Context:
Evaluate the improper integral \( \int_0^\infty \frac{x\sin x}{1+x^2} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For real numbers \(x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty)\), let \(S_k = \sum_{i = 1}^{333} x_i^k\) for each \(k\). Given that \(S_2 = 777\), find the minimum possible value of \(S_3\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the closed-form expression for the constant $c_n$, defined as follows?
$$ c_n := n! \sum_{i+j+k+l=n} \binom{2i}{i}\binom{2j}{j}\binom{2k}{k}\binom{2l}{l} $$
Here, $i, j, k, l$ are nonnegative integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
123. In reverse order. What nine-digit number, when multiplied by 123456 789, gives a product where the nine least significant digits are 9,8,7,6,5,4,3, 2,1 (in that exact order)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve for $z$ in the following equation: $2-iz = -1 + 3iz$.

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


Context:
Let $\mathcal{A} = A_0A_1A_2A_3 \cdots A_{2013}A_{2014}$ be a regular 2014-simplex, meaning the 2015 vertices of $\mathcal{A}$ lie in 2014-dimensional Euclidean space and there exists a constant $c > 0$ such that $A_iA_j = c$ for any $0 \le i < j \le 2014$. Let $O = (0,0,0,\dots,0)$, $A_0 = (1,0,0,\dots,0)$, and suppose $A_iO$ has length $1$ for $i=0,1,\dots,2014$. Set $P=(20,14,20,14,\dots,20,14)$. Find the remainder when $PA_0^2 + PA_1^2 + \dots + PA_{2014}^2$ is divided by $10^6$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(p(x) = x^5 - 833x^4 + ax^3 + bx^2 + cx + d\) be a polynomial whose roots are in geometric progression. If the sum of the reciprocals of the roots is 17, determine all possible values of \(d\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest integer $N$ such that the sequence $\{ N, N+1, N+2, \ldots, N+a\}$ consists entirely of composite numbers. Provide a general formula for $N$ in terms of $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given integers $a$, $b$, $c$, and $d$ such that $a$ and $b$ are relatively prime, $d \leq 2022$, and $a + b + c + d = ac + bd = 0$, determine the largest possible value of $d$. \( \text{(Express your answer as an integer.)} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that $f$ and $g$ are continuous functions on $(0, \\infty)$ such that $f(x \\cdot t) = f(x)g(t)$ for all $x, t > 0$. Given that $g'(1) = 2$, find $f'(1)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the ring of integers of an imaginary quadratic field, determine the number of elements with norm equal to a prime number $q$, given that $q$ splits in the field and $N(q) = q^2$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the degree of the splitting field of the polynomial \( p(x) = x^4 - 2x^2 - 8 \) over \( \mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a group \(G\), it is given that \(a^5 = e\), where \(e\) is the identity element of \(G\). If \(a, b \in G\) and \(aba^{-1} = b^2\), determine the order of the element \(b\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of real solutions $a$ to the equation:
\[ \left\lfloor \frac{1}{2}a \right\rfloor + \left\lfloor \frac{1}{3}a \right\rfloor + \left\lfloor \frac{1}{5}a \right\rfloor = a. \]
Here, if $x$ is a real number, then $\left\lfloor x \right\rfloor$ denotes the greatest integer that is less than or equal to $x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a discrete-time Markov Chain described by the equation $X(n+1) = X(n) - 1 + A(n)$, where $A(n)$ is a random variable taking values $0, 1, 2$ with probabilities $p_0, p_1, p_2$, respectively, and $X(n) = n$ with $X(n) \geq 1$, determine the condition on $p_0, p_1, p_2$ for the stability of the system.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the general solution of the differential equation $y = xy' - {y'}^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a set of n natural numbers, determine the number of possible order relations among them. Express your answer in terms of Fubini numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many trees \( T = \langle V, E \rangle \) exist such that \( V = \{ v_1, v_2, \dots, v_7 \} \), with \( \deg(v_1) = 1 \) and \( \deg(v_j) \leq 5 \) for every \( j = 2, \ldots, 7 \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square is divided by horizontal and vertical lines that form \( {n}^{2} \) squares each with side 1. What is the greatest possible value of
 \( n \) such that it is possible to select \( n \) squares such that any rectangle with area \( n \) formed by the horizontal and vertical lines
 would contain at least one of the selected \( n \) squares.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit \[ \lim_{b\to \infty}\frac{\vert \text{erf}(\sqrt{a+\mathrm{i} b})\vert^{2}}{\sqrt{a^2+b^2}} \] where \[ \text{erf}(a+\mathrm{i} b)=\frac{2}{\sqrt{\pi}}\int_{0}^{a+\mathrm{i} b}e^{-t^2}~dt \] is the error function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
## Task B-4.4.

From point $A$ outside a circle, tangents are drawn to circle $k$ with points of tangency at points $B$ and $C$. A line parallel to $A C$ passes through point $B$ and intersects circle $k$ again at point $D$. If $|A B|=49$ and $|C D|=28$, determine the length of $|A D|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the area enclosed by the curves \(y=\frac{4}{x}\), \(y=\frac{6}{x}\), \(y=4x\), and \(y=8x\) using a suitable coordinate transformation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of all seven-digit numbers that start with the digit 1 and end with the digit 9, such that the difference between any two adjacent digits is either 1 or 2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve for \( x \) in degrees in the equation: \[ \sin 84^\circ \sin(54^\circ-x) = \sin 126^\circ \sin x. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $\triangle ABC$, $\angle B$ is obtuse. The interior bisector $BM$ is drawn from point $B$, along with the altitudes $AN$ and $CQ$. Given that $AN = 8$ cm and $CQ = 12$ cm, determine the length of the perpendicular drawn from $M$ to line $\overline{AC}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n \to \infty} \left[ \frac{1}{n+1} + \frac{1}{n+2} + \ldots + \frac{1}{kn} \right].$$ Determine if the limit is 0 or another value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x\rightarrow0}\frac{1-\frac{1}{2}x^2-\cos(\frac{x}{1-x^2})}{x^4}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the first common value of the sequences defined by the expressions \(m^2 + 928m + 687\) and \(n^2 + 54n + 729\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the determinant of the matrix \(A_{2n}:=\begin{pmatrix} a &0 &\cdots & \cdots &0 &b \\ 0& a & & & b&0 \\ \vdots &\vdots &\ddots &\ddots &\vdots &\vdots \\ \vdots&\vdots & \ddots & \ddots &\vdots &\vdots \\ 0& b & & & a&0 \\ b&0 &\cdots &\cdots &0 &a \end{pmatrix}\in \mathbb{R}^{2n \times 2n}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine a formula to calculate the number of distinct accepted $2 \times n$ matrices, where the first row is a permutation of the numbers $1$ to $n$, and the second row is also a permutation such that no numbers are the same within rows or columns, and no column has identical numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest non-negative integer \(a\) such that \(\lim_{x \to 1} \left(\frac{-ax+\sin(x-1)+a}{x+\sin(x-1)-1}\right)^{\frac{1-x}{1-\sqrt{x}}} = \frac{1}{4}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Fill the integers from 1 to \(2016^2\) into a \(2016 \times 2016\) grid, ensuring each cell contains a unique number. Draw a vector from the center of one cell to the center of another if the number in the first cell is smaller than the number in the second cell. Given that the sum of the numbers in each row and each column of the grid is the same, determine the sum of all the vectors drawn.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a coffeehouse, the number of guests follows a Poisson distribution with parameter 10, i.e., $X \sim \text{Poi}(10)$. Each guest independently drinks a Poisson(1) number of cups of coffee and raises their hand a Poisson(1) number of times. The number of guests, the number of cups of coffee each guest drinks, and the number of hand raises each guest performs are all independent. Given that no cups of coffee have been sold, calculate the expected value of the total number of hand raises.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \oint_C f(z) \, dz \) where \( f(z) = \pi e^{\pi \cdot \overline{z}} \) and \( C \) is the square contour with vertices at \( 0 \), \( 1 \), \( 1+i \), and \( i \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the general solution to the ordinary differential equation \( x\frac{dy}{dx} = y - \frac{1}{y} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can a plane intersect a regular dodecahedron in space to form a regular hexagon?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( Z = \{ (1,0), (0,1) \} \subset \mathbb{A}_k^2 \). Find the ideal \( \mathbb{I}(Z) \) in the polynomial ring \( k[x, y] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum number of attracting and neutral orbits for a complex polynomial of degree $d$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Provide an example of an ideal $I$ in the polynomial ring $R[x]$ such that the set $X = \{ x \in R : f(x) = 0, \forall f \in I \}$ is empty.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the infinity norm \(||B^{-1}||_{\infty}\) for the square real matrix \(B\) defined by:
\[ b_{ii}=1, \quad b_{ij}=-1 \text{ for } i<j, \quad b_{ij}=0 \text{ for } i>j. \]

Given that \(B = I + N\), where \(N\) is a strictly lower triangular matrix, find \(||B^{-1}||_{\infty}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the tighter bounds for $|\sin z|$ given that $|z|=1$.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\sin 1 \leq |\sin z| \leq \sinh 1}


