounded by the path, the \(x\)-axis, and the line \(x=6\) over all such paths.
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Context:
Given that $x, y, z$ are positive real numbers satisfying $\frac{1}{3} \le xy + yz + zx \le 3$, determine the range of possible values for the product $xyz$. Express your answer as an interval.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the Riemann-Stieltjes integral \( \int_{-1}^1 \frac{1}{1+x^2} \, dg(x) \), where \( g(x) = \begin{cases} 1 & \text{if } x<0, \\ 0 & \text{if } x=0, \\ -1 & \text{if } x>0. \end{cases} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the vector \( \boldsymbol{a}_{k}=\left(\cos \frac{k \pi}{6}, \sin \frac{k \pi}{6}+\cos \frac{k \pi}{6}\right) \), where \( k=0,1,\cdots,12 \), find the value of \( \sum_{k=0}^{11} \boldsymbol{a}_{k} \cdot \boldsymbol{a}_{k+1} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be a set with $\text{card}(A) = a$. Determine the cardinal number of the set of countably infinite subsets of $A$. Express your answer in terms of $a$. \( \text{Hint: Consider the set of injective functions from } \mathbb{N} \rightarrow A. \)
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Context:
A circular sector with radius \( R \) and an opening angle of \( \frac{\pi}{2} \) is rotated around its axis of symmetry (the x-axis) to form a homogeneous body. Determine the position of the center of gravity of this body.
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Context:
Let $ S(N) $ be the number of 1's in the binary representation of an integer $ N $, and let $ D(N) = S(N + 1) - S(N) $. Compute the sum of $ D(N) $ over all $ N $ such that $ 1 \leq N \leq 2017 $ and $ D(N) < 0 $.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the set of discontinuities of the function:
\[ f(x) = \lim_{n \to \infty} \left( \lim_{t \to 0} \frac{(\sin(n!\pi x))^2}{(\sin(n!\pi x))^2 + t^2} \right) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the asymptotic approximation for the expression \( \left(1 - \sqrt{ \frac{2}{n-1} } \cdot \frac{ \Gamma(n/2) }{ \Gamma( \frac{n-1}{2} ) } \right) \) as \( n \to \infty \), where \( n \) is a positive integer, using the Taylor series expansion.
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Context:
Let \( f \) be a function such that \( f'(x) \) is continuous on \([0,1]\) and satisfies the following conditions: \( f(1) = 0 \), \( \int_0^1 [f'(x)]^2 \, dx = 7 \), and \( \int_0^1 x^2 f(x) \, dx = \frac{1}{3} \). Calculate \( \int_0^1 f(x) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $a$ for which the given population model exhibits cyclical behavior.
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Context:
Calculate the sum of the series $\sum\limits_{-\infty}^{\infty} \frac{\sin^3 3^k}{3^k}$.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{
The sum of the series is \( \frac{3}{4} \).

}


Context:
Determine the probability that a random polynomial of degree \( n \), with integer coefficients between \(-L\) and \(L\), is irreducible over the rationals and has the symmetric group \( S(n) \) as its Galois group.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $p$ for which the Fourier transform $\mathcal{F}: L^p(\mathbb{T}) \rightarrow l^{p'}(\mathbb{Z})$ is a continuous and surjective mapping.
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Context:
Determine the number of zeros of the entire analytic function $g(z)$ inside the unit square in the first quadrant, given the conditions $g(z+1) = g(z)$, $g(z+i) = e^{\pi}e^{-2\pi iz}g(z)$, and that $g(z)$ has no zeros on the boundary of the square.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of partitions of the set \( \{0, 1, 2, 3, 4, 5, 6, 7\} \) that have an even number of parts. What is the cardinality of this set of partitions?
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Context:
Determine the number of Hausdorff topologies on a set with 5 elements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a compact connected Riemannian 2-manifold $M$ with positive curvature and diameter 1. Determine the supremum of the average distance between two randomly chosen points on $M$, distributed according to the volume form on $M$. Assume $M$ is diffeomorphic to a 2-sphere. What is this supremum?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{\mathbb{R}} (F(x+a) - F(x)) \, dx \) for any real number \( a \), where \( F \) is a distribution function.
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Context:
8. Let $A B C$ be an equilateral triangle with side length 8 . Let $X$ be on side $A B$ so that $A X=5$ and $Y$ be on side $A C$ so that $A Y=3$. Let $Z$ be on side $B C$ so that $A Z, B Y, C X$ are concurrent. Let $Z X, Z Y$ intersect the circumcircle of $A X Y$ again at $P, Q$ respectively. Let $X Q$ and $Y P$ intersect at $K$. Compute $K X \cdot K Q$.
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Context:
Sari and Ken climb up a mountain. On the day they are supposed to reach the summit, Sari wakes up at 08:00 and starts climbing at a constant pace. Ken starts climbing only at 10:00, when Sari is already 700 meters ahead of him. Nevertheless, Ken climbs at a constant pace of 500 meters per hour, and reaches the summit before Sari. If Sari is 50 meters behind Ken when he reaches the summit, at what time did Ken reach the summit?
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Context:
Evaluate the integral \( A = \int_0^1 \{\operatorname{arccosh}(x)\} \, dx \), where \( \{x\} \) denotes the fractional part of \( x \) and for complex \( x, \{x\} = \{\Re x\} + i\{\Im x\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the supremum of the set of maximum eigenvalues of correlation matrices, where a correlation matrix is defined as a positive semidefinite matrix $B \in M_{n}(\mathbb{C})$ with the identity matrix $I$ as its diagonal.
Please reason step by step, and put your final answer within \boxed{}.