Context:
Find the greatest common divisor of all numbers of the form $p^{12} - 1$, where $p$ is a prime number with a units digit of one.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a,b,c,d$ be positive real numbers such that $abcd=4$ and
\[a^2+b^2+c^2+d^2=10.\]Find the maximum possible value of $ab+bc+cd+da$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the derivative of the sign function with respect to $y$ for the expression $\mathrm{sgn}(x-y)$. That is, find \( \frac{\mathrm{d}}{\mathrm{d}y}\mathrm{sgn}(x-y) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X_1, X_2, \ldots$ be a sequence of independent exponential random variables, each with mean 1. Given a positive real number $k$, define $N$ as $N=\min\left\{ n: \sum_{i=1}^n X_i >k \right\}$. Compute $E[N]$, the expected value of $N$. \(N\) is the smallest number for which the sum of the first \(N\) of the \(X_i\) is larger than \(k\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest value of $k$ such that for all real numbers $a_1, a_2, \ldots, a_n$, the inequality \( \sum_{i=1}^n a_i^3 \leq k \sqrt{\sum_{i=1}^n a_i^6} \) holds, where $n$ is an integer and $n \geq 3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an expression for the number of binary strings $b_k \cdots b_0$ such that the number of 0s, $m$, and the number of 1s, $n$, satisfy $m + 2n \leq L$. Provide a simplified or alternative expression to the following sum:
$$\sum_{n=0}^{\lfloor L/2 \rfloor} \sum_{m=0}^{L-2n} {n+m \choose n}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the rank of the tensor product of the integers \( \mathbb{Z} \) and the rational numbers \( \mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the operator norm of the linear functional \( T_n: \ell_q(\mathbb{N}) \to \mathbb{C} \) defined by \( T_n(x) = \sum_{i=2n+1}^{3n} x_i \) for \( q \in [1, \infty] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a strictly increasing function $f: [0, 1] \to \mathbb{R}$ that is continuous on $[0,1]$ and differentiable on $(0,1)$, with $f(0)=0$, and satisfying the condition $f'(x) \geq xf(x) + x$ for $x \in (0,1)$, find the minimum value of $f(1)$. Assume $f'(x) \geq 2xf(x)$ for $x \in (0,1)$ as well.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For what values of \( p > 0 \) is the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) defined by \( f(x) = |x|^p \) if \( x \in \mathbb{Q} \) and \( f(x) = 0 \) if \( x \in \mathbb{R} \setminus \mathbb{Q} \) differentiable?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a symmetric function $f(x_1, x_2, \ldots, x_n)$, it is known that \( \int_0^\infty \int_0^\infty f(x_1,x_2) \, dx_1 \, dx_2 = 2 \int_0^\infty \int_0^{x_2} f(x_1,x_2) \, dx_1 \, dx_2 \). Determine if there exists a constant $C$ such that \( \int_0^\infty \cdots \int_0^{\infty} f(x_1, \ldots, x_n) \, dx_1 \cdots \, dx_n = C \int_0^\infty \int_0^{x_n} \cdots \int_0^{x_n} f(x_1, \ldots, x_n) \, dx_1 \cdots \, dx_n \) for any symmetric function $f$ and any integer $n$. If such a $C$ exists, express $C$ in terms of $n$. \(n\) is the number of integrals.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compare the expressions \(8^{9^{8^{9}}}\) and \(9^{8^{9^8}}\). Determine which is greater, or if they are equal.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a grid with coordinates, you start at (0,0) and aim to reach (8,14) by moving only up or right. You cannot pass through any points where both coordinates are odd, such as (1,1), (1,3), (3,1), etc. Calculate the total number of valid paths from (0,0) to (8,14) under these conditions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the rate of convergence for the sample mean $\bar{X}_n$ to the population mean $\mu$ in terms of probability, expressed as $\bar{X}_n - \mu = o_p(n^\alpha)$. What is the value of $\alpha$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the system of stochastic differential equations given by:
\[
\begin{cases}
dX_t = a X_t \, dt + Y_t \, dW_t & ,X_0 = x\\
dY_t = a Y_t \, dt - X_t \, dW_t & ,Y_0 = y
\end{cases}
\]
where \(W=\{W_t:t \geq 0\}\) is the standard Brownian motion. Determine \(\mathbb{E}[X_t]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \(a\) such that \[ L = \lim_{n\to\infty}\frac{1^a+2^a+\cdots+n^a}{(n+1)^{a-1}[(na+1)+(na+2)+\cdots+(na+n)]} = \frac{1}{60} \] for \(a \in \mathbb{R} \setminus \{-1\}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a game with successive levels, the probability of a player passing the first level is \( \frac{1}{2} \), the probability of passing the second level given they passed the first is \( \frac{1}{3} \), and the probability of passing the third level given they passed the second is \( \frac{1}{4} \). What is the expected level at which a random player stops?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^{\pi/2}\frac{\cos 3x+1}{\cos 2x-1}dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a set of real numbers \( A \), define \( A - A \) as the set of distinct pairwise differences of \( A \), given by:
\[ A - A := \{ a - b : a, b \in A \} \]
If \( |A - A| = 25 \), determine the sum of all possible values of \( |A| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute \( \operatorname{Ext}_{k[x] / \langle x^2 \rangle}(k,k) \) where \( k \) is a field.
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Context:
What is the smallest non-trivial $E_8$-module?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the square root of $e$ to within an error of $10^{-4}$ using the Beuker-style integration by parts method.
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Context:
Let $f(x)$ be an irreducible polynomial of degree $p$, where $p$ is a prime number, over the field of rational numbers $\mathbb{Q}$. If $f(x)$ has exactly two nonreal roots in the complex numbers $\mathbb{C}$, determine the Galois group of $f(x)$ over $\mathbb{Q}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the growth order of the entire function \( f(z) = \sum \frac{z^n}{n^{bn}} \), where \( b \) is a positive constant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_\gamma \frac{dz}{z} \), where \( \gamma = -i + 5e^{2it} \) for \( t \in [0, 2\pi] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a single scaled Cauchy distribution \( f_{X|\theta}(s)=\frac{2}{\tau} \cdot \frac{\theta}{\theta^2+s^2} \) and the hypotheses \( H_0: \theta=1.4 \) versus \( H_1: \theta=2.3 \), calculate the Bayes Factor for \( H_0 \) versus \( H_1 \) given that a single data point \( X = 4.1 \) is drawn.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of positive integer triples \((x, y, z)\) such that \([x, y] = 72\), \([x, z] = 600\), and \([y, z] = 900\), where \([a, b]\) denotes the least common multiple of \(a\) and \(b\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The incircle \(\omega\) of an equilateral triangle \(ABC\) has a radius of 1. Three smaller circles are inscribed such that each is tangent to \(\omega\) and to two sides of \(ABC\). Then, three more circles are inscribed, each tangent to one of the previous circles and to two sides of \(ABC\). This process is repeated infinitely. Calculate the total length of the circumferences of all the circles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\to\infty}\frac{\sum_{k=0}^n\log\binom{n}{k}}{n^2}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a circle with a circumference of 2013 units, there are 2013 equally spaced points. A piece is placed at each point. The distance between two points is defined as the length of the shorter arc between them. Determine the maximum value of $n$ such that the pieces can be rearranged so that each point has a piece again, and the distance between any two pieces that were initially no more than $n$ units apart increases.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of points on the curve given by \(x^3 + x + 1 = y^2\) over the finite field \(\mathbb{F}_{125}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the set of numbers \( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \), how many ways can you choose 5 numbers such that their sum is 22?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equation \( \theta+\tan^{-1}\left(\frac{6}{4-x}\right)+\tan^{-1}\left(\frac{3}{x}\right)=\pi \), find \( \frac{d\theta}{dt} \) when \( x=4 \) and \( \frac{dx}{dt}=2 \text{ cm/sec} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f(x) = x^6 + ax^4 + bx^2 + c \), where \( a, b, c \in \mathbb{Q} \), be an irreducible polynomial in \( \mathbb{Q}[x] \). Determine the upper bound for the order of the Galois group \( G = \text{Gal}[K:\mathbb{Q}] \), where \( K \) is the splitting field of \( f(x) \) over \( \mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the name of a linear bijection whose transpose equals its inverse?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ways to schedule 10 interviews for 5 candidates, ensuring each candidate is interviewed at least once and no more than 4 times, with the order of interviews being important.
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Context:
Find the equation of the locus of the vertex of a variable parabola that touches a fixed parabola given by \(y^2 = 4ax\), assuming the variable parabola's axis is parallel to the x-axis and both parabolas are equal.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \(a < 0\) for which the inequalities \(2\sqrt{ax} < 3a - x\) and \(x - \sqrt{\frac{x}{a}} > \frac{6}{a}\) have at least one solution in common.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\pi}^{\pi} \frac{\sin\left(e^{ix}\right)}{e^{ix}} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the digits of \( l \), where \( l \) is the number of integers \( n \) such that the expression \( k^2 + \left\lfloor \frac{n}{k^2} \right\rfloor \) has a minimum value of 1991, with \( k \) being any positive integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \frac{1}{2\pi i}\int_{\gamma} \frac{1}{z^2(e^{z}-1)}dz \) where \( \gamma=\{z: z=\frac{1}{2}e^{2\pi i t},0\leq t\leq1 \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the largest factor of 10 for which the Collatz Conjecture has been verified to hold for all natural numbers up to that factor?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x_0, x_1, x_2, \ldots$ be the sequence defined by
$x_i= 2^i$ if $0 \leq i \leq 2003$
$x_i=\sum_{j=1}^{2004} x_{i-j}$ if $i \geq 2004$
Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by 2004.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two mirrors intersect at an acute angle at point A. A beam of light parallel to one mirror is reflected at point C, then again at points E and F. The beam finally intersects the other mirror perpendicularly at point G. If the distance from A to G is 5, find the length of the line segment from G to the intersection point of the mirrors.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all natural numbers \( n \) that satisfy the inequality: \( n \leqslant n! - 4^{n} \leqslant 4n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \( \triangle ABC \), point \( D \) is on side \( AC \). Given that \( \overline{AB} = \overline{DC} = 1 \), \( \angle DBC = 30^{\circ} \), and \( \angle ABD = 90^{\circ} \), find the length of \( \overline{AD} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the equation: \(8.469 \sin^{4} x + 2 \cos^{3} x + 2 \sin^{2} x - \cos x + 1 = 0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int _0^{2\pi} y e^{-a\cos(y)}\, dy \) where \( a \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f(x) \) be a continuously differentiable function on \([0,1]\) satisfying the following conditions: \( f(1) = 0 \), \( \int_0^1 [f'(x)]^2 \, dx = 7 \), and \( \int_0^1 x^2 f(x) \, dx = \frac{1}{3} \). Compute \( \int_0^1 f(x) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the matrices \(A = \begin{bmatrix}0 & 1 & 0 & 0 & 1\\ 0 &0& 1& 1& 0\\ 1& 0& 0& 0& 0\\ 1& 0& 0& 0& 1\\ 0& 1& 0& 1& 0\end{bmatrix}\) and \(M=A+A^2 = \begin{bmatrix}0& 2& 1& 2& 1\\ 2& 0& 1& 1& 1\\ 1& 1& 0& 0& 1\\ 1& 2& 0& 1& 2\\ 1& 1& 1& 2& 1\end{bmatrix}\), determine the minimum number of zeroes in \(A\) that must be replaced by ones so that all elements of \(M\) are non-zero.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $r$ for which a superstable period-2 orbit exists in the logistic map defined by $G(x) = rx(1-x)$. Provide your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can you rearrange the letters of ‘Alejandro’ such that it contains one of the words ‘ned’ or ‘den’?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\to\infty}\frac{\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}}{\sqrt[n]{n+1}-\sqrt[n]{n}} \]
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Context:
$\begin{aligned} & \text{Find the number of integers }n\mathrm{~with~}1\leq n\leq2017\text{ so that }(n-2)(n-0)(n-1)(n-7)\text{is an integer} \\ & \text{multiple of 1001.} \end{aligned}$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a divisor of the number $4^{1000} - 1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For what values of \( p \) is the limit comparison test invalid for the sequences \( a_n = 1 - 2^{-1/n} \) and \( b_n = \frac{1}{n^p} \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume of the solid generated by revolving the region under the first arc of the cycloid, given by the parametric equations \(x(t) = a(t - \sin t)\) and \(y(t) = a(1 - \cos t)\), about the \(x\)-axis. Use the formula \(V = \pi \int [f(y)]^2 \, dy\) to find the volume.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an integer $k \geq 1$, determine the minimal integer $n$ such that there exist matrices $A_1, A_2, \ldots, A_k \in M_n(\mathbb{C})$ satisfying the conditions $A_i A_j = -A_j A_i$ for all $i \neq j$, and $A_1 A_2 \cdots A_k \neq 0$. Additionally, each $A_i^2$ must be zero. Find this minimal $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the work done against the force field \( F(x, y) = -\frac{(x, y)}{\sqrt{x^2 + y^2}} \) when a particle is moved from the point \((2a, 0)\) to the origin along the upper half of the circle with the equation \((x - a)^2 + y^2 = a^2\). Use the parametrization \(x = a + a\cos(t)\), \(y = a\sin(t)\) for \(t \in [0, \pi]\) to determine the work done using a line integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest positive integer \( n \) such that the number of distinct groups of order \( n \) is greater than \( n \) itself.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the value of $p + q$ if the integral $$\int_{0}^{\infty} \frac{\log(x)}{x^2 + 2x + 4} \, \mathrm dx$$ equals $\frac{\pi \ln(p)}{\sqrt{q}}$, where $p$ and $q$ are coprime integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{x\to\infty} \left(\frac{1}{x}\frac{a^x - 1}{a - 1} \right)^\frac{1}{x}, \] where \(0 \leq a\) and \(a \neq 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the ratio of the height to the radius of a cone such that the cone's volume is maximized and its surface area is minimized.
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Context:
Given the functions $f(x)=2x-4,g(x)=-x+4$, then the interval of monotonic increase for the function $y=f(x)g(x)$ is ______.
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Context:
Determine the basis or dimension of the vector space $A_k$, consisting of polynomials $f(x_1,x_2,\ldots,x_n)$ in $\mathbb{Q}[x_1,x_2,\ldots,x_n]$ that are symmetric in the first $k$ variables, i.e., $$f(x_{\sigma(1)},\ldots, x_{\sigma(k)},x_{k+1}, \ldots,x_n)=f(x_1,x_2,\ldots,x_n),\,\, \sigma \in S_k.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( N \) be the number of ordered pairs of nonempty sets \( \mathcal{A} \) and \( \mathcal{B} \) such that:

- \( \mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\} \),
- \( \mathcal{A} \cap \mathcal{B} = \emptyset \),
- The number of elements in \( \mathcal{A} \) is not an element of \( \mathcal{A} \),
- The number of elements in \( \mathcal{B} \) is not an element of \( \mathcal{B} \).

Find \( N \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the angle between the planes represented by the factors of the expression \(6x^2+4y^2-10z^2+3yz+4zx-11xy\).
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Context:
In the Cartesian coordinate system \(xOy\), the set of points \(K=\{(x, y) \mid x, y=-1,0,1\}\). Three points are randomly selected from \(K\). What is the probability that the distance between any two of these three points does not exceed 2?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the greatest integer \( n \) such that, regardless of the 100 distinct integers written by Peter, Basil can fill a \( 100 \times 100 \) table with integers so that each number from 1 to \( n \) appears at least once, while ensuring the sum in each \( 1 \times 3 \) rectangle (either vertical or horizontal) equals one of Peter's numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of the real number $a$ such that the inequality \(x^{2} + 25 + \left|x^{3} - 5 x^{2}\right| \geq a x\) holds for all \(x \in [1,12]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $z = 5 + 12i$ be a complex number and $w$ be a complex number such that $|w| = 13$. What is the probability that $|z - w| > 13\sqrt{3}$?
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Context:
Find the minimum value of \( \sum_{1\le i<j\le 6}[a_{i}+a_{j}] \) where \( a_{1},a_{2},\cdots,a_{6} \) are real numbers such that \( a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}=2014 \) and \( [x] \) denotes the greatest integer less than or equal to \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the norm of the differential operator \( \frac{d}{dx} : P^n[0,1] \rightarrow P^{n-1}[0,1] \) in the space of polynomials of degree \( \leq n \) on \([0,1]\), equipped with the sup norm.
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 $0 \leq i < j \leq 2014$. Let $O = (0, 0, 0, \ldots, 0)$, $A_0 = (1, 0, 0, \ldots, 0)$, and suppose $A_iO$ has length 1 for $i = 0, 1, \ldots, 2014$. Set $P = (20, 14, 20, 14, \ldots, 20, 14)$. Find the remainder when
$$PA_0^2 + PA_1^2 + \cdots + PA_{2014}^2$$
is divided by $10^6$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the interval of convergence for the Maclaurin series of the function \( f(x) = \ln(1+x) \) using the formula for the remainder term in Taylor's Theorem.
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Context:
Find the fundamental group \( \pi_1(X) \) of the topological space \( X = (S^1 \times S^2) \setminus (A \cup B) \), where \( A = \{(0,1)\} \times S^2 \) and \( B = S^1 \times \{(0,0,1)\} \).
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Calculate the limit: $$\lim_{x\to 0}\dfrac{1}{1-\cos(x^2)}{\sum_{n=4}^\infty {n^5 x^n} }$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the supremum of the function \( \frac{f(x)}{g(x)} \), where
\[
f(x) = \dfrac{\beta \exp(-\beta x) \int\limits_{-\infty}^{x} \exp\left(-\frac{(x-\mu)^2}{2 \sigma^2}\right) \,\mathrm{d}x}{\int\limits_{-\infty}^{\infty}\beta \exp(-\beta x) \,\mathrm{d}x \int\limits_{-\infty}^{x} \exp\left(-\frac{(x-\mu)^2}{2 \sigma^2}\right) \,\mathrm{d}x}
\]
and
\[
g(x)=e^{-x} \;.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ways to divide 12 people into groups such that person A and person B are never in the same group, considering all possible group configurations.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit of the sequence \( b_n = \frac{(n+2)!}{\sqrt{(2(n+2)+1)!}} \cdot \frac{\sqrt{(2n+1)!}}{n!} \).
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Context:
A teacher has 2010 exam questions and wants to divide them into three folders, each containing 670 questions. Each folder is to be given to a student who has solved all 670 questions in that folder. Determine the minimum number of students required in the class, given that for any question, at most two students did not solve it.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( n \geq 3 \) and consider the linear transformation \( T : p(x) \rightarrow 6p(x) + p'(x) - x^2p''(x) \) on the vector space \( \mathbb{R}[x]_n \). Determine the nullity of \( T \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the expected value of the ratio between the second largest and the largest number in a set of $n$ independent, identically distributed random variables, each following a uniform distribution from $0$ to $1$. Specifically, given $X_1, X_2, \ldots, X_n \sim U[0,1]$, where $x_p = \max_i X_i$ is the largest number and $x_q = \max_{i \neq p} X_i$ is the second largest number, calculate $\mathbb{E}\left[\frac{x_q}{x_p}\right]$. \(n\) is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \( f(n) \) defined as follows: \( f(n) = n - 2 \) for \( n > 3000 \) and \( f(n) = f(f(n+5)) \) for \( n \leq 3000 \), find the value of \( f(2022) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: \[ \int_0^{\ln 2}\left(\frac{1}{e^x-1} - \left\lfloor\frac{1}{e^x-1}\right\rfloor\right)dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways are there to divide 30 objects, consisting of 5 pears, 5 apples, 5 doughnuts, 5 lollipops, 5 chocolate cats, and 5 candy rocks, into two unordered piles of 15 objects each?
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Context:
Calculate the limit: \[ \lim_{k\to\infty}\int_{\mathbb{R}^2}\frac{(x^2+y^2)^{k/2}}{1+(x^2+y^2)^{(k+3)/2}}\,dx\,dy \] without using the change of variable theorem.
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Context:
Determine an upper bound on the number of iterations required for the recurrence relation \( R(0) = n \) and \( R(t) = R(t-1) - \max\left(1, \left\lfloor\sqrt{R(t-1)}\right\rfloor\right) \) to terminate, where the recurrence stops when \( R(\cdot)=0 \).
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Context:
Calculate the inverse Fourier transform of \( \frac{1}{a+\mathrm{j}w} \) using the definition:
\[
f(t)=\frac{1}{2\pi}\int_{-\infty}^\infty\frac{1}{a+\mathrm{j}w}\mathrm{e}^{\mathrm{j}wt}\,\mathrm{d}w.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dimension of the smallest faithful irreducible representation of the direct product $(C_3)^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the omega limit set of a trajectory in the dynamical system given by \( \dot x = -y, \dot y = x, \dot z = -x^2-y^2 \) that lies on a helix and travels downward at a constant speed.
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Context:
Let \( p = 101 \). The sum \( \sum_{k=1}^{10}\frac{1}{\binom{p}{k}} \) can be expressed as a fraction of the form \( \frac{a}{p!} \), where \( a \) is a positive integer. Compute \( a \mod p \).
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Context:
In an isosceles trapezoid with bases \( AD \) and \( BC \) where \( AD > BC \), a circle is inscribed such that it touches side \( CD \) at point \( M \). If segment \( AM \) intersects the circle at point \( N \) and the ratio \( AN : NM = k \), determine the ratio of \( AD \) to \( BC \).
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Context:
Find the larger of the two positive integers \(n \ge 3\) for which the expression \(\frac{\log 2 \times \log 3 \times \log 4 \times \cdots \times \log(n-1)}{10^n}\) takes its smallest value.
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Context:
You are at location \((0,0)\) and know the treasure is within 100 meters of you. Person A claims that they walked 108 meters to the treasure, with their distance calculated by \(|x| + |y|\). Person B claims that they walked 105 meters to the treasure, with their distance calculated by \(\frac{|x|}{\sqrt{3}} + \max\left\{\frac{|x|}{\sqrt{3}}, |y|\right\}\). Determine the number of possible locations for the treasure.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the kernel of the ring homomorphism \( f: k[x,y] \to k[t] \) defined by \( f(x) = t \) and \( f(y) = q(t) \), where \( q(t) \in k[t] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple integral \( \int_{-2}^{-1} \int_{-2}^{-1} \int_{-2}^{-1} \frac{x^2}{x^2+y^2+z^2} \, dx \, dy \, dz \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit: $$\lim_{z\to 0}\frac{d^2}{dz^2}\left(\frac{e^zz}{\sin z}\right)$$ without using l'Hôpital's rule or Taylor series.
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Context:
Simplify the expression \((\lambda y. z)((\lambda x. x x)(\lambda x. x x))\).
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Context:
Evaluate the integral: $$\int\limits_{\frac{1}{e}}^{1} \frac{dx}{x\sqrt{\ln{(x)}}} $$
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Context:
Find the spectrum of the C*-algebra \( B = \bigoplus_{k=1}^\infty \mathbb{C} \).
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Context:
Determine the degree of the expression $\rho_7+\rho_7^2+\rho_7^4$ over $\mathbb{Q}$, where $\rho_7$ is a primitive seventh root of unity.
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Context:
A mother with blood type O and a father with blood type AB have twins, both sons, with blood type B. Given that 32% of all twins have different genders, calculate the probability that the twins are identical.
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Context:
Evaluate the improper integral: $$\int_0^{\infty}\frac{1}{x(1+x^2)}\,dx$$
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Context:
Express the truncated Riemann zeta function in terms of generalized harmonic numbers.
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Context:
In a scalene triangle $ABC$, the angle bisector $AD$ intersects the circumcircle of the triangle at point $P$. Point $I$ is the incenter of the triangle $ABC$. Given that $ID = DP$, determine the ratio $AI: ID$. (20 points)
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Context:
Evaluate the integral \( \int_{0}^{\pi/2}\ln(4\sin^2\frac{u}{2})\,du \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all real values of \( \alpha \) such that the integral \( \int_{0}^{\infty} x(x^2/2 + 1)^{\alpha} \, dx \) converges.
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Context:
Evaluate the integral \( \int_{-\pi/2}^{\pi/2}\log(\cos(\theta))\cos(\theta)\,\mathrm{d}\theta \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[\lim_{n\to +\infty}\left(\frac{f(a+\frac{1}{n})}{f(a)}\right)^{\frac{1}{n}}\] where \(f: \mathbb{R} \to \mathbb{R}\) is differentiable at \(a \in \mathbb{R}\) with \(f(a) > 0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The price of an article is cut \( {10}\% \) . To restore it to its former value, the new price must be
 increased by:
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an $11\times 11$ diagonal matrix $A$ with the characteristic polynomial $(x-3)^4(x-4)(x-7)^4(x-9)^2$, find the dimension of the space $V$ of $11\times 11$ matrices $B$ such that $AB = BA$. Determine the dimension of $V$.
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Context:
Compute the Lebesgue integral \( \int_{0}^{\pi} f(x) \, dx \) where \( f(x) = \begin{cases} \sin x, & \text{if } x \in \mathbb{I} \\ \cos x, & \text{if } x \in \mathbb{Q} \end{cases} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the sign of the integral \( \int_{0}^{2\pi} \frac{\sin x}{x} \, dx \).
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Context:
Let $ p = 2017 $. If $ A $ is an $ n \times n $ matrix composed of residues $ (\mod p) $ such that $ \det A \not\equiv 0 \pmod{p} $ then let $ \text{ord}(A) $ be the minimum integer $ d > 0 $ such that $ A^d \equiv I \pmod{p} $, where $ I $ is the $ n \times n $ identity matrix. Let the maximum such order be $ a_n $ for every positive integer $ n $. Compute the sum of the digits when $ \sum\limits_{k=1}^{p+1} a_k $ is expressed in base $ p $.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 2010 lemmings. At each step, we may separate the lemmings into groups of 5 and purge the remainder, separate them into groups of 3 and purge the remainder, or pick one lemming and purge it. Find the smallest number of steps necessary to remove all 2010 lemmings.
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Context:
Evaluate the limit: $$\lim_{x\to 0}\dfrac{1}{1-\cos(x^2)}{\sum_{n=4}^\infty {n^5 x^n} }$$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum sample size \( n \) such that the probability that the sample mean \( \frac{Y_1 + Y_2 + \cdots + Y_n}{n} \) is within two standard deviations of the mean \( \mu \) is at least 98%, given that \( Y_1, Y_2, \ldots, Y_n \) are independent and identically distributed random variables.
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Context:
Compute the second-order mixed partial derivative \( \frac{\partial^2 f}{\partial y \partial x} \) at the point \((0,0)\) for the function \( f(x,y) = \frac{xy(x^2-y^2)}{x^2+y^2} \) when \((x,y) \neq (0,0)\), and \( f = 0 \) otherwise.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Fiona has an open chain with 2018 links. She can cut out links one by one, each time creating 2 or 3 new chains. What is the minimum number of links she needs to remove to be able to pay for any amount from 1 to 2018 links using some combination of her chains?
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Context:
Evaluate the integral: \[ \int_{0}^{\infty}\dfrac{1-e^{-2t}}{t}\sin{2t}\operatorname d\!t \]
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Context:
Consider a tetrahedron inscribed in a cube. Determine the number of rotation matrices that correspond to rotations of the tetrahedron fixing two of its vertices.
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Context:
Determine the dimension of the $R$-module $N$ over the ring $R = \mathbb{R}[T]$, where $N$ is generated by elements $u$ and $v$ with the relations $Tu = v$ and $T^{3}u = v$. What is the dimension of $N$?
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Context:
For how many integers $n\neq1$ does $\left(n-1\right)^3$ divide $n^{2018\left(n-1\right)}-1$?
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Context:
Diagonalize the quadratic form \( q(x_1, x_2, x_3, x_4) = x_1x_2 + x_2x_3 + x_3x_4 \) and determine its signature.
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Context:
Determine the values of $m$ for which the equation \((a_{m-1}+a_1)+\ldots+ 10^{2m-2}(a_{m-1}+a_1) = \frac{10^{2m}-1}{9}\) has a solution, where $a_1$ and $a_{m-1}$ are digits from 0 to 9.
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Context:
Let \( n \) be the least positive integer for which \( {149}^{n} - {2}^{n} \) is divisible by \( {3}^{3} \cdot  {5}^{5} \cdot  {7}^{7} \) . Find the number of positive divisors of \( n \) .
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Context:
Find the real value(s) of \(a\) (where \(a \neq -1\)) for which the limit \( \lim_{ n \to \infty} \frac{ 1^a+2^a+\cdots+n^a}{(n+1)^{a-1}[ (na+1)+(na+2)+\cdots+(na+n)]} = \frac{1}{60} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the Shapley-Shubik power index of player 1 in a weighted majority game with weights \([q;w_1,1,\dots,1]\), where \(q\) and \(w_1\) are whole numbers satisfying \(w_1\leq q\leq n-1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a medical diagnostic scenario, consider the following probabilities: $\Pr(C)$ (condition present), $\Pr(N)$ (condition absent), $\Pr(+)$ (test positive), and $\Pr(-)$ (test negative). Additionally, there are eight conditional probabilities related to these events. Given the cyclical pattern in these conditional probabilities, determine the number of degrees of freedom in this framework. How many probabilities must be specified to determine the rest?
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Context:
Find the minimum number of distinct non-trivial solutions for the Diophantine equation \(\frac{1}{a} = \frac{1}{x_1} + \cdots + \frac{1}{x_n}\) for any positive integers \(a\) and \(n\).
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Context:
Determine the dimension of a maximal commutative vector subspace of $M_3(\mathbb{C})$. A vector subspace $V$ of $M_3(\mathbb{C})$ is commutative if for every $A, B \in V$, $AB = BA$. Additionally, $V$ is maximal if any larger subspace $W$ containing $V$ has some $A, B \in W$ such that $AB \neq BA$. What is $\dim V$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the discriminant of the number field $K = \mathbb{Q}(\alpha)$, where $\alpha$ is a root of the minimal polynomial $x^3 - x - 1$. The polynomial has degree 3 over $\mathbb{Q}$, with 2 complex roots and 1 real root.
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Context:
Evaluate the limit: \[ \lim_{x\rightarrow0}\frac{\int_{\cos(\sqrt{x})}^{1}\frac{\cos(t)}{1-t}dt}{\sqrt{x}} \]
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Context:
Let $A=\{a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n\}$ be a set with $2n$ distinct elements, and $B_i\subseteq A$ for any $i=1,2,\cdots,m.$ If $\bigcup\limits_{i=1}^m B_i=A,$ we say that the ordered $m-$tuple $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A.$ If $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A,$ and for any $i=1,2,\cdots,m$ and any $j=1,2,\cdots,n,$ $\{a_j,b_j\}$ is not a subset of $B_i,$ then we say that ordered $m-$tuple $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A$ without pairs. Define $a(m,n)$ as the number of the ordered $m-$coverings of $A,$ and $b(m,n)$ as the number of the ordered $m-$coverings of $A$ without pairs.
Let $m\geq2,$ and there is at least one positive integer $n,$ such that $\dfrac{a(m,n)}{b(m,n)}\leq2021,$ Determine the greatest possible values of $m.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( A = \{1, 2, \ldots, 2002\} \) and \( M = \{1001, 2003, 3005\} \). A non-empty subset \( B \) of \( A \) is called an \( M \)-free set if the sum of any two elements in \( B \) does not belong to \( M \). If \( A = A_1 \cup A_2 \), \( A_1 \cap A_2 = \varnothing \), and both \( A_1 \) and \( A_2 \) are \( M \)-free sets, then the ordered pair \(\left(A_{1}, A_{2}\right)\) is called an \( M \)-partition of \( A \). Determine the number of all \( M \)-partitions of \( A \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Randy has a deck of $29$ distinct cards. He chooses one of the $29!$ permutations of the deck and then repeatedly rearranges the deck using that permutation until the deck returns to its original order for the first time. What is the maximum number of times Randy may need to rearrange the deck?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum order of the automorphism group of an elliptic curve over a field of characteristic 0, when restricted to torsion subgroups.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider an infinitely differentiable function $y=f(x)$ defined on the real numbers, such that $f(0) \neq 0$ and the $n$th derivative of $y$ with respect to $x$ evaluated at $x=0$ is nonzero for $n=1,2,3,4$. Determine the value of $25a + 50b + 100c + 500d$, where $a$, $b$, $c$, and $d$ are constants, given that the following limit exists:

$$\lim_{x \to 0} \frac{f(4x) + af(3x) + bf(2x) + cf(x) + df(0)}{x^4} = L$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be a finite abelian group of order $360$ which does not contain any elements of order $12$ or $18$. How many elements of order $6$ does $A$ contain?
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Context:
Let $f$ be a function from the set $\{1,2,3,\dots,10\}$ to the real numbers $\Bbb{R}$, satisfying the condition:

$$\left(\sum\limits_{i=1}^{10}{\frac{|f(i)|}{2^i}}\right)^2 = \left(\sum\limits_{i=1}^{10}{|f(i)|^2}\right)\left(\sum\limits_{i=1}^{10}{\frac{1}{4^i}}\right)$$

Determine the number of distinct functions $f$ that satisfy this condition.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the function:

$$f(x, y) = \sqrt{4 + y^2} + \sqrt{(x - 2)^2 + (2 - y)^2} + \sqrt{(4 - x)^2 + 1}$$

where \(x\) and \(y\) are real numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim _{x\to \infty }\left(x\sqrt{x^2-x}-x^2\cos\left(\frac{1}{\sqrt{x}}\right)\right)$$
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Context:
Given a ring \( R \) and an ideal \( \mathbf{a} \), let \( \pi: R \rightarrow R/\mathbf{a} \) be the natural projection map. Define \( \pi^*:\mathrm{Spec}(R / \mathbf{a}) \rightarrow \mathrm{Spec}(R) \) as the pullback of \( \pi \). Determine the image of \( \pi^* \) in \( \mathrm{Spec}(R) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^1 \frac{x^{10}-1}{\ln x} \, dx \) using Feynman's method.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \(a\) for which the integral \(\int_{0}^{\infty}\frac{dx}{a^2+\left(x-\frac{1}{x}\right)^2}\) equals \(\frac{\pi}{5050}\) for \(a \geq 2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_R (y+2x^2)(y-x^2) \, dA \), where the region \( R \) is bounded by the curves \( xy=1 \), \( xy=2 \), \( y=x^2 \), and \( y=x^2-1 \), with both \( x \) and \( y \) positive.
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Context:
In triangle $ABC$, a point $O$ is located such that $OA = 15$, $OB = 12$, and $OC = 20$. The feet of the perpendiculars from $O$ to the sides of triangle $ABC$ form an equilateral triangle. Determine the measure of angle $BAC$. \( \text{(in degrees)} \)
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Context:
A unit circle is centered at(0,0)on the(x,y)plane. A regular hexagon passing through(1,0)is inscribed in the circle. Two
 points are randomly selected from the interior of the circle and horizontal lines are drawn through them, dividing the hexagon
 into at most three pieces. The probability that each piece contains exactly two of the hexagon's original vertices can be
 written as
 \( \frac{2{\left( \frac{m\pi }{n} + \frac{\sqrt{p}}{q}\right) }^{2}}{{\pi }^{2}} \)
 for positive integers \( m,n,p \) ,and \( q \) such that \( m \) and \( n \) are relatively prime and \( p \) is squarefree. Find \( m + n + p + q \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest integer \( n \) such that for any two sets \( \{ x_1, x_2, \ldots, x_n \} \) and \( \{ y_1, y_2, \ldots, y_n \} \) of \( n \) distinct points on the unit circle \( \mathbb{S}^1 \), there exists a unique automorphism \( f: \mathbb{D} \rightarrow \mathbb{D} \) satisfying \( f(x_i) = y_i \) for all \( i \).
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Context:
Determine the number of terms needed in the series \( \sum_{n=0}^\infty \frac{(-1)^n(7)^{2n+1}}{(2n+1)(2n+1)!} \) to approximate the integral \( \int_0^1 \frac{\sin(7x)}{x} \, dx \) to four decimal places.
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Context:
The inclination angle of the line $x-y+1=0$ is what value?
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Context:
Calculate the minimum value of the function \( f(z) = |z^2 + z + 1| + |z^2 - z + 1| \) for \(z \in \mathbb{C}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the equation for real numbers $x_1, x_2, \ldots, x_n$:

$$
\sqrt{x_{1}-1^{2}}+2 \sqrt{x_{2}-2^{2}}+\cdots+n \sqrt{x_{n}-n^{2}}=\frac{x_{1}+x_{2}+\cdots+x_{n}}{2}
$$
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Context:
Determine all values of \( a \) for which the equation \( a\sin{x}+(a+1)\sin^2{\frac{x}{2}} + (a-1)\cos^2{\frac{x}{2}} =1 \) has at least one solution for \( x \).
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Context:
Evaluate the integral: $$\int x^{x^2+1}(2\ln x+1)dx.$$
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Context:
A mouse is trapped in a maze and can initially choose to go right or left. If it goes left, it will walk for 3 minutes and return to the initial position. If it goes right, with probability \( \frac{1}{3} \) it will exit the maze after 2 minutes, and with probability \( \frac{2}{3} \) it will return to the initial position after 5 minutes. Assuming the mouse randomly chooses left or right, calculate the variance of the time it remains in the maze.
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Context:
Determine the convergence domain of the series \( \sum_{n=1}^{\infty}\frac{(2n-1)^{n}(x+1)^n}{2^{n-1}n^n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a random sample of size \(n\) from a distribution with the probability density function given by:
\[
f(x;\theta)=\begin{cases}
 \sqrt{\frac{2}{\pi}}e^{-\frac{1}{2}(x-\theta)^2} & \text{if } x \geq \theta \\
 0 & \text{elsewhere}
\end{cases}
\]
Find the maximum likelihood estimator of \(\theta\).
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Context:
For a given natural number $n$, determine the greatest possible value of $k$ such that there exist subsets $A, B, C, D$ of the set of the first $n$ natural numbers with the property that any two elements of the set are in at least one of the subsets, and at least one of $|A|,|B|,|C|,|D|$ is greater than $k|S|$. Find the maximum value of $k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \(a\) such that for \(a \in \mathbb{R}\) and \(a \neq 1\), the following limit holds:
\[ \lim_{n \to \infty} \frac{1^a + 2^a + \cdots + n^a}{(n+1)^{a-1} \left[ (na+1) + (na+2) + \cdots + (na+n) \right]} = \frac{1}{60}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of continuous involutions on the set of real numbers, where an involution is a function \( f \) such that \( f = f^{-1} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of inequivalent complete norms that exist on an infinite-dimensional vector space $X$.
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Context:
Calculate the probability that five randomly chosen points form a hyperbola.
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Context:
A quadrilateral that has consecutive sides of lengths $70,90,130$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length $x$ and $y$. Find $|x-y|$.
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Context:
Evaluate the limit: $$L = \lim_{x\rightarrow\infty}xe^{x^2} \left(\int_0^xe^{-t^2}dt-\int_0^\infty e^{-t^2 } dt\right)$$
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Context:
Solve the equation for real numbers \(x\), given that \(a\) and \(b\) are real:
\[
\left| a^2-2a-b^2-4b-x \right| + \left| a^2-2a-b^2-4b-3x+2 \right| + \left| a^2-2a+b^2+4b+2x \right| + a^2-2a+b^2+4b+18 \left| x-2 \right| + 11x = 20.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of matrices in $GL(2,\mathbb{F_{25}})$ that have a determinant equal to 3.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the codimension of the maximal ideal $J = (x,y,z)$ in the ring $R = k[x,y,z]/(x^2y^2z^2, xy^3z^2)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$\int_0^{2\pi}\cot(e^{it})ie^{it}dt$$
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Context:
Calculate the limit of the sequence defined by:
\[ x_n = \frac{1}{n}\left(n + \frac{n-1}{2} + \frac{n-2}{3} + \ldots + \frac{2}{n-1} + \frac{1}{n} - \log(n!)\right) \]
Use asymptotic equivalences to find the limit, considering that \( \log(n!) \sim n\log(n) \).
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Context:
Let $A$ be a complex unital Banach algebra and consider the matrix $R_t=\begin{pmatrix} \cos\frac{\pi t}{2} & -\sin\frac{\pi t}{2} \\ \sin\frac{\pi t}{2} & \cos\frac{\pi t}{2} \end{pmatrix}$. Determine the norm of $R_t$ as an element of $M_2(A)$, where $M_2(A)$ is equipped with the operator norm derived from $A^2$ with the norm $||(a,b)||=\sqrt{||a||^2+||b||^2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of the real constant \(a\) for which the series \[ 1 + \frac{1}{\sqrt{3}} - \frac{a}{\sqrt{2}} + \frac{1}{\sqrt{5}} + \frac{1}{\sqrt{7}} - \frac{a}{\sqrt{4}} + \frac{1}{\sqrt{9}} + \frac{1}{\sqrt{11}} - \frac{a}{\sqrt{6}} + \cdots \] converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $G=(\Sigma,N,R,S)$ be a context-free grammar where $n = |N|$ is the number of non-terminal symbols and $k$ is the maximal length of a production rule. Assume $L(G)$, the language generated by $G$, is not empty. Find a tight upper bound expression $f(n,k)$ for the length of the shortest word in $L(G)$. The expression should satisfy that the shortest word in $L(G)$ is at most $f(n,k)$, and there exists another context-free grammar $F$ such that the shortest word in $L(F)$ is exactly $f(n,k)$. Provide the expression $f(n,k)$. \( \text{Express your answer in terms of } n \text{ and } k. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the sequence \( \{ a_{n} \} \) in the Cantor space with respect to the Cantor metric, where each term \( a_n \) consists of the first \( n \) digits being 0 followed by 1s. Does this sequence converge? If so, determine the limit.
Please reason step by step, and put your final answer within \boxed{}.


Context:
John’s birthday cake is a scrumptious cylinder of radius $6$ inches and height $3$ inches. If his friends cut the cake into $8$ equal sectors, what is the total surface area of a piece of birthday cake?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all increasing functions \( g: \mathbb{Z}^+ \to \mathbb{Z} \) such that \( g(1) = 1 \) and \( g(ab) = g(a) + g(b) + g(a)g(b) \) for all positive integers \( a \) and \( b \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral \( \oint_{L}{\frac{e^{\frac{1}{z-a}}}{z}}dz \) using the Residue Theorem, where \( L=\{z\in\mathbb{C}:|z|=r\} \) for some \( r>|a| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
$k$ stones are put into $2012$ boxes in such a way that each box has at most $20$ stones. We are chosing some of the boxes. We are throwing some of the stones of the chosen boxes. Whatever the first arrangement of the stones inside the boxes is, if we can guarantee that there are equal stones inside the chosen boxes and the sum of them is at least $100$, then $k$ can be at least?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a function $g(t) \in L^1(\mathbb{R})$ that satisfies the integral equation:

$$ g(t) = \int_{-\infty}^{t}e^{u-t}g(u)\,du + f'(t) $$

where $f \in L^1(\mathbb{R})$ and its derivative $f'$ is both continuous and in $L^1(\mathbb{R})$. Use the Fourier transform to determine $g(t)$. Provide your answer as a function $g(t)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rhombus \( \mathcal{R} \) has a short diagonal of length 1 and a long diagonal of length 2023. Let \( \mathcal{R}' \) be the rotation of \( \mathcal{R} \) by \( 90^\circ \) about its center. Define \( \mathcal{U} \) as the set of all points contained in either \( \mathcal{R} \) or \( \mathcal{R}' \) (the union of \( \mathcal{R} \) and \( \mathcal{R}' \)), and \( \mathcal{I} \) as the set of all points contained in both \( \mathcal{R} \) and \( \mathcal{R}' \) (the intersection of \( \mathcal{R} \) and \( \mathcal{R}' \)). Compute the ratio of the area of \( \mathcal{I} \) to the area of \( \mathcal{U} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two random vectors \( X = (X_1, \ldots, X_n) \) and \( Y = (Y_1, \ldots, Y_n) \), with a covariance matrix \( G \) such that the covariance between \( X_i \) and \( Y_j \) is \( G_{ij} \), and two \( n \times n \) matrices \( A \) and \( B \), find the covariance matrix between \( AX \) and \( BY \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of terms, n, that need to be summed in the Maclaurin series for the arctangent function to ensure that \(|4P_n(1) − π| < 10^{-3}\), given that \(\tan \frac{π}{4} = 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum area bounded by the curves \(y^2=4ax\), \(y=ax\), and \(y=\frac{x}{a}\) for \(a \in [1,2]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of non-isomorphic semisimple $\mathbb{C}$-algebras of dimension 8.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the equation: \[ 8x^2 + 2\ln x - 6 = 0 \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ways to arrange the first 9 positive integers on a 3x3 chessboard such that the largest of the smallest numbers in each column is equal to the smallest of the largest numbers in each row, and both are equal to 4.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of bits Alice must transmit to Bob so that he can compute the approximate Hamming distances between a binary string $A$ of length $n$ and every substring of length $n$ of a binary string $B$ of length $2n$, within a factor of 2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest integer $n$ for which the number of pairs $(x, y)$ satisfying $x \ge y$, $\operatorname{lcm}(x, y) = n!$, and $\gcd(x, y) = 2023$ is greater than 2023. Note that $2023 = 7 \times 17^2$. What is this value of $n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{\alpha\to\infty} \int_{0}^{\delta}\frac{\sin(\alpha x) \sin(\lambda\alpha x)}{x^2}dx$$ where $\lambda,\delta \in \mathbb{R}, \lambda,\delta > 0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the complexity of the recurrence relation \( T(n) = T(n - \sqrt{n}) + n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $a,b,c,d \in \mathbb{R}$ such that $c^2+d^2 \neq 0$, consider the Cauchy problem $au_x+bu_y=e^{x+y}$ for $x,y\in \mathbb{R}$ with the condition $u(x,y)=0$ on $cx+dy=0$. Using Lagrange's equations, we have $bx-ay=C_1$ and $u-\frac{a}{a+b}e^{x+y}=C_2$. Determine the condition for a unique solution to this problem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a Markov chain with state space $S = \{0, 1, 2, \dots \}$ and given transition probabilities. Determine the values of $p$ for which the chain is recurrent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x \to 0} \frac{(x-\sqrt{2})^4}{\sin^3(x)\cos^2(x)}\int_{0}^x \frac{t^2}{t^4+1}dt$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x \to 0^+} \lim_{n \to \infty} \frac{1}{1 + n \sin^2 x}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integer \( \lambda \) such that the group homomorphism \( \mathbb{Z} \cong \pi_{n+1}(S^{n+1}) \rightarrow \pi_n(SO_{n+1}) \rightarrow \pi_n(S^n) \cong \mathbb{Z} \) is multiplication by \( \lambda \). The map is induced by the matrix multiplication map \( \alpha: SO_{n+1} \rightarrow S^n \) and the connecting homomorphism of the fibration \( SO_{n+1} \rightarrow SO_{n+2} \rightarrow S^{n+1} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
## 11. "Wipers"