Context:
Determine the smallest integer $a$ such that the inequality \(x^2 - ax - \ln x + e^{\sin x} - 1 < 0\) is satisfied for at least one positive value of $x$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the equation of the locus of the foot of the perpendicular drawn from the origin to a line that makes intercepts on the coordinate axes, given that the square of the length of the perpendicular is equal to the product of the lengths of the intercepts.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the expression:
\[
\left[\cos ^{2}\left(\dfrac{\theta }{2}\right) + \,\sqrt{\,{1 - \gamma}\,}\,\sin^{2}\left(\dfrac{\theta }{2}\right)\right] ^{2} + \dfrac{\gamma }{4}\,\sin^{2}\left(\theta\right)\quad \text{where}\quad 0 \le \gamma \le 1.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the product as \( n \) approaches infinity:
\[ P=\lim_{n \to \infty}\prod_{a=5}^{n}\frac{a^3-64}{a^3+64} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A frog makes 3 jumps, each exactly 1 meter long, with the directions of the jumps chosen independently at random. What is the probability that the frog's final position is no more than 1 meter from its starting position?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the behavior of the iterated composition of the function \( f(x) = \sin(\cos(x)) \) as the number of iterations approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle ABC, points E, F, and C are collinear, D, E, and B are collinear, and A, D, and F are collinear. Given that |CF| = |FE|, |DE| = |EB|, and |AD| = |DF|, with |AC| = |CB| = \( \sqrt{14} \) dm and the area of triangle DEF is 0.01 m², calculate the length of |AB| in centimeters.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the genus $g$ of a closed Riemann surface if its tangent bundle $T$ is trivial.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X$ and $Y$ be two random variables such that for all $1 \le m < k$, the joint probability is given by: \[ P(X=k, Y=m) = \frac{1}{16}\left(\frac{3}{4}\right)^{k-2} = \frac{1}{9}\left(\frac{3}{4}\right)^{k} \] Calculate the covariance $\text{Cov}(X,Y)$ using the formula: \[ \text{Cov}(X,Y) = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y] \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the ratio of the minimum to the maximum area of a circle that is concentric with the ellipse \(x^2 + 2y^2 = 2\), given that tangents are drawn from a point \((h, k)\) such that \(h^2 + k^2 = 3\), and the chord of contact touches the circle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( F_1 = \mathbb{Q}(e^{\frac{2\pi i}{5}}) \) and \( F_2 = \mathbb{Q}(\cos(\frac{2\pi}{5})) \). Calculate the degree of the field extension \([F_1F_2:F_2]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim _{n \rightarrow \infty}\left(\frac{\sin \left\{\frac{2}{n}\right\}}{\left[2 n \tan \frac{1}{n}\right]\left(\tan \frac{1}{n}\right)}+\frac{1}{n^{2}+\cos n}\right)^{n^{2}} \] where \(\{\}\) and \([\]\) denote the fractional part function and the greatest integer function, respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f:[0,1]\rightarrow \mathbb{R} \) be a differentiable function such that \( f(0)=f(1)=0 \) and \( f''(x)-4f'(x)+4f(x)\ge e^{2x} \) for all \( x \in [0,1] \). Determine the minimum value of \( \frac{f'(x)}{f(x)} \) on the interval \( (0,\frac{1}{2}) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the range of values for the parameter $a$ such that the function \( g(x) = \frac{ax^2-4x+2}{a-4x+2x^2} \) is onto.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a constant \( A > 0 \) such that for all real \( x \geqslant A \), we have

\[ x^{3} + x + 1 \leq \frac{x^{4}}{1000000} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest constant \( C \) such that for all non-negative real numbers \( x, y, z \), the inequality \((yz + zx + xy)^{2}(x + y + z) \geq C \,xyz\,(x^{2} + y^{2} + z^{2})\) holds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of the region in the first quadrant bounded by the parabolas $y=x^2$, $y=2x^2$, $x=y^2$, and $x=2y^2$ using a change of variables.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \iint\limits_D \max\{\sin x, \sin y\} \, dx \, dy \), where \( D = \{0 \le x \le \pi, \ 0 \le y \le \pi\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many integers $n$, satisfy $|n|< 2020$ and the equation $11^3|n^3+3n^2-107n+1$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two players play the following game. On the lowest left square of an $8\times 8$ chessboard there is a rook. The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second player is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{Player $2$ has a winning strategy.}


Context:
Five dice are rolled. The product of the faces are then computed. Which result has a larger probability of occurring; $180$ or $144$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In chess tournament participates $n$ participants ($n >1$). In tournament each of participants plays with each other exactly $1$ game. For each game participant have $1$ point if he wins game, $0,5$ point if game is drow and $0$ points if he lose game.
If after ending of tournament participant have at least $ 75 %
$ of maximum possible points he called $winner$ $of$ $tournament$.
Find maximum possible numbers of $winners$ $of$ $tournament$.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{ \left\lfloor \frac{n+1}{2} \right\rfloor }


Context:
Calculate the discriminant of the number field \( \mathbb{Q}(\sqrt{-13}) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A board has six coins arranged such that every alternate coin shows heads and tails initially. A die is thrown, and the coin corresponding to the number rolled is flipped. Compute the expected number of die throws required for all coins to show the same side.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two independent random variables, $X$ and $Y$, with distribution function $F_X(x) = \frac{1}{2}x$ for $0 < x < 2$ and density function $f_Y(y) = \frac{2}{9}y$ for $0 < y < 3$, respectively, find the probability that their product $XY$ is greater than or equal to 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the Fejer kernel on the real line, defined as \( \mathcal{F}_R(t) = R\left(\frac{\sin(\pi t R)}{\pi t R}\right)^2 \), as \( R \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: \[ \int_0^1 \frac{dx}{\sqrt{x + \sqrt{x^2 + \sqrt{x^3}}}}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral of the function \( f(z) = \frac{2z + j}{z^3(z^2 + 1)} \) along the positively oriented circle \( \Gamma \) defined by \( |z - 1 - i| = 2 \) using residue theory.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral expression: $$\int_0^4\sqrt{x^2+4}\,dx+\int_2^{2\sqrt{5}}\sqrt{x^2-4}\,dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be an $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^p(x-b)^q$, where $a$ and $b$ are distinct real numbers. Let $V$ be the real vector space of all $n \times n$ matrices $B$ such that $AB=BA$. Determine the dimension of $V$. \( \text{Express your answer as a single integer.} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an infinite binary sequence, each bit is independently '0' or '1' with a probability of \( \frac{1}{2} \). However, if three consecutive '0's appear, the next bit must be '1'. Given this rule, what is the probability that a randomly chosen bit in the sequence is '1'? Express your answer as a limit as the sequence length approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of all positive integers that cannot be expressed as the sum of two abundant numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f: D(0,1) \to \mathbb{C} \) be a holomorphic function, where \( D(0,1) = \{ z \in \mathbb{C} \mid |z| < 1 \} \), and assume that \( |f(z)| \leq 2 \) for all \( z \) in \( D(0,1) \). Estimate \( |f^{(3)}(i/3)| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $σ(n)$ be the number of positive divisors of $n$, and let $rad n$ be the product of the distinct prime divisors of $n$. By convention, $rad 1 = 1$. Find the greatest integer not exceeding
$$100 \left( \sum\limits_{n=1}^{\infty} \frac{σ(n)σ(n rad n)}{n^2 σ(rad n)} \right)^{\frac{1}{3}}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum possible order of a group $G$ in which every element $g$ satisfies $g^3 = e$, where $e$ is the identity element.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\infty}^{+\infty} \left[ f(t+1) - f(t) \right] dt \) for a continuous function \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that \( \lim_{t \rightarrow -\infty} f(t) = l_1 \) and \( \lim_{t \rightarrow +\infty} f(t) = l_2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Provide an example of a set $G$ with a binary operation $\cdot$ such that $G$ has a two-sided identity element $e$, every element in $G$ has a left inverse, and the operation $\cdot$ is associative but not commutative.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral: \[ \int_{0}^{\infty} \frac{\log(1+x^2)}{1+x^2}~dx. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of free indices in the result when taking the derivative of a matrix-valued function with respect to a matrix.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a study, 100 people are randomly selected each day for 1095 days. Calculate the probability that at least 50% of these people have been selected at least twice during this period.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $\triangle ABC$, points $X$, $Y$, and $Z$ are located on sides $BC$, $CA$, and $AB$, respectively, such that lines $AX$, $BY$, and $CZ$ intersect at point $P$. Given that $AX = 10$, $BY = 8$, $CZ = 6$, $XP = 5$, $YP = 4$, and $ZP = 3$, determine the ratio of the area of triangle $\triangle XYZ$ to the area of triangle $\triangle ABC$. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p(x)$ and $q(x)$ be two polynomials with positive integer coefficients, each of degree 4, such that $p(1) + q(1) = 26$. Determine the number of unordered tuples $(p(x), q(x))$ for which the polynomial $W(x) = (p(x)q(x))^7$ has exactly one odd coefficient.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a set with 25 elements, determine the minimum number of subsets of size 5 required such that every possible subset of size 3 is included in at least one of these subsets of size 5.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the residue of the function \( f(z) = \frac{1}{z(z^2+4)(z+2i)} \) at \( z = -2i \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of square roots of matrices in a neighborhood of the identity matrix $I$ that are not reflections, for the function $\mathbf{f}: A \mapsto A^2$ where $A \in \mathbb{R}^{n \times n}$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the derivative of the functional \( J[f] = \int_{0}^{1} f'(x)^2 \, \mathrm{d}x \) with respect to the function \( f \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum \(1^{2017} + 2^{2017} + \cdots + 1000^{2017} \pmod{2016}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A quadratic polynomial $f(x) = Ax^2 + Bx + C$ is small if $A, B, C$ are single-digit positive integers. It is full if there are only finitely many positive integers that cannot be expressed as $f(x) + 3y$ for some positive integers $x$ and $y.$ Find the number of quadratic polynomials that are both small and full.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Construct a polynomial with integer coefficients \( f(x) \) such that \( f: \mathbf{Q} \rightarrow \mathbf{Q} \) is injective, but \( f: \mathbf{R} \rightarrow \mathbf{R} \) is not injective.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose you have $10,000 and invest in the stock market by buying 500 shares of DGCo at $10 each. You trade every day, buying 10 shares when the price increases by $1 and selling 10 shares when it decreases by $1. The stock price is equally likely to go up or down by $1 at any time. Calculate the expected amount of money you will have after trading for one year.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \iint \dfrac {\mathrm dx\mathrm dy}{\sqrt {x^{2}-3y^{2}}} \) over the region defined by \( x^{2}+y^{2}\leq 4x-3 \). Use polar coordinates to simplify the integration.
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Context:
Three identical spheres are placed inside a cone such that the diameter of the base of the cone is equal to the slant height. The spheres touch each other externally, with two spheres touching the lateral surface and the base of the cone, and the third sphere touching the lateral surface at a point in the same plane as the centers of the spheres. Determine the ratio of the radius of the base of the cone to the radius of the spheres.
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Context:
If $x^3-3\sqrt3 x^2 +9x - 3\sqrt3 -64=0$ find the value of $x^6-8x^5+13x^4-5x^3+49x^2-137x+2015$ .
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Context:
Trapezoid $EFGH$ has sides $EF=105$, $FG=45$, $GH=21$, and $HE=80$, with $EF$ parallel to $GH$. A circle with center $Q$ on $EF$ is drawn tangent to $FG$ and $HE$. Find the exact length of $EQ$ using fractions.
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Context:
Find the number of distinct normal lines to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) that pass through the point \((p,q)\), where \(a, b, p, q\) are real numbers.
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Context:
Determine the convergence radius of the power series \(\sum_{n=0}^\infty a_nb_nx^n\), given that the convergence radii of \(\sum_{n=0}^\infty a_nx^n\) and \(\sum_{n=0}^\infty b_nx^n\) are \(R_1>0\) and \(R_2=\infty\), respectively.
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Context:
Evaluate the integral \( \int s_{4} \, d\mu \) for the Lebesgue measure \( \mu \), where \( s_{n} = \sum^{n2^{n}}_{i=1} \frac{i-1}{2^{n}} \mathbb{1}_{E_{n,i}} + n\mathbb{1}_{A_{n}} \) and \( n=4 \). The sets \( E_{n,i} \) are defined as \( E_{n,i} = \left\{ x : \frac{i-1}{2^{n}} \leq f(x) < \frac{i}{2^{n}} \right\} \) and \( A_{n} = \{ x : f(x) \geq n \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
You are planning to invest in a selection of stocks:

- Stock A yields 21%, is quoted at 10%, has an annual dividend growth rate of 5%, is considered low risk, and has taxes on dividends at 10%.
- Stock B yields 15%, is quoted at 20%, has an annual dividend growth rate of 8%, is considered high risk, and has taxes on dividends at 20%.

If you invest $10,000 in Stock A and $15,000 in Stock B, taking into account their risk levels, dividend growth rates, and taxes on dividends, what will be the effective weighted average yield of your portfolio for the next year?

Note: The problem assumes the investment amounts will not change over time.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum number of planes in the space, such there are $ 6$ points, that satisfy to the following conditions:
1.Each plane contains at least $ 4$ of them
2.No four points are collinear.
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Context:
A dog has three trainers:

- The first trainer gives him a treat right away.
- The second trainer makes him jump five times, then gives him a treat.
- The third trainer makes him jump three times, then gives him no treat.

The dog will keep picking trainers with equal probability until he gets a treat. (The dog's memory isn't so good, so he might pick the third trainer repeatedly!) What is the expected number of times the dog will jump before getting a treat?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\to \infty} \sum_{r=0}^{n} \frac{\binom{n}{r}}{(r+4)n^r} \]
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Context:
Given that $a > b > c$ are the roots of the polynomial $P(x) = x^3 - 2x^2 - x + 1$, find the value of $K = a^2b + b^2c + c^2a$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume under the surface \( z = x^2 + 2xy + y^2 \) over the triangular region with vertices \((0,0)\), \((1,0)\), and \((1,1)\).
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Context:
A point is randomly dropped on the interval $[5, 7]$, and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-3k-4\right)x^{2}+(3k-7)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$.
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Context:
Given a three-dimensional Lie algebra \( L \) with its derived algebra \( L' \) such that \( \dim(L') = 2 \), determine the smallest natural number \( m \) such that the derived series \( L^{(m)} = 0 \), where \( L^{(1)} := L' \) and \( L^{(n)} := [ L^{(n-1)}, L^{(n-1)}] \).
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Context:
If the function $f(x) = (m-1)x^2 + mx + 3$ ($x \in \mathbb{R}$) is an even function, then the decreasing interval of $f(x)$ is.
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Context:
Positive integers $a, b,$ and $c$ have the property that $a^b, b^c,$ and $c^a$ end in $4, 2,$ and $9,$ respectively. Compute the minimum possible value of $a+b+c.$
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Context:
What is the logical negation of the proposition '$\exists x\geqslant 3$, $x^{2}-2x+3 \lt 0$'?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{$\forall x\geqslant 3$, $x^{2}-2x+3\geqslant 0$}


Context:
A circle with radius $r$ is contained within the region bounded by a circle with radius $R$. The area bounded by the larger circle is $a/b$ times the area of the region outside the smaller circle and inside the larger circle. Then $R:r$ equals:
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Context:
Let $\lfloor z \rfloor$ denote the greatest integer less than or equal to $z.$ Compute $$\sum\limits_{j=-1000}^{1000} \left\lfloor \frac{2025}{j+0.5}\right\rfloor.$$
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Context:
Determine the eccentricity of the ellipse formed when a cylinder of radius \( r \) is cut at an angle \( \alpha \) to its axis.
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Context:
Determine the smallest cardinal \( \kappa \) such that \( \kappa^{<\kappa} = \kappa \) in models of ZFC without large cardinal assumptions.
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Context:
Let $ABCD$ be a convex quadrilateral such that $AB=4$, $BC=5$, $CA=6$, and $\triangle{ABC}$ is similar to $\triangle{ACD}$. Let $P$ be a point on the extension of $DA$ past $A$ such that $\angle{BDC}=\angle{ACP}$. Compute $DP^2$.
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Context:
Determine the number of index 3 normal subgroups of the group given by the presentation \( \langle a, b, c \mid aba^{-1}b^{-1}c^2 = 1 \rangle \).
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Context:
A teacher intends to give the children a problem of the following type. He will tell them that he has thought of a polynomial \( P(x) \) of degree 2017 with integer coefficients and a leading coefficient of 1. Then he will provide them with \( k \) integers \( n_{1}, n_{2}, \ldots, n_{k} \), and separately provide the value of the expression \( P(n_{1}) \cdot P(n_{2}) \cdot \ldots \cdot P(n_{k}) \). From this data, the children must determine the polynomial that the teacher had in mind. What is the minimum value of \( k \) such that the polynomial found by the children will necessarily be the same as the one conceived by the teacher?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of real values of $x$ that satisfy the equation: \[ \left| 2 - x \right|^{ \log_5(x^2) - 5\log_x (25)} = (2-x)^2 \]
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Context:
Determine the number of idempotent elements in the ring $\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{Z}[i]$. An element $e$ is idempotent if $e^2 = e$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the vertices of a convex 2020-gon, numbers are placed such that among any three consecutive vertices, there is both a vertex with the number 7 and a vertex with the number 6. On each segment connecting two vertices, the product of the numbers at these two vertices is written. Andrey calculated the sum of the numbers written on the sides of the polygon and obtained the sum \( A \), while Sasha calculated the sum of the numbers written on the diagonals connecting vertices one apart and obtained the sum \( C \). Find the largest possible value of the difference \( C - A \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p, q, r$ be distinct real numbers such that $ap^2 + bp + c = (\\sin(\\theta))p^2 + (\\cos(\\theta))p$, and similarly for $q$ and $r$. Here, $a, b, c \in \mathbb{R}$. Find the maximum value of the expression \( \frac{a^2 + b^2}{a^2 + 3ab + 5b^2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A train 100 meters long takes 27.997760179185665 seconds to cross a bridge of a certain length if the speed of the train is 36 kmph. How long is the bridge?
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Context:
Let \( a_{0}=0 \), \( a_{1}=1 \), and then move one square at a time in a counterclockwise direction on the complex plane. If \( a_{n+1}-a_{n} = \mathrm{i}^{f(n)} \), where \( \mathrm{i} \) is the imaginary unit, find the simplest unified expression for \( f(n) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the area of the projection of the quadrilateral $PQRS$ onto the $xz$ plane, where $P(1, 1, 2)$, $Q(3, 2, 1)$, $R(4, 0, 3)$, and $S(2, -1, 4)$ are the vertices, and the acute angle between the plane containing $PQRS$ and the $yz$ plane is $\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \( \iint_K {x \, dS} \) over the part of the parabolic cylinder \( z = \frac{x^2}{2} \) that lies inside the first octant part of the cylinder \( x^2 + y^2 = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the equation of the curve that intersects all level curves of the function \( g(x,y) = \ln(x) + \cos(y) \) orthogonally and passes through the point (2,0).
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Context:
Evaluate the limit $$\lim_{r \to \infty} \frac {r^C \int_0^{\frac{\pi}{2}} x^r \sin(x)\, dx}{\int_0^{\frac{\pi}{2}} x^r \cos(x)\, dx}$$ where $C$ is a real constant, and determine the value of $\pi L - C$, given that $L > 0$ is the limit's value.
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Context:
Calculate the line integral $ \int_\gamma{\frac{d\xi}{\xi^4+1}}$ where $\gamma(t)=1+e^{it}, \,\,\,0\leq t\leq 2\pi$.
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Context:
The region below $$y = \sqrt{\ln (x+1)}$$ and above the $x$-axis, $ 0 \leq x \leq 1$, is rotated about the $x$-axis. Find the mass centrum for this solid.
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Context:
A collection of coins includes only pennies (1 cent), nickels (5 cents), dimes (10 cents), and quarters (25 cents). Using the coins in this collection, it is possible to create any amount of money less than one dollar (100 cents). What is the smallest possible number of coins in the collection? Express your answer as a single integer.
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Context:
Given a sequence of positive terms $\{a\_n\}$ that satisfies $2a\_1+3a\_2+a\_3=1$, find the minimum value of the arithmetic mean between $\frac{1}{a\_1+a\_2}$ and $\frac{1}{a\_2+a\_3}$.
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Context:
In the triangle \( \triangle ABC \), if \( \log \tan A + \log \tan C = 2 \log \tan B \), what is the range of angle \( B \)?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\frac{\pi}{3} \leq B < \frac{\pi}{2}}