On the windshield of a car, there are two "wipers" of length $L$ each, rotating around two points that are also $L$ apart. Each "wiper" "sweeps" one semicircle. What area do both wipers sweep?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the constant $p$ such that the geometric mean of any positive number $N_0$ multiplied by successive random numbers between $0$ and $p$ will, on average, neither diverge to infinity nor converge to zero.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the conditional expectation \( E[W|X>Y] \) where \( W = X+Y \) and \( X, Y \sim \exp(2) \) are independent exponential random variables.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \int_C y^2 \, dx + x \, dy + z \, dz \), where \( C \) is the curve formed by the intersection of the surfaces \( x^2 + y^2 = x + y \) and \( 2(x^2 + y^2) = z \). The curve \( C \) is oriented in the positive direction as viewed from the point \( (0, 0, 2R) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{|z|=2} \log\frac{z+1}{z-1} \, dz \), assuming that for \( w \in \mathbb{C} - \{z; \text{Re}(z) \leq 0\} \), we have \(-\pi < \text{Im}(\log w) < \pi \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For integers $n > 1$, let $a_n$ represent the number of trailing zeroes in the factorial $n!$ when expressed in base $n$. Determine the maximum value of the ratio $\frac{a_n}{n}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The plane given by the equation \(x + 2y + 3z = 7\) is rotated about the line where it intersects the yz-plane by an angle \(m\). In its new position, the plane contains the point \((-1, 0, 2)\). Find \(\cos m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a stationary and asymptotic Markov chain $\{X_t; t = 0,1,...\}$ with state space $S_x = \{1,2,3\}$ and transition matrix $P = \begin{bmatrix}0.2&0.5&0.3\\0.4&0.3&0.3\\1&0&0\end{bmatrix}$, and initial distribution $\overline{p} = \begin{bmatrix}1/3\\1/3\\1/3\end{bmatrix}$, compute $P(X_3 = 3, X_2 = 1 | X_1 = 2, X_0 = 2)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $p$ and $q$ for which the inequality \((2^{q-1}x^q+2^{q-1}y^q)^{1/q} \leq (2^{p-1}x^p+2^{p-1}y^p)^{1/p}\) holds for all positive real numbers $x$ and $y$. Provide your answer in terms of $p$ and $q$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the recurrence relations:

\[ a_{2n} = a_{n} - 1 \]
\[ a_{2n+1} = 2a_{n} + 1 \]

and the condition \( a_{20} = 1 \), find the value of \( \sum_{n=1}^{63} a_n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of subgroups isomorphic to \( \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} \) in the group \( G \) isomorphic to \( \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/9\mathbb{Z} \times \mathbb{Z}/81\mathbb{Z} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two spheres are inscribed in a dihedral angle such that they touch each other. The radius of one sphere is 1.5 times the radius of the other, and the line connecting the centers of the spheres forms an angle of $30^{\circ}$ with the edge of the dihedral angle. Determine the cosine of the dihedral angle, rounding to two decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral sum given that \( a^5 - b^5 = 1 \):
\[ \int_1^a \sqrt[5]{x^5-1}\,dx + \int_0^b \sqrt[5]{x^5+1}\,dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the expansion of \((x^3 + x^2 + x)^{36}\), how many coefficients of the resulting polynomial are not divisible by 2?
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Find all values of $x$ such that
\[3^x + 4^x + 5^x = 6^x.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the definite integral \( \int_{1}^{2} \int_{1}^{2} \int_{1}^{2} \int_{1}^{2} \frac{x_{1}+x_{2}+x_{3}-x_{4}}{x_{1}+x_{2}+x_{3}+x_{4}} \, d x_{1} \, d x_{2} \, d x_{3} \, d x_{4} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of conjugacy classes of subgroups of the symmetric group $S_5$ that are isomorphic to the non-cyclic group of order 4.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit using the Taylor series approach: $$\lim_{x\to 0^+} \, \frac{-\tan ^{-1}\left(2 x^3\right)+\frac{(x+2) x^5}{x^2+3}+2 x \sin ^2(x)}{x \exp \left(-\frac{1}{x}\right)-2 x^2+\log \left(\frac{x^2+1}{1-x^2}\right)}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \(18! \mod 437\) given that \(437 = 19 \cdot 23\), \(18!\equiv -1 \pmod {19}\), and \(22!\equiv -1 \pmod {23}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum value of $k$ for which the sum \( \sum_{i=0}^k \binom {10}{i} \binom{15}{k-i} +\sum_{i=0}^{k+1} \binom {12}{i} \binom {13}{k+1-i} \) exists.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of solutions for the equation \( \cot^{-1}{\sqrt{4-x^2} + \cos^{-1}{(x^2-5)}} = \frac{3\pi}{2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum value of \( \alpha \) such that for any continuously differentiable function \( f: [0,1] \rightarrow \mathbb{R} \), the following inequality holds:
\[
\left|\frac{f(0)+f(1)}{2}-\int_{0}^{1} f(x) \, dx\right| \leq \alpha \max_{x \in [0,1]}\left|f^{\prime}(x)\right|.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the equation of the tangent line to the curve \((x^2 + y^2 - 3)^3 = (xy)^3 - x^2 - y^2\) at the point \((1, 1)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of magnitude of the number of involutions in the symmetric group $S_{256}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
(1901 + 1902 + 1903 + \cdots + 1993) - (101 + 102 + 103 + \cdots + 193) =
Please reason step by step, and put your final answer within \boxed{}.


Context:
Provide an example of a Young function \( \Phi: \mathbb{R} \to \mathbb{R}^+ \) that is convex, satisfies \( \Phi(0) = 0 \), \( \Phi(-x) = \Phi(x) \), and \( \lim_{x \to \infty} \Phi(x) = +\infty \), but does not satisfy the \( \Delta_2 \) condition, which is \( \Phi(2x) \leq K\Phi(x) \) for some \( x \geq x_0 \geq 0 \) and a constant \( K > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( A \) as \( n \) approaches infinity:

\[ A = \lim_{n \to \infty} \left(\binom{n}{0} \binom{n}{1} \binom{n}{2} \cdots \binom{n}{n}\right)^{\frac{1}{n(n+1)}} \]

Given that the product can be rewritten as \( \prod_{k=0}^{k=n} \prod_{r=1}^{r=k} \frac{n-k+r}{r} \), find the value of \( A \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient $c_{-2}$ of the Laurent series expansion of the function $g(z)$ in the region $\Omega=\{|z|>2\}$, where $g(z)$ is a holomorphic branch of the square root of $f(z)=(z-1)(z-2)$. Express your answer in terms of known constants or functions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the power set of \( \{0, 1, 2, 3\} \), and let \( C = \{ \{0\}, \{1, 2\}, \{1, 3\}, \{2, 3\} \} \). Calculate the intersection number of \( C \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct values the function \( f(z) = \sqrt{\frac{z^2-1}{\sqrt{z}}}(z-i)^{1/3} \) can take for a given value of \( z \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all real pairs \((p, q)\) such that the inequality \(|x^2 + px + q| > 2\) has no solutions in the interval \([1, 5]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $z$ be a non-real complex number with $z^{23}=1$. Compute\[\sum_{k=0}^{22}\dfrac{1}{1+z^k+z^{2k}}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral \( \int_{0}^{1} \sin(x^2) \, dx \) such that the error is less than or equal to \( 10^{-3} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two normally distributed random variables, X and Y, where X has a mean of 50 and a standard deviation of 10, and Y has a mean of 60 and a standard deviation of 12, with a correlation coefficient of -0.6, calculate the probability that \(2X < 0.5Y\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Points $A$, $B$, and $C$ lie on a line, in that order, with $AB = 8$ and $BC = 2.B$ is rotated $20^\circ$ counter-clockwise about $A$ to a point $B'$, tracing out an arc $R_1$. $C$ is then rotated $20^\circ$ clockwise about $A$ to a point $C'$, tracing out an arc $R_2$. What is the area of the region bounded by arc $R_1$, segment $B'C$, arc $R_2$, and segment $C'B$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the domain of convergence for the series \( \sum_{n=1}^{\infty} \frac{x^n}{n^x} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x, y, z$ be the sides of a triangle such that $x \neq z$. Determine the range of values for $p$ such that the equation \( t^2 + 2(x + y + z)t + 3p(xy + yz + zx) = 0 \) has real roots, given that $x, y, z$ satisfy the triangle inequality.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum step size for the forward Euler method such that the solution of the system of ODEs \( a' = -a - 2b \) and \( b' = 2a - b \) with initial conditions \( a(0)=1 \) and \( b(0)=-1 \) goes to zero. Use the eigenvalues of the matrix \( \begin{pmatrix} -1-h & -2-2h \\ 2+2h & -1-h \end{pmatrix} \) to find the solution stability.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $(a^2 + b^2) : c^2 = 2019 : 1$, find the ratio $(\cot A + \cot B + \cot C) : (\cot A + \cot B)$. Provide your answer as a simplified ratio.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_a^b \delta(t-a)e^{-iwt}dt \) and determine its result in terms of the sifting property of the Dirac delta function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of intersection points between a circle of radius $R$ centered at the origin in 3D space and a Cassini oval defined by two fixed points $(x_{1},y_{1},z_{1})$ and $(x_{2},y_{2},z_{2})$ such that the product of the distances from any point on the oval to these two points equals a constant $C$. How many points can this circle and Cassini oval have in common?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ n\in N^*$. A permutation $ (a_1,a_2,...,a_n)$ of the numbers $ (1,2,...,n)$ is called quadratic iff at least one of the numbers $ a_1,a_1+a_2,...,a_1+a_2+a+...+a_n$ is a perfect square. Find the greatest natural number $ n\leq 2003$, such that every permutation of $ (1,2,...,n)$ is quadratic.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the value of the improper integral: $$\int_{0}^{\infty }\frac{(\ln x)^{40021}}{x}dx$$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a queuing system with two facilities, A and B, each facility has a single server with service times that are negative exponential random variables. The mean service times are 1 minute for facility A and 4 minutes for facility B. A total of 60 customers per hour arrive, with arrivals following a Poisson process. Each customer independently chooses a facility without knowing the queue status, with probability \( p \) for facility A and \( 1-p \) for facility B. Determine the value of \( p \) that minimizes the expected time a customer spends in the system, assuming the system reaches a steady state.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the number of spanning trees in a complete graph with 10 vertices, given that vertex 2 has a degree of 5.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Big-O notation for the expression \( \prod_{i=2}^n \log(i) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of parameter-free definable subsets of the structure $(\mathbb{R}, *)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the quadratic variation \([X,X]_t\) of the zero-mean Ornstein-Uhlenbeck process \((X_t)_{t\geq 0}\) given by the representation:
\[ X_t = \sigma e^{-\alpha t}\int_0^t e^{\alpha s}\,dB_s \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( \{X_i\}_{i=0}^{\infty} \) be a sequence of IID random variables, which can be either discrete or continuous. Determine the expected value of \( \alpha \), where \( \alpha \) is the smallest index \( i \) such that \( X_i > X_0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a vector space $V$ of $3 \times 3$ matrices and a matrix $A \in V$ with $\det(A) = 32$ and minimal polynomial $(x-4)(x-2)$, find the trace of the linear map $L_A: V \to V$ defined by $L_A(B) = AB$ for $B \in V$. What is the trace of $L_A$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an equilateral triangle ABC, point D is inside the triangle such that \(DB^2 + DC^2 + BC^2 = 120\). If the area of triangle DBC is \(6\sqrt{3}\), find the length of side BC.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 10 balls numbered 1 to 10. Two students each have two identical bags. In how many ways can the students distribute the balls into their bags such that no bag is empty?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all the prime number $p, q$ and r with $p < q < r$, such that $25pq + r = 2004$ and $pqr + 1 $ is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an ideal \( I \) of \( \mathbb{Z}[\sqrt{3}] \) generated by an integer prime such that \( \mathbb{Z}[\sqrt{3}]/I \) is not an integral domain.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of subgroups of the group \( \Bbb Z_{10} \oplus \Bbb Z_{15} \) using Goursat's Theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the inequality
\[(a_1+a_2+\dots+a_n)^2\ge 4(a_1a_2+a_2a_3+\cdots+a_na_1).\]
 Find all $n\ge 3$ such that the inequality is true for reals.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the larger base of an isosceles trapezoid equals a diagonal and the smaller base equals the altitude, then the ratio of the smaller base to the larger base to the larger base is ：
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an equilateral triangle $ABC$, the distances from a fixed point $P$ on the plane to vertices $A$ and $B$ are $AP=2$ and $BP=3$. What is the maximum possible length of the line segment $PC$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the complexity of the function $f(n)$ given by the recurrence relation $f(0) = 3$, $f(1) = 3$, and $f(n) = f(\lfloor n/2 \rfloor) + f(\lfloor n/4 \rfloor) + cn$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the differential equation \( \frac{dP}{dt} = 0.8P \left(1 - \frac{P}{350}\right) \), where \( P \) is the number of people infected per week after the outbreak of a disease, and knowing that at the outbreak, 7 people were infected, determine the initial condition for \( P \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a cube ABCD.EFGH with a side length of 12 cm, let R be the intersection of lines AD and CE. Let S be the intersection of segment ER with plane ACGE. Find the length of segment ES.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $(X,Y)\sim N(\mu_x=1,\mu_y=1,\sigma^2_x=4,\sigma^2_y=1,\rho=1/2)$, compute the probability $P(X+2Y\leq 4)$. Provide your answer to four decimal places.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $O$ be the center of the concentric circles $C_1,C_2$ of radii $3$ and $5$ respectively. Let $A\in C_1, B\in C_2$ and $C$ point so that triangle $ABC$ is equilateral. Find the maximum length of $ [OC] $.
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Given two independent random samples $X_1, \ldots, X_{100}$ and $Y_1, \ldots, Y_{50}$ from a normal distribution $N(\mu, 1)$, calculate the probability that the 80% confidence intervals for $\mu$ constructed from these samples are disjoint.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim\limits_{n\to\infty} \frac{\sum\limits_{k=1}^{n} \frac{n^{k-1}}{(k-1)!}}{e^n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of 2-Sylow subgroups of the group $S_4 \times S_4$. Assume that any such subgroup is isomorphic to $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the integral: $$ \lim_{n\to\infty}\int_A\frac{\ln(n+y^3)-\ln n}{\sin(x/n)}d\lambda_2 $$ where $A=\{(x,y)\in\mathbb{R}^2_+:1<xy<4,1<y/x<4\}$. Determine the value of this limit.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral \( I = \int_{C} ze^{\frac{1}{z-1}} \, dz \) where \( C \) is the circle centered at 1 with radius \( \frac{1}{2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given distinct proper rigid transformations \(A, B \in \operatorname{SE}(n)\), determine the maximum dimension of the nullspace of \(A - B\). Specifically, find the maximum dimension of \(\operatorname{Eq}(A, B) = \{x \in \mathbb{R}^n\ |\ A(x) = B(x)\}\) for all integers \(n \geq 2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X$ and $Y$ be independent random variables, each uniformly distributed on $[0,1]$. Define $Z = \max(X, Y)$. Calculate the expected value of $X$ given $Z$, expressed as $\mathbb{E}[X|Z]$ in terms of $Z$. Express your answer as a function of $Z$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For large values of $x$, approximate the expression $\ln \frac{1+e^{ax}}{1+e^{-bx}}$. Given that the leading term is $ax$, determine the value of $c$ in the approximation $ax + c + \frac{c_1}{x} + \dots$. What is the value of $c$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A plane contains points \( A \) and \( B \) with \( {AB} = 1 \) . Let \( S \) be the union of all
 disks of radius 1 in the plane that cover \( \overline{AB} \) . What is the area of \( S \) ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let a sequence be defined as follows: $a_{1}= 3$, $a_{2}= 3$, and for $n \ge 2$, $a_{n+1}a_{n-1}= a_{n}^{2}+2007$. Find the largest integer less than or equal to $\frac{a_{2007}^{2}+a_{2006}^{2}}{a_{2007}a_{2006}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the angle between the vector \( v = (2,2,1,1) \) and the linear subspace \( S \) spanned by the vectors \( u_1 = (3,4,-4,-1) \) and \( u_2 = (0,1,-1,2) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the constant \( d \) to which the sequence \( \frac{1}{n} \sum_{i=1}^n X_i \) converges almost surely, given that \( \{X_i, i \geq 1\} \) is a sequence of independent random variables with \( \Bbb P(X_i=i^3 -1)=i^{-3} \) and \( \Bbb P(X_i=-1)=1-i^{-3} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the units digits in the decimal expansion of
$$\left(2008 + \sqrt{4032000}\right)^{2000} + \left(2008 + \sqrt{4032000}\right)^{2001} + \left(2008 + \sqrt{4032000}\right)^{2002} + \cdots + \left(2008 + \sqrt{4032000}\right)^{2007} + \left(2008 + \sqrt{4032000}\right)^{2008}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \(a > 0\) for which the sequence \(Y_n = \prod_{j=1}^n X_j\), where \(X_n\) are independent random variables with \(X_n \sim \text{Bern}(1-\frac{1}{n^a})\), converges in probability to zero.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(f\) be a function on the nonnegative integers defined by the following properties:
\[ f(2n) = f(f(n)) \]
and
\[ f(2n + 1) = f(2n) + 1 \]
Determine all nonnegative integers \(k\) such that \(f(0) = 2^k\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equation \(x^3 = (3+4i)\overline{x}\), where \(x\) is a complex number, find the sum of the moduli of its roots.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient of \(x^{62}\) in the expansion of \((x^3 - \frac{1}{x^5})^{50}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of mappings \( \phi: \mathbb{N} \cup \{0\} \to \mathbb{N} \cup \{0\} \) such that \( \phi(ab) = \phi(a) + \phi(b) \) for all non-negative integers \( a \) and \( b \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the group obtained by imposing the relations $a^2 = e$ and $b^3 = e$ on the amalgamated free product $G = \mathbb{Z}_4 \ast_{\mathbb{Z}_2} \mathbb{Z}_6$, where $a$ and $b$ are the generators of $\mathbb{Z}_4$ and $\mathbb{Z}_6$ respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A, B, C$ be three collinear points with $B$ between $A$ and $C$. Consider all circles passing through $B$ and $C$. Determine the type of curve on which the points of contact of the tangents from $A$ to these circles lie.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the computational complexity of deciding whether $a^b = c$, given that $a, c \leq 2^p$ and $b \leq 2^q$, without explicitly computing $a^b$, when the size of $q$ is not restricted.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In parallelogram \(ABCD\), the angle between diagonals \(AC\) and \(BD\) is \(30^\circ\). The ratio of \(AC\) to \(BD\) is \(2 : \sqrt{3}\). Let \(B_1\) be the reflection of vertex \(B\) across line \(AC\), and \(C_1\) be the reflection of vertex \(C\) across line \(BD\). Determine the ratio of the areas of triangle \(AB_1C_1\) to parallelogram \(ABCD\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\to\infty}\left(2n\int_{0}^{1}\frac{x^n}{1+x^2}dx\right)^n \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two angles of an isosceles triangle measure $80^o$ and $x^o$. What is the sum of all the possible values of $x$?
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Find all polynomials P(x) satisfying the equation
$(2x-1)P(x) = (x-1)P(2x), \forall x.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( C \) be the disk of radius 1 centered at the origin in \( \mathbb{R}^2 \). Consider the linear transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) given by:
\[ T(x, y) = \left( \frac{5x + 3y}{4}, \frac{3x + 5y}{4} \right) \]
What is the smallest natural number \( n \) such that \( T^n(C) \) contains at least 2019 points \( (a, b) \) where \( a, b \in \mathbb{Z} \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{x \to 0} \frac{x\sin(\sin x) - \sin^2 x}{x^6} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
For what values of \( V \) does the function \( y(t) = -(V+2)e^{-3t} + (V+3)e^{-2t} \) remain nonnegative for all \( t \geq 0 \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a chess tournament with 12 participants, each participant creates 12 lists after the tournament. The first list includes only themselves, the second list includes themselves and those they defeated, the third list includes everyone from the second list and those they defeated, and so on. The 12th list includes everyone from the 11th list and those they defeated. It is known that for any participant, there is a person in their 12th list who was not in their 11th list. Determine the number of drawn games played in the tournament.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of continuous functions \( f: [0,1] \rightarrow \mathbb{R} \) that satisfy the equation \[ \int_{0}^{1} f(x) \, dx = \frac{1}{3} + \int_{0}^{1} f^2(x^2) \, dx. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ and $B$ be $3 \times 3$ matrices such that $A^3 + 5AB = I$ and $A^3 - 5BA = 2I$. Find $\det(A)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine if there exists a function $f$ such that for all $a, b, c, d$, the following equations hold: \[ f(a)f(b)-f(c)f(d) = (f(a)-f(c))(f(b)-f(d)) \] and \[ f(a)f(b)+f(c)f(d) = (f(a)+f(c))(f(b)+f(d)). \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the recurrence relation \( A(n) = 2A(n/2) + O(1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of elements that the subset \(X\) of the set \(\{00, 01, \ldots, 98, 99\}\) must contain so that in any infinite sequence of numbers, there are always two adjacent numbers that form an element of \(X\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimal polynomial of \( \overline{x+1} \) over the field \( \mathbb{F}_2 \) in the field \( K = \mathbb{F}_2[x] / (x^3+x+1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( A = (a_{ij}) \) be the \( n \times n \) matrix, where \( a_{ij} \) is the remainder of the division of \( i^j + j^i \) by \( 3 \) for \( i, j = 1, 2, \ldots, n \). Find the greatest \( n \) for which \( \det A \neq 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{x\rightarrow 0} \frac{\sin x^2}{\ln(\cos x^2 \cos x + \sin x^2 \sin x)} \) without using L'Hôpital's rule, by applying trigonometric identities and known limits.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cylindrical fuel tank with hemispherical ends must hold 300 liters of jet fuel. The side wall of the cylinder costs $0.1 per cm² to manufacture, while the hemispherical ends cost $0.3 per cm². Determine the optimal radius of the tank that minimizes the total manufacturing cost.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of equivalence relations on a set with 7 elements such that the equivalence relation contains exactly 29 ordered pairs.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\Delta ABC$, $\angle ABC = 90^\circ$ and $BA = BC = \sqrt{2}$. Points $P_1, P_2, \dots, P_{2024}$ lie on hypotenuse $\overline{AC}$ so that $AP_1= P_1P_2 = P_2P_3 = \dots = P_{2023}P_{2024} = P_{2024}C$. What is the length of the vector sum
\[ \overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \dots + \overrightarrow{BP_{2024}}? \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A total fixed amount of $N$ thousand rupees is given to $A,B,C$ every year, each being given an amount proportional to her age. In the first year, A got half the total amount. When the sixth payment was made, A got six-seventh of the amount that she had in the first year; B got 1000 Rs less than that she had in the first year, and C got twice of that she had in the first year. Find N.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a function $g(x)$ such that $\lim_{x\to\infty}\frac{g(x^2)}{g(x)}=1$ and $\lim_{x\to\infty}g(x)=\infty$. Provide your answer as a function expression.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of equally spaced subintervals, N, required such that the difference between the Upper and Lower Riemann sums for the integral \( \int_{1}^{17} \frac{1}{x^3} \, dx \) is less than \( \frac{1}{100} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $T_0$ is the minimum variance unbiased estimator (MVUE) for $\gamma(\theta)$ and $T_1$ is another unbiased estimator for $\gamma(\theta)$ with an efficiency of 0.0169, find the correlation coefficient between $T_0$ and $T_1$. \( \text{corr}(T_0, T_1) = ? \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum possible price difference between two American-style call options on a stock with different strike prices, assuming no dividends are paid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the matrix \( A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix} \), where \( A = B + C \), \( B \) is symmetric \( (B = B^T) \), and \( C \) is skew-symmetric \( (C = -C^T) \). If \( (CB)^T = D \) and \( D = [ d_{ij} ]_{3 \times 3} \), find the value of \( d_{12} + d_{21} + d_{13} + d_{31} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many integers $n$ with $10 \le n \le 500$ have the property that the hundreds digit of $17n$ and $17n+17$ are different?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( P \) and \( Q \) be two points on the curve \( y = \log_{\frac{1}{2}}\left(x - \frac{1}{2}\right) + \log_2\left(\sqrt{4x^2 - 4x + 1}\right) \). Point \( P \) is also on the circle \( x^2 + y^2 = 10 \). Point \( Q \) lies inside the circle such that its abscissa is an integer. Find the smallest possible value of \( \overrightarrow{OP} \cdot \overrightarrow{OQ} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two positive number $a$, $b$ such that $a<b$. Let A.M. be their arithmetic mean and let G.M. be their positive geometric mean. Then A.M. minus G.M. is always less than:
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two positive integers $a$ and $b$, where $a$ has 20,041 '1's and $b$ has 20,051 '1's in their binary representations, determine the smallest possible number of '1's in the binary representation of $a + b$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the cubic equation \(x^3 + 2x^2 + 3x + 1 = 0\) with roots \(a, b, c\), it is known that \(\frac{1}{a^3} + \frac{1}{b^3} - \frac{1}{c^3}\), \(\frac{1}{a^3} + \frac{1}{c^3} - \frac{1}{b^3}\), and \(\frac{1}{c^3} + \frac{1}{b^3} - \frac{1}{a^3}\) are the roots of another cubic equation \(px^3 + qx^2 + rx + s = 0\). Find the value of \(p + q - 12r + s\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Géza starts with the sequence \((1, 2, \dotsc, 9)\) and extends it 2017 times. After the final extension, he randomly selects one element from the resulting sequence. What is the probability that the chosen element is 1?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a Markov chain with state space $S=\{0, 1, 2, \ldots \}$ and transition probabilities $p(0,0)=3/4$, $p(0,1)=1/4$, $p(x, x-1)=3/4$, and $p(x,x+1)=1/4$ for $x \geq 1$. Let $e(x)$ denote the expected