Context:
The angle between vector $\overrightarrow{a}=(\sqrt{3},\;1)$ and vector $\overrightarrow{b}=(\sqrt{3},\;-1)$ is _______.
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Context:
Consider the hyperbola $M$: ${x^{2}} - \frac{{y^{2}}}{{{b^{2}}}} = 1$. Its left and right foci are $F_{1}$ and $F_{2}$, respectively, with $|F_{1}F_{2}|=2c$. A circle with center at the origin $O$ and radius $c$ intersects with the hyperbola $M$ at point $P$ in the first quadrant. If $|PF_{1}|=c+2$, find the x-coordinate of point $P$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
一年以 365 天计算, 则甲、乙、丙三人恰有两人在国庆节过生日的概率等于
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum value of the function

$$
f(x)=\sin (x+\sin x)+\sin (x-\sin x)+\left(\frac{\pi}{2}-2\right) \sin (\sin x)
$$
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Context:
Suppose $p$: the function $y = \log_a(x+1) (a > 0$ and $a \neq 1)$ is monotonically decreasing on $(-1, +\infty)$; $q$: the curve $y = x^2 + (2a - 3)x + 1$ intersects the $x$-axis at two distinct points. If $p \land q$ is false and $p \lor q$ is true, find the range of values for the real number $a$.
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Context:
In a certain table tennis singles tournament, it was originally planned for every two players to compete exactly once. However, three players each played two matches before withdrawing from the tournament, resulting in a total of 50 matches. Calculate the number of matches played among these three players. Express your answer as a whole number.
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Context:
Determine the value of $g'(2)$ given that $g$ is a real-valued function defined on the interval $(-1,1)$ such that $$e^{-x}(g(x)-2e^x)=\int_0^x \sqrt{y^4+1}\,dy$$ for all $x\in (-1,1)$, and $f$ is another function such that $$f(g(x))=g(f(x))=x.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sixth graders were discussing how old their principal is. Anya said, "He is older than 38 years." Borya said, "He is younger than 35 years." Vova: "He is younger than 40 years." Galya: "He is older than 40 years." Dima: "Borya and Vova are right." Sasha: "You are all wrong." It turned out that the boys and girls were wrong the same number of times. Can we determine how old the principal is?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the sixth digit after the decimal point in the number $(\sqrt{1978} + \lfloor \sqrt{1978} \rfloor)^{20}$.
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Context:
Given that $x=\ln π$, $y=\log _{5}2$, $z=e^{- \frac {1}{2}}$, compare the values of $x$, $y$, and $z$.
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Context:
In $\triangle{ABC}$, $\angle{C} = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. The original answer for the perimeter of the triangle is in the form $k + m\sqrt{2}$. Please provide the value of $k + m$.
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Context:
Find the covariance between two random variables X and Y, given their joint probability density function (pdf) is \( f(x,y) = 6x \) for \( 0 < x < y < 1 \) and \( 0 \) otherwise.
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Context:
Every square of $1000 \times 1000$ board is colored black or white. It is known that exists one square $10 \times 10$ such that all squares inside it are black and one square $10 \times 10$ such that all squares inside are white. For every square $K$ $10 \times 10$ we define its power $m(K)$ as an absolute value of difference between number of white and black squares $1 \times 1$ in square $K$. Let $T$ be a square $10 \times 10$ which has minimum power among all squares $10 \times 10$ in this board. Determine maximal possible value of $m(T)$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The axial section of a cone is an equilateral triangle with a side length of 1. Find the radius of the sphere that is tangent to the axis of the cone, its base, and its lateral surface.
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Context:
Evaluate the surface integral \( \int_{S}\nabla\times\vec{F}\cdot{\rm d}\vec{S} \) where \( \vec{F}(x,y,z) = y^2\vec{i} + 2xy\vec{j} + xz^2\vec{k} \) and \( S \) is the surface of the paraboloid \( z = x^2 + y^2 \) bounded by the planes \( x = 0, y = 0 \), and \( z = 1 \) in the first quadrant, with the surface normal pointing upward.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set of subgroups of the free group on two generators, $F_2 = \mathbb{Z} * \mathbb{Z}$.
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Context:
Calculate the limit: $$\lim_{n\to \infty}\left((1+\frac{1}{n^2})(1+\frac{2^2}{n^2})^2\cdots(1+\frac{n^2}{n^2})^n\right)^{\frac{1}{n}}$$
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Context:
Evaluate the following limit: $$\lim_{n\to \infty}\frac{1}{n^2}\sum_{i=1}^{n}\log\left(\frac{n!^n}{\left(\prod_{i=1}^{n}i!\right)\cdot\left(\prod_{i=1}^{n-1}i!\right)}\right).$$
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Context:
Determine the dimension of the vector space of 3x3 matrices where the sum of the elements in each column, each row, and both diagonals is zero.
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Context:
Find the area of the region bounded by a function $y=-x^4+16x^3-78x^2+50x-2$ and the tangent line which is tangent to the curve at exactly two distinct points.

Proposed by Kunihiko Chikaya
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Context:
Calculate the condition number of the matrix \( A = \begin{bmatrix} 0 & 0 & -10^4 & 0 \\ 0 & 0 & 0 & -10 \\ 0 & 10^{-3} & 0 & 0 \\ 10^{-2}& 0& 0& 0 \\ \end{bmatrix} \).
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Context:
Evaluate the definite integral: \[ \int_{0}^{3} (x^2+1) \, d[x] \] where \([x]\) is the integer part of \(x\).
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Context:
Determine the number of Sylow 2-subgroups in the symmetric group $S_5$.
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Context:
A four-digit positive integer is called *doubly* if its first two digits form some permutation of its last two digits. For example, 1331 and 2121 are both *doubly*. How many four-digit *doubly* positive integers are there?
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Context:
Given that $20^{22} + 1$ has exactly four prime divisors $p_1 < p_2 < p_3 < p_4$, determine $p_1 + p_2$.
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Context:
A broken clock moves the minute hand each minute randomly one minute ahead or one minute behind with equal probabilities, independently of previous movements. The hour hand is synchronized with the minute hand. The clock starts running at noon. Let T denote the first time that the clock shows either 1 o'clock or 11 o'clock, whichever occurs first. Compute the expected value of T, E(T).
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Context:
Let $a$, $b$, $c$ be positive integers and $p$ be a prime number. Assume that
\[ a^n(b+c)+b^n(a+c)+c^n(a+b) \equiv 8 \pmod{p} \]
for each nonnegative integer $n$. Let $m$ be the remainder when $a^p+b^p+c^p$ is divided by $p$, and $k$ the remainder when $m^p$ is divided by $p^4$. Find the maximum possible value of $k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_R x^4 y^7 \, dx \, dy \) over the region \( R \) in the positive quadrant bounded by the hyperbolas \( xy = 1 \), \( xy = 4 \), \( xy^2 = 1 \), and \( xy^2 = 5 \).
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Context:
实系数方程 $x^{4}+a x^{3}+b x^{2}+c x+d=0$ 的根都不是实数，其中两个根的和为 $2+\mathrm{i}$, 另两个根的积为 $5+6 \mathrm{i}$, 求 $b$ 的值。
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Context:
A triangle \( \triangle ABC \) has its vertices lying on a circle \( C \) of radius 1, with \( \angle BAC = 60^\circ \). A circle with center \( I \) is inscribed in \( \triangle ABC \). The line \( AI \) meets circle \( C \) again at \( D \). Find the length of the segment \( ID \).
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Context:
In a ring $R$, where $a^2 = a$, find the inverse of $1+a$, given that the inverse exists.
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Context:
Evaluate the integral \( \frac{1}{5} \int_0^5 (4 + 2^{0.1t^2}) \, dt \).
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Context:
n and r are positive integers. Find the smallest k for which we can construct r subsets \( A_1, A_2, \ldots, A_r \) of \( \{0, 1, 2, \ldots, n-1\} \) each with k elements such that each integer \( 0 \le m < n \) can be written as a sum of one element from each of the r subsets.
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Context:
Let \( a_1 = 1 \), and let \( a_n = \left\lfloor \frac{n^3}{a_{n-1}} \right\rfloor \) for \( n > 1 \). Determine the value of \( a_{999} \).
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Context:
On the sides  $ AB $  and  $ AC $  of the triangle  $ ABC $  consider the points  $ D, $  respectively,  $ E, $  such that  $$  \overrightarrow{DA} +\overrightarrow{DB} +\overrightarrow{EA} +\overrightarrow{EC} =\overrightarrow{O} .  $$
If  $ T $  is the intersection of  $ DC $  and  $ BE, $  determine the real number  $ \alpha $  so that:  $$  \overrightarrow{TB} +\overrightarrow{TC} =\alpha\cdot\overrightarrow{TA} .  $$
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Context:
Point  $P$  and equilateral triangle  $ABC$  satisfy  $|AP|=2$ ,  $|BP|=3$ . Maximize  $|CP|$ .
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Context:
Determine the range of values for \( p \) such that the interval \((1,2)\) lies between the roots of the equation \(2^x + 2^{2-x} + p = 0\).
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Context:
Exactly at noon, Anna Kuzminichna looked out the window and saw Klava, the saleswoman of the village store, going on a break. At two minutes past one, Anna Kuzminichna looked out the window again and still saw no one in front of the closed store. Klava was gone for exactly 10 minutes, and when she returned, she found Ivan and Foma waiting at the door, with Foma evidently arriving after Ivan. Determine the probability that Foma had to wait for the store to open for no more than 4 minutes.
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Context:
Let  $P$  be the portion of the graph of  $$ y=\frac{6x+1}{32x+8} - \frac{2x-1}{32x-8} $$  located in the first quadrant (not including the  $x$  and  $y$  axes). Let the shortest possible distance between the origin and a point on  $P$  be  $d$ . Find  $\lfloor 1000d \rfloor$ .

*Proposed by **Th3Numb3rThr33***
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Context:
Consider a 1998-digit number where each pair of adjacent digits forms a two-digit number divisible by either 17 or 23. If the last digit of this number is 1, determine the first digit of the number.
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Context:
The sequence starts with 12, followed by a number, then 15, 17, 111, 113, 117, 119, 123, 129, and 131. What is the second number in the sequence?
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Context:
Find the greatest real  $k$  such that, for every tetrahedron  $ABCD$  of volume  $V$ , the product of areas of faces  $ABC,ABD$  and  $ACD$  is at least  $kV^2$ .
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Context:
In an acute triangle with an area of 1, the altitudes divide it into six smaller triangles labeled counterclockwise as $t_1$, $t_2$, $t_3$, $t_4$, $t_5$, and $t_6$. Given that $t_1t_3t_5 + t_2t_4t_6 = 5$, determine the value of $t_1t_2t_3t_4t_5t_6$. Provide your answer as a single number.
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Context:
A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least x times as large as the shorter piece is
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Context:
Snorlax's weight is modeled by the function $w(t)=t2^t$ where $w(t)$ is Snorlax's weight at time $t$ minutes. Find the smallest integer time $t$ such that Snorlax's weight is greater than $10000.$
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Context:
Select several numbers from the numbers 1, 2, 3, ..., 20, so that for any two numbers chosen, their least common multiple is also among the selected numbers. What is the maximum number of numbers that can be selected?
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Context:
Two points, A and B, are 30 cm apart. Two strings, A and B, start simultaneously from A and B respectively and move towards each other. String A is 151 cm long and moves at 2 cm per second; String B is 187 cm long and moves at 3 cm per second. If both strings are ignited at their tail ends upon departure, string A burns at a rate of 1 cm per second and string B burns at a rate of 2 cm per second. How many seconds will it take for the two strings to completely pass each other from the moment they meet?
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Context:
21 In a regular tetrahedron wooden block $ABCD$ with edge length 2, there is a point $P (AP<1)$ on the edge $AB$. A cross-section perpendicular to the edge $AB$ is to be sawed through point $P$. When the sawing stops at a certain position, it is measured that the saw cut $PM=1$ on the face $ABD$. The saw cut $PN=\frac{2}{3}$ on the face $ABC$, then $MN=$
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the graph of the function $f(x)=\sin \omega x+\sin (\omega x- \frac {\pi}{2})$ ($\omega > 0$) is symmetric about the point $\left( \frac {\pi}{8},0\right)$, and there is a zero point within $\left(- \frac {\pi}{4},0\right)$, determine the minimum value of $\omega$.
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Context:
a soccer store typically sells replica jerseys at a discount of 30 percent to 70 percent off list price . during the annual summer sale , everything in the store is an additional 20 percent off the original list price . if a replica jersey ' s list price is $ 80 , approximately what percent of the list price is the lowest possible sale price ?
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Context:
Mrs. Fredrickson has 80 chickens, where 1/4 are roosters and the rest are hens. Only three-fourths of those hens lay eggs. However, 20% of the egg-laying hens have gone on vacation, and they won't contribute to egg production during this period. Additionally, 10% of the remaining egg-laying hens have fallen ill, further reducing the number of hens that lay eggs. How many chickens does Mr. Fredrickson have that do not lay eggs while considering the egg-laying hens on vacation and the ill hens?
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Context:
$ f(x)$  is a given polynomial whose degree at least 2. Define the following polynomial-sequence:  $ g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x))$ , for all  $ n \in N$ . Let  $ r_n$  be the average of  $ g_n(x)$ 's roots. If  $ r_{19}\equal{}99$ , find  $ r_{99}$ .
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Context:
Archimedes drew a square with side length $36$ cm into the sand and he also drew a circle of radius $36$ cm around each vertex of the square. If the total area of the grey parts is $n \cdot \pi$ cm$^2$, what is the value of $n$?
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Context:
\[\log _{3} \log _{4} \frac{4 x-1}{x+1} - \log _{\frac{1}{3}} \log _{\frac{1}{4}} \frac{x+1}{4 x-1} < 0.\]
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\boxed{x \in \left( \frac{2}{3}, \infty \right)}


Context:
Given 5000 cards, each containing a unique natural number from 1 to 5000 (each number appears exactly once). You need to select two cards such that the sum of the numbers on them is divisible by 100. In how many ways can this be done?
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Context:
Find the closure of the interval \((0,1)\) in the lower-limit topology on \(\mathbb{R}\), where the topology is generated by the subbasis consisting of all half-open intervals \([a,b)\).
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Context:
A malfunctioning digital clock shows the time $9: 57 \mathrm{AM}$; however, the correct time is $10: 10 \mathrm{AM}$. There are two buttons on the clock, one of which increases the time displayed by 9 minutes, and another which decreases the time by 20 minutes. What is the minimum number of button presses necessary to correctly set the clock to the correct time?
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Context:
If the plane vectors \(\vec{a}, \vec{b}, \vec{c}\) satisfy \(\vec{a} \cdot (\vec{a} + \vec{c}) = 0\) and \(|\vec{a} + \vec{b} - 2\vec{c}| = 2\), find the maximum value of \(\vec{a} \cdot \vec{b}\).
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Context:
Determine the maximum variance of a distribution with support in the interval $[0, 1]$ and a fixed mean $\mu \in [0, 1]$. Is there an upper bound on this variance?
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Context:
In quadrilateral \(ABCD\), the diagonals intersect at point \(M\) with an angle \(\alpha\) between them. Let \(O_{1}, O_{2}, O_{3}, O_{4}\) be the centers of the circumcircles of triangles \(ABM, BCM, CDM,\) and \(DAM\), respectively. Find the ratio of the areas of quadrilateral \(ABCD\) to quadrilateral \(O_{1} O_{2} O_{3} O_{4}\).
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Context:
Evaluate the integral \( \int_0^{2\pi} \frac{\sin^2x}{5+4\cos x} \ dx \) using complex integration techniques.
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Context:
Let $b(n)$ be the number of digits in the base -4 representation of $n$. Evaluate $\sum_{i=1}^{2013} b(i)$.
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Context:
Let \( h(x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} \) and let \( k(x) = h^{-1}(x) \). Find \( k^{(4)}(0) \).
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Context:
Given that an equilateral triangle \( \triangle ABC \) has sides \( AB \) and \( AC \) that are tangent to circle \( \odot O \) at points \( B \) and \( C \) respectively, what is the ratio of the area of \( \triangle ABC \) that lies outside the circle to the total area of the triangle?
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Context:
Find the critical points of the function defined by the following system of equations: \(2(x-1)=-2\lambda x\), \(2(y-3)=-2\lambda y\), \(2(z-1)=2\lambda z\), and \(z^2 - x^2 - y^2 =0\). Determine the minimum distance between these critical points.
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Context:
Dean ordered 2 large pizzas that were each cut into 12 slices. One pizza was Hawaiian and the other was cheese. He invited his friends Frank, Sammy, Nancy, and Olivia to enjoy some pizza and watch a movie. The pizzeria had a promotion for a free medium veggie pizza with 10 slices if you ordered at least 2 large pizzas, so they received an additional veggie pizza as well.

Dean was hungry and ate half of the Hawaiian pizza. Frank only ate 3 slices of Hawaiian pizza and Sammy ate a third of the cheese pizza. Nancy, who is a vegetarian, ate 2 slices of the cheese pizza and a slice of the veggie pizza. She gave her pepperoni from her veggie pizza slice to Olivia. Olivia, having received the extra pepperoni from Nancy, added it to her slice of Hawaiian pizza and then ate that slice, along with 2 slices of the veggie pizza.

During the movie, Frank wanted to taste the cheese pizza, so he decided to swap 1 slice of his remaining Hawaiian pizza with 1 slice of cheese pizza from Sammy. Meanwhile, Dean craved another type of pizza, so he took 1 slice from the veggie pizza and exchanged it with 1 leftover slice of Hawaiian pizza with Olivia.

How many total slices were left over?
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Context:
Given the equation in terms of $x$, $x^{2}+2(a-1)x+a^{2}-7a-4=0$, with roots $x_{1}$ and $x_{2}$, and it satisfies $x_{1}x_{2}-3x_{1}-3x_{2}-2=0$. Find the value of $(1+ \dfrac {4}{a^{2}-4})\cdot \dfrac {a+2}{a}$.
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Context:
Elsa is venturing into the unknown. She stands on $(0,0)$ in the coordinate plane, and each second, she moves to one of the four lattice points nearest her, chosen at random and with equal probability. If she ever moves to a lattice point she has stood on before, she has ventured back into the known, and thus stops venturing into the unknown from then on. After four seconds have passed, the probability that Elsa is still venturing into the unknown can be expressed as $\frac{a}{b} $ in simplest terms. Find $a+b$.(A lattice point is a point with integer coordinates.
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Context:
Find the following limit.

$ \lim\limits_{n\to\infty} \int_{-1}^1 |x|\left(1+x+\frac{x^2}{2}+\frac{x^3}{3}+\cdots +\frac{x^{2n}}{2n}\right)\ dx$.
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Context:
Find all values of $c \in \mathbb{R}$ such that for the map $F: \mathbb{R} \to \mathbb{R}^2$ satisfying the differential equation $D[F] = XF$, where $X = \begin{bmatrix} -1 & 1 \\ c & -1 \end{bmatrix}$, the limit $\lim_{t \to \infty} F(t) = 0$.
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Context:
Find the sum of the alternating series \( \sum_{n=1}^{\infty}(-1)^n \frac{16^n \pi^{2n+1}}{(2n)!} \).
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Context:
Evaluate the integral expression: $$2^{2010} \frac{\int_0^{1} x^{1004}(1-x)^{1004} \, dx}{\int_0^{1} x^{1004}(1-x^{2010})^{1004} \, dx}$$
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Context:
Determine the values of \(a\) for which the improper integral \(I = \int_{0}^\infty x^a \ln{\left(1+\frac{1}{x^2}\right)} \, dx\) converges.
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Context:
Let $U=\{ (x,y,z) \: : \: x-z=0\}$ and $V=\{(x,y,z) \: :\: z=0\}$ be two subspaces of $\mathbb{R}^3$. Define $W=\{f:\mathbb{R}^3 \rightarrow \mathbb{R}^3 \: : f(U) \subseteq V \: \: f(V) \subseteq U \}$. Find the dimension of $W$ using its matrix representation.
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Context:
Batteries A and B have lifetimes that are independent and exponentially distributed with a common mean of m years. Given that the probability that battery B outlasts battery A by more than one year is 0.33, calculate the value of m.
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Context:
The line $l: x - 2y + 2 = 0$ passes through the left focus F1 and a vertex B of an ellipse. Find the eccentricity of the ellipse.
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Context:
Determine the number of altitudes of a tetrahedron that can lie outside the tetrahedron.
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Context:
What is the result of the sum of three Cantor sets, denoted as $C+C+C$, where $C$ is the usual Cantor set obtained from the interval $[0,1]$?
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Context:
Find the exponential generating function (EGF) for the sequence $(a_n)_n$ where $a_0 = 1$ and $a_n = \alpha(\alpha-1)\cdots(\alpha - (n-1))$ for $n \geq 1$, with $\alpha$ being a real number.
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Context:
Evaluate the limit: \[ \lim_{m \to \infty} m \cdot E \left[ \log\left( \frac{N+\frac{1}{2}}{m+1} \right) \right] \] where \( N \) is a Poisson random variable with mean \( m \).
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Context:
Evaluate the integral using the Cauchy Principal Value: \[ \int_{-\infty}^\infty \frac{\cos(ax) - \cos(bx)}{x^2} \, dx \]
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Context:
In a right triangle $ABC$, point $P$ is on $AC$ such that $AP = PQ = QC$. Given that $QB = 67$ and $PB = 76$, find the length of $AB$. Express your answer in terms of the given lengths.
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Context:
Find the limit of the integral \( \int_{0}^{1}u^k\cot{\frac{\pi(1-u)}{m}}\sin{\frac{2\pi n(1-u)}{m}}\,du \) as \( n \) approaches infinity, where \( k \) is a non-negative integer and \( m \) is a real number greater than or equal to 1.
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Context:
Given the function defined by the following conditions:
\[
\begin{align}
f'(x) &= \frac{1}{2} f(x)\quad\text{for all } x,\\
f(0) &= 1,\\
|f(x)| &\le 6\quad\text{for all } x \in [-3, 3],
\end{align}
\]
find the higher error bound of the approximation of \(f(2)\) using the 6th degree MacLaurin polynomial \(T_6(2)\).
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Context:
Given that $Y$ is uniformly distributed over $(0,2)$ and $X$ is exponentially distributed with mean $Y$, find the expected value of the output $XY^3$ of the process.
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Context:
Let $\{e_i\}$ be a basis for a complex linear space $E$ and $h$ a Hermitian metric on $E$. Denote by $H=\{h_{ij}\}$ the matrix representation of $h$ with respect to $\{e_i\}$, where $h_{ij}=h(e_i,e_j)$. Given a complex automorphism $g\in GL(E)$ and the transformed basis $\{e'_i\}=\{ge_i\}$, find the relationship between the matrices $H$ and $H'=\{h'_{ij}\}$, representing $h$ with respect to $\{e'_i\}$.
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Context:
Determine the number of distinct equivalence classes for logical statements composed of 5 variable letters, considering logical equivalence.
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Context:
Find the value of \( \alpha \) that satisfies the equation: \( \sin^{-1}(2 \alpha) \sin^{-1}(60^{\circ} - 2 \alpha) \sin^{-1}(60^{\circ} + 2 \alpha) = 4 \sin^{-1}(6 \alpha) \).
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Context:
Find the constant \(c\) such that \(\max_{a\le n/2}\frac{C_n^a}{\sum_{k=0}^{\lfloor{a/3}\rfloor}C_n^k}=(c+o(1))^n.\)
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Context:
Evaluate the limit: \[ \lim_{y \to +\infty} y \int_0^{+\infty} e^{-x^2} \sin(2xy) \, dx \]
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Context:
Two points rotate uniformly around two concentric circles. One completes a revolution 5 seconds faster than the other, resulting in two additional revolutions per minute. Initially, the rays from the center of the circles to these points coincide. Calculate the magnitude of the angle between the rays after 1 second.
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Context:
How many natural numbers greater than 200 are formed by using the natural numbers 0, 1, 3, 5, and 6 without overlapping?
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Context:
Polycarp wrote a multiplication problem of two three-digit numbers on the board and mistakenly wrote 0 instead of the multiplication sign, resulting in a seven-digit number. This seven-digit number is an integer multiple of the product of the two numbers. Determine the exact factor by which the seven-digit number is a multiple of the product.
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Context:
Find the number of positive integers less than 1000000 which are less than or equal to the sum of their proper divisors. If your answer is $X$ and the actual value is $Y$, your score will be $\max \left(0,20-80\left|1-\frac{X}{Y}\right|\right)$ rounded to the nearest integer.
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Context:
Given a discrete random variable $X$ that can take integer values from $0$ to $20$, and the mean of $X$ is $0.005$, determine the maximum possible variance of $X$. Provide your answer as a single number.
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Context:
Let $G$ be a group and $H$ be a normal subgroup of $G$ generated by an element $a$. If $a$ has order $n$, determine the possible orders of $bab^{-1}$ for $b \in G$.
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Context:
How many $6$-digit positive integers whose digits are different from $0$ are there such that each number generated by rearranging the digits of the original number is always divisible by $7$?
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Context:
Evaluate the line integral \( \int_{C_1} \frac{y\vec{e_1}-x\vec{e_2}}{x^2+y^2} d\vec{r} \), where \( C_1 \) is a circle of radius 3 centered at the origin in \( \mathbb{R}^2 \).
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Context:
Determine the order of the element $a^{p^{t-2}}$.
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Context:
Consider the region $D$ in $\mathbf{R}^3$ that lies above the paraboloid $z = x^2 + y^2 - 1$ and below the semi-sphere $z = \sqrt{1 - x^2 - y^2}$. Determine the volume of $D$ by evaluating the integral $\iiint z \, dV$. The intersection of the surfaces can be found by equating them: $$\sqrt{1 - x^2 - y^2} = x^2 + y^2 - 1$$ Squaring both sides to eliminate the square root: $$1 - x^2 - y^2 = (x^2 + y^2 - 1)^2$$ Expanding and simplifying: $$-1 = x^4 + y^4 - 2x^2 - 2y^2 + 1$$ Rearranging the terms: $$x^4 + y^4 - 2x^2 - 2y^2 = 0$$ This equation represents the intersection of the surfaces. Calculate the volume of the region $D$.
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Context:
Evaluate the limit: $$\lim_{n \rightarrow \infty} \sum_{i=(n+1)/2}^n {n \choose i} \times 0.51^i \times 0.49^{n-i}$$
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Context:
Find a lower bound for the function \( g(x) = \frac{x + (1-x)q_1}{1+3x}\cdot \frac{x + (1-x)q_2}{1+2x} \), where \( x \in (0,1) \), and \( q_1 \) and \( q_2 \) are constants such that \( q_1 + q_2 \leq 1 \) and \( q_1, q_2 > 0 \).
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Context:
Evaluate the limit: \[ \lim_{k \to 0^{+}} \frac{(k+1)A-3kA+2A^3}{(k-A^2)^{\frac{3}{2}}} \] where \( A = -\sqrt{2} \frac{\Gamma(\frac{k+1}{2})}{\Gamma(\frac{k}{2})} \) and \( \Gamma \) is the Gamma function.
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Context:
Find the integrating factor that makes the differential equation \( y^{2}dy = x\left ( x dy - y dx \right ) e^{\frac{x}{y}} \) exact.
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Context:
Find the positive constant \( c \) such that the sum \( \frac{S_n}{n^c} \) of independent and identically distributed random variables \( X_1, X_2, \ldots, X_n \) converges in distribution to a random variable \( A \), where the characteristic function of each \( X_i \) is given by \( \phi(t) = 1 - \sqrt{|t|(2-|t|)} \) for \( t \in [-1, 1] \) and zero elsewhere.
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Context:
Given the quadratic equation x^2 - 3x = a^2 and the root x = 4, find the value of the constant a. Express your answer as ±value, where value is a whole number.
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Context:
Given the force of interest \( F(t) = \frac{0.02 + 0.01t}{1 + 0.02t + 0.005t^2} \), where \( t \) is the number of years starting from March 31, 2001, calculate the accumulated value of two investments: $100 invested on March 31, 2002, and $200 invested on March 31, 2006. Find the total accumulated value of these investments on March 31, 2010.
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Context:
Consider the operator \( T: L^2([0,1]) \to L^2([0,1]) \) defined by \( Tf(t) = \int_0^t f(s) \, ds \). Find \( \alpha < 1 \) such that \( ||T|| \leq \alpha \).
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Context:
Determine the number of integer solutions \((x, y, z)\) to the congruence equation \(x^5 + y^5 \equiv z^5 \pmod{25}\).
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Context:
Let \(a, b\) be nonzero complex numbers such that \(\frac{a}{b}\) is not a real number. Define the set:
\[
L_{a, b}=\{r a+s b \mid r, s \in \mathbf{Z}\},
\]
and the set:
\[
R_{a, b}=\left\{z \mid z \text{ is a nonzero complex number, and } L_{a, b} = L_{z a, z}\right\}.
\]
Determine the maximum number of elements in the set \(R_{a, b}\) as \(a, b\) vary.
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Context:
Calculate the area enclosed by the curve defined by the equation \( \sin^4(x) + (\cos(y) - 3)^2 - 16 = 0 \).
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Context:
Let \( \mathbf{G}: \mathbb{R}^2 \to \mathbb{R}^2 \) be defined by \( G(x,y) = (\cos(x-y), \sin(x+y)) \). Assume \( f: \mathbb{R}^2 \to \mathbb{R} \) is differentiable with gradient \( \nabla f(0,1) = (2,3) \). Define \( h: \mathbb{R}^2 \to \mathbb{R} \) by \( h(x) = f(\mathbf{G}(\mathbf{x})) \). Find the gradient \( \nabla h\left( \frac{\pi}{2}, 0 \right) \).
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Context:
Determine the order of the element $x + x^2$ in the multiplicative group of the field $\mathbb{F}_2[x]/(x^4 + x^3 + 1)$. Provide your answer as a single integer.
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Context:
Given a random sample $X_1$ and $X_2$ of size $n=2$ from an exponential population, find the efficiency of the estimator $2Y_1$ relative to the estimator $\bar{X}$, where $Y_1$ is the first order statistic and both $2Y_1$ and $\bar{X}$ are unbiased estimators of the parameter $\theta$. Provide your answer as a numerical value.
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Context:
Find the smallest positive integer \( n \) such that:
\[ 1,000,000^{{1,000,000}^{1,000,000}} < n^{n^{n^n}} \]
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Context:
Determine the total charge on the spherical surface defined by \(x^2+y^2+z^2=R^2\) with a charge density given by \(\rho(x,y,z)=|z|\) C/m\(^2\).
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Context:
Evaluate the integral \( I = \int_0^{\pi} e^{\cos\theta}\cos(\sin\theta)\ d\theta \).
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Context:
Find the maximum value of the function \( f(x) = 4\sin x + 48\sin x\cos x + 3\cos x + 14\sin^2x \).
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Context:
What maximal number of the men in checkers game can be put on the chess-board \( 8 \times  8 \) so,that every man can be taken by at
 least one other man?
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Context:
Find the minimal polynomial of $\sqrt{3}$ over the field $\mathbb{Q}(\sqrt[6]{3})$. Provide your answer as a polynomial with rational coefficients.
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Context:
Let $P = (0, q)$ be the hyperbolic midpoint of $A = (0, a)$ and $B = (0, a + b)$ in the Hyperbolic/Half-Plane Model. Compute the limit of $q$ as $a \rightarrow \infty$, as a function of the fixed number $b$.
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Context:
Find an integer $n$ such that the multiplicative group $(\mathbb{Z}/n\mathbb{Z})^*$ contains elements of order $4$ and $5$, but no elements of order $3$. What is the value of $n$?
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Context:
Evaluate the line integral \( \int_C G(x,y,z) \, ds \) for the curve parameterized by \( x= \frac{1}{3} t^3, \quad y=t^2, \quad z=2t \), where \( t \) ranges from 0 to 1, and \( G(x, y, z) = 4xyz \). Round your answer to the nearest hundredth.
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Context:
Given a sequence \( a_1, a_2, a_3, \ldots, a_n \) of non-zero integers such that the sum of any 7 consecutive terms is positive and the sum of any 11 consecutive terms is negative, what is the largest possible value for \( n \)?
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Context:
Approximate \( \sqrt{8} \) using the Taylor series expansion of \( \sqrt{1+x} \) around zero up to the fourth term. Use the Lagrange Remainder to ensure the approximation is accurate to within \( 10^{-4} \).
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Context:
Determine the largest natural number \( n \) such that the group \( \mathbb{Z}_{180} \times \mathbb{Z}_{30} \) is isomorphic to the direct product of \( n \) cyclic groups.
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Context:
Find the ratio of the bases of the trapezoid, given that its midline is divided into three equal parts by the diagonals.

#
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Context:
Count the number of 10-digit numbers using the digits from \( \{1, 2, 3, 4\} \) such that no two adjacent digits differ by 1.
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Context:
## Task Condition

Find the derivative of the specified order.

$$
y=(5 x-1) \ln ^{2} x, y^{\prime \prime \prime}=?
$$
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Context:
Determine all integer values of \( n \) such that the equation \( 2\sin{nx} = \tan{x} + \cot{x} \) has real solutions.
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Context:
An equilateral triangle with a side length of 8 is divided into smaller equilateral triangles with side lengths of 1 (see the figure). What is the minimum number of these smaller triangles that need to be shaded so that every intersection point of the lines (including those on the edges) is a vertex of at least one shaded triangle?
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Context:
Determine the minimum layer width required for universal approximation in deep feedforward networks with ReLU activation function, given that the input dimension is $\nu$. Provide your answer in terms of $\nu$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of automorphisms of the symmetric group $S_6$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest value of $C$ such that any shape in the plane with an area less than $C$ can be translated and rotated to avoid touching any point in the integer lattice $\mathbb{Z}^2$. What is this largest value of $C$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \( \int_M (\nabla \times F) \cdot N \, dS \) where \( F = (e^x + z^2, \sin(yz), x^3) \) and \( M = \{(x,y,z) : y = \frac{x^2}{2} + z^2 - 4, y \leq 0\} \). The vector \( N \) is the outward-pointing unit normal to the surface \( M \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two circles with radius 1 intersect at points $X$ and $Y$, with a distance of 1 between them. From point $C$ on one circle, tangents $CA$ and $CB$ are drawn to the other circle. Line $CB$ intersects the first circle again at point $A'$. Find the distance $AA'$.
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Context:
Let $V$ be a finite dimensional vector space over the finite field $\mathbb{F}_q$ with $q$ elements, and let $U$ be a subspace of $V$. Determine the number of subspaces $W \subset V$ such that $W \cap U = \{0\}$ and $V = W + U$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a function \( f : \mathbb{N} \rightarrow  \mathbb{N} \) such that for any two positive integers \( x,y \) , the equation \( f\left( {{xf}\left( y\right) }\right)  = {yf}\left( x\right) \) holds. Find the smallest possible value of \( f\left( {2007}\right) \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
In trapezoid \(ABCD\), \(\overrightarrow{AB} = 2 \overrightarrow{DC}\), \(|\overrightarrow{BC}| = 6\). Point \(P\) is a point in the plane of trapezoid \(ABCD\) and satisfies \(\overrightarrow{AP} + \overrightarrow{BP} + 4 \overrightarrow{DP} = 0\). Additionally, \(\overrightarrow{DA} \cdot \overrightarrow{CB} = |\overrightarrow{DA}| \cdot |\overrightarrow{DP}|\). Point \(Q\) is a variable point on side \(AD\). Find the minimum value of \(|\overrightarrow{PQ}|\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the intersection multiplicity of the curves $V(f)$ and $V(g)$ at the point $p = [-2:1:4]$ in $\mathbb{P}^2$, where $f = x^2 - yz$ and $g = (x+z)^2 - yz$. Use the fact that the intersection multiplicity is the Hilbert polynomial of the $I(p)$-primary component of the ideal $(f) + (g)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many distinct necklaces can be made using exactly three red beads and three black beads?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\rightarrow\infty} n^2 \int_{\frac{1}{n+1}}^{\frac{1}{n}}\frac{\tan^{-1}nx}{\sin^{-1}nx}dx \]
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Context:
Let \( \alpha \) denote \( \cos^{-1}(\tfrac 23) \). The recursive sequence \( a_0,a_1,a_2,\ldots \) satisfies \( a_0 = 1 \) and, for all positive integers \( n \), \[ a_n = \dfrac{\cos(n\alpha) - (a_1a_{n-1} + \cdots + a_{n-1}a_1)}{2a_0}. \] Suppose that the series \[ \sum_{k=0}^\infty\dfrac{a_k}{2^k} \] can be expressed uniquely as \( \tfrac{p\sqrt q}r \), where \( p \) and \( r \) are coprime positive integers and \( q \) is not divisible by the square of any prime. Find the value of \( p+q+r \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the oscillation, $O(f, (0,0))$, of the function $f$ defined by:

$$ f(x,y)= \begin{cases} \frac{xy}{x^2+y^2} & (x,y)\neq(0,0) \\ 0 & (x,y)=(0,0) \end{cases} $$

at the point $(0,0)$. Use the definition of oscillation at a point $c$:

$$ O(f,c) = \inf_{c\in U}\sup_{x_1,x_2\in U}|f(x_1)-f(x_2)| $$

where $U$ is an open subset containing $c$. Note that $f$ is not continuous at $(0,0)$ since:

$$ \lim_{x\rightarrow0}f(x,0) = 0 \neq \frac{1}{2} = \lim_{x \rightarrow 0}f(x,x) $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( \Gamma\left(-\frac{1}{2}\right) \) directly from the definition \( \Gamma(s) = \int_0^\infty e^{-t}t^{s-1}dt \), without using the known values \( \Gamma\left(\frac{1}{2}\right) = \sqrt{\pi} \) and \( \Gamma(s+1) = s\Gamma(s) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Construct a sequence \((y_n)\) that is not bounded, but satisfies \(\lim_{n\to+\infty}|y_n-y_{n+1}|=0\).
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Context:
Find the conditions on a matrix \( A \) such that the matrix \( (I+A)(I-A)^{-1} \) is orthogonal.
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Context:
Given the equation \( x^a \cdot y^b = (x+y)^{(a+b)} \), where \( a \) and \( b \) are positive constants, find an expression for \( \frac{dx}{dy} \) by differentiating implicitly, assuming that \( ay \neq bx \).
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Context:
Evaluate the integral \( \int_{|z| = 106} \cos\left(\frac{1}{6z+1}\left(\frac{1}{(z-1)(z-2)\cdots(z-105)}\right)\right)\,dz \).
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Context:
Find the values of $x$ for which the series \( \sum_{n=1}^{\infty}\frac{x^n}{1-x^n} \) converges absolutely.
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Context:
Find the angle \( \theta \) in the range \( 0 \leq \theta < 360 \) degrees that minimizes the x-coordinate of the Archimedean spiral given by \( r = 1 - \frac{0.5}{2\pi}\theta \).
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Context:
Consider a rectangle with dimensions $m \ge 3$ and $n \ge 3$, consisting of $mn$ unit squares. In each unit square, exactly one number is written such that any 3x3 sub-square forms a magic square. Determine the maximum number of different numbers that can be written in the entire rectangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum step size $h$ for the forward Euler method such that the solution's norm approaches zero for the system of ODEs given by:
\[
a' = -a - 2b \\
b' = 2a - b
\]
with initial conditions $a(0)=1$ and $b(0)=-1$. The eigenvalues of the coefficient matrix for the forward Euler method are $(-1 + 2i)(1 + h)$ and $(-1 - 2i)(1 + h)$. Find the largest $h$ that ensures stability.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $x$ and $y$ are elements in a group $G$ where the order of $y$ is not equal to $1$, $yx = xy^2$, and the order of $x$ is $3$. Determine the order of $y$.
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Context:
Given the differential equation \( x' + x^3 = 2x \), find the differential equation for \( v = \frac{1}{x^2} \).
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Context:
Evaluate the limit: $$\lim_{n \to \infty} \left[\frac{(n+2)^{n + 2}}{(n+1)^{n+1}} - \frac{(n+1)^{n+1}}{n^n} \right]$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the oscillation of the function \( f(x,y) = \begin{cases} \frac{xy}{x^2+y^2} & (x,y)\neq(0,0) \\ 0 & (x,y)=(0,0) \end{cases} \) at the point \((0,0)\). The oscillation of \(f\) at a point \(c\) is defined as: \[ O(f,c) = \inf_{c\in U}\sup_{x_1,x_2\in U}|f(x_1)-f(x_2)| \], where \(U\) is an open subset containing \(c\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How to integrate: \(\int_0^T\cos(T-s)\sin(s)ds\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sequence \( \{b_n\} \) is defined by the recurrence relation \( b_n = 2b_{n-1} - b_{n-2} + b_{n-3} \) with initial conditions \( b_1 = 1 \), \( b_2 = 1 \), and \( b_3 = 2 \). Calculate the value of the infinite series:
\[
b_1 + \frac{b_2}{3} + \frac{b_3}{3^2} + \cdots
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a given fixed integer $N$, find the largest integer $n$ such that the symmetric group $\Sigma_n$ can be embedded as a subgroup of the orthogonal group $O(N)$. What is the maximum value of $n$?
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Context:
A set $S$ consisting of $2019$ different positive integers has the following property: the product of every $100$ elements of $S$ is a divisor of the product of the remaining $1919$ elements. What is the maximum number of prime numbers that $S$ can contain?
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Context:
我们称置换群 $S_{9}$ 中的两个元素 $\sigma_{1}, \sigma_{2}$ 是共轭的，如果存在 $S_{9}$ 中的元素 $\tau$使得 $\tau^{-1} \sigma_{1} \tau=\sigma_{2}$ 。设 $\sigma=(12345)$ (4567), 则 $S_{9}$ 中有 $\qquad$个元素与 $\sigma$ 共轭。
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a school, the decision to balance the menu with both meat and vegetables was agreed upon by the following percentage of students from different grade levels:

- 60% of the third-grade students
- 45% of the fourth-grade students
- 35% of the fifth-grade students
- 55% of the sixth-grade students

There are 256 third-graders, 525 fourth-graders, 410 fifth-graders, and 600 sixth-graders in the school. Calculate the total number of students from all four grade levels who agreed with the decision.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f:\mathbb{N}\mapsto\mathbb{R} \) be the function \[f(n)=\sum_{k=1}^\infty\dfrac{1}{\operatorname{lcm}(k,n)^2}.\] It is well-known that \( f(1)=\tfrac{\pi^2}{6} \). Determine the smallest positive integer \( m \) such that \( m\cdot f(10) \) is the square of a rational multiple of \( \pi \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a random variable xi that follows a normal distribution with mean 1 and variance 36, and P(-2 <= xi <= 1) = 0.4, calculate the probability P(xi > 4). Express your answer as a decimal value between 0 and 1.
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Context:
Bonnie constructs a frame for a cube using 12 pieces of wire that are each eight inches long. Meanwhile, Roark uses 2-inch-long pieces of wire to create a collection of unit cube frames that are not connected. The total volume of Roark's cubes is the same as the volume of Bonnie’s cube. What is the ratio of the total length of Bonnie's wire to the total length of Roark's wire?
Please reason step by step, and put your final answer within \boxed{}.


Context:
7. If the solution set of the inequality $k x^{2}-2|x-1|+6 k<0$ with respect to $x$ is an empty set, then the range of values for $k$ is $\qquad$

Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
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Context:
Given the system of differential equations \( x' = xy + t^2 \) and \( y' = \frac{-y^2}{2} \), with initial conditions \( x(1) = 3 \) and \( y(1) = 2 \), find \( \frac{d x}{d y_0} \) at \( x_0 = 3 \) and \( y_0 = 2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
在 $1,2, \cdots, 10$ 中随机选出三个不同的数, 它们两两互素的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Let \( X \sim \mathrm{Geometric}(1/11) \). Use Jensen's inequality to determine a lower bound on \( \mathrm{E}(X^4) \) by applying Jensen's inequality to \( X^2 \) with \( f(x) = x^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of six-digit numbers of the form $abcabc$ that have exactly four prime factors, with one of these factors having a power of 3. Express your answer as an integer $n$. Note that $abcabc$ can be expressed as $7 \times 11 \times 13 \times (100a + 10b + c)$. The prime factor with a power of 3 must be $100a + 10b + c$. Find the number of such numbers $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the degree of the algebraic number $\xi_{7} = e^{2\pi i/7}$ over the field $\mathbb{Q}(\xi_{3})$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the degree of the polynomial \( \deg(\rho_7+\rho_7^2+\rho_7^4) \) over the field of rational numbers \( \mathbb{Q} \), where \( \rho_7 \) is a primitive seventh root of unity.
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Context:
For the equation $6 x^{2}=(2 m-1) x+m+1$ with respect to $x$, there is a root $\alpha$ satisfying the inequality $-1988 \leqslant \alpha \leqslant 1988$, and making $\frac{3}{5} \alpha$ an integer. How many possible values are there for $m$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a given integer $n \geq 2$, determine how many integers $x$ in the set $\{1, 2, \ldots, n\}$ satisfy the congruence $x^2 \equiv x \pmod{n}$.
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Context:
Let $V = M_{3 \times 3}(\mathbb{C})$, the set of $3 \times 3$ matrices with complex entries. Consider the matrix $A = \begin{pmatrix} 0 & -2 & 0 \\ 1 & 3 & 0 \\ 0 & 0 & 2 \end{pmatrix}$. Define $W = \{ p(A) \mid p(t) \in \mathbb{C}[t] \}$, where $\mathbb{C}[t]$ is the set of polynomials with complex coefficients. Determine the dimension of the subspace $W$ in $V$. \(\dim W\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $M$ be a Riemannian manifold with submanifolds $S$ and $T$. Suppose $	heta$ is a differential $k$-form on $M$ such that \( \oint_{[S]} \theta = r \neq 0 \). If \( \eta \) is the Hodge star of \( \theta \) and the \( \mathbb{Z}\pi \)-intersection number of \( S \) and \( T \) is \( \pm 1 \), determine the value of \( \oint_{[T]} \eta \).
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Context:
Let  $ A \equal{} \{(a_1,\dots,a_8)|a_i\in\mathbb{N}$  ,  $ 1\leq a_i\leq i \plus{} 1$  for each  $ i \equal{} 1,2\dots,8\}$ .A subset  $ X\subset A$  is called sparse if for each two distinct elements  $ (a_1,\dots,a_8)$ , $ (b_1,\dots,b_8)\in X$ ,there exist at least three indices  $ i$ ,such that  $ a_i\neq b_i$ .
   Find the maximal possible number of elements in a sparse subset of set  $ A$ .
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Context:
Let $a,b,c$ be positive real numbers such that $a+b+c=2005$. Find the minimum value of the expression:
$$E=a^{2006}+b^{2006}+c^{2006}+\frac{(ab)^{2004}+(bc)^{2004}+(ca)^{2004}}{(abc)^{2004}}$$
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Context:
Determine the function \( h(t) \) for the partial differential equation \( u_{xt}(x,t) + u(x,t)u_{xx}(x,t) = h(t) \) with the boundary conditions \( u(\pi,t) = u_x(0,t) = u_x(\pi,t) = 0 \), where \( x \in [0,\pi] \) and \( t > 0 \). Assume all computations and regularity conditions are valid.
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Context:
Determine all positive integers \( n \) such that if \( 1 = d_1 < d_2 < \ldots < d_k = n \) are the divisors of \( n \), then \( n = d_6^2 + d_7^2 - 1 \).
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Context:
在 $4 n \times 4 n\left(n \in \mathbf{N}_{+}\right)$实数表中, 每个数的绝对值均不超过 1 , 所有数的和为 0 . 已知数表中的每行各数之和的绝对值、每列各数之和的绝对值均不小于 $c$. 求 $c$ 的最大可能值.
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Context:
Given proposition $p$: The equation $\frac{x^2}{m} + \frac{y^2}{6-m} = 1$ represents an ellipse with foci on the $x$-axis; proposition $q$: The eccentricity $e$ of the hyperbola $\frac{y^2}{5} - \frac{x^2}{m} = 1$ belongs to $(\frac{\sqrt{6}}{2}, \sqrt{2})$. If the proposition “$p \lor q$” is true, and “$p \land q$” is false, find the range of values for $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $U$ and $V$ be the null spaces of matrices $A = \begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 2 & 3 & 2 \\ 0 & 1 & 2 & 1 \end{bmatrix}$, respectively. Determine the dimension of the vector space $U + V$. \( \text{Provide your answer as a single integer.} \)
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Context:
现有 $6$ 人会英语， $4$ 人会日语， $2$ 人都会（共 $12$ 人），从其中选出 $3$ 人做翻译，若要求两种语言都有人翻译，则符合条件的选法共 $\qquad$种。
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Context:
In triangle \(ABC\), if the triangle formed by its medians and the triangle formed by its altitudes are both similar to triangle \(ABC\), what is the relationship between the side lengths of triangle \(ABC\)?
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Context:
Given the function f(x) = x^2 * e^|x|, a = f(log_2(3)), b = f(-log_5(8)), and c = f(-2^1.001), determine the relationship between a, b, and c. Express your answer using > or < to describe the order of a, b, and c.
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Context:
Given a triangle \( \triangle ABC \), which fulfills the equations \( 6\cos{A}+2\cos{B}=7 \) and \( 6\sin{A}-2\sin{B}=\sqrt{3} \), find the value of \( \angle C \).
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Context:
Determine the amortized time complexity of performing a series of union and find operations on disjoint sets, assuming that all union operations occur after all find operations.
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Context:
Calculate the limit \( \lim_{x \to \infty, n \to \infty} f^n(x) \), where \( f(x) = \ln(1+x) \) and \( f^n(x) = f(f^{n-1}(x)) \) for all integers \( n \geq 2 \) with \( f^1(x) = f(x) \).
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Context:
Let $B$ be a $3\times3$ invertible matrix with real entries such that $\det(B)=m$. Given that $\det(B+madj(B))=0$, find the value of $\det(B-madj(B))$. \(\text{Express your answer as a single number.}\)
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Context:
Suppose $x$ is a real number such that $\sin(1 + \cos^2 x + \sin^4 x) = \tfrac{13}{14}$. Compute $\cos(1 + \sin^2 x + \cos^4 x)$.
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Context:
Find the sum of the values of $x$ such that $\cos^3 3x+ \cos^3 5x = 8 \cos^3 4x \cos^3 x$, where $x$ is measured in degrees and $100< x< 200.$
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Context:
Find the volume of the solid generated by rotating the region bounded by $x^2-y^2 = 7$, $x = 4$, and $y = 5$ about the $y$-axis using the shell method.
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Context:
Isabella must take four 100-point tests in her math class. Her goal is to achieve an average grade of 95 on the tests. Her first two test scores were 97 and 91. After seeing her score on the third test, she realized she can still reach her goal. What is the lowest possible score she could have made on the third test?
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Context:
Find the value of \(x\) such that \(e^{e^{10^{10^{2.8}}}} = 10^{10^{10^{10^{x}}}}\).
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Context:
Evaluate the following limit.$\begin{aligned} \lim_{n\to\infty}\sqrt{n}\int_0^\pi\sin^nxdx \end{aligned}$
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Context:
What conditions must the elements of a $3 \times 3$ matrix $\mathbf{R}$ satisfy for the matrix to be orthogonal, thereby preserving the length of any vector $\mathbf{A}$?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\mathbf{R}^\top \mathbf{R} = \mathbf{I}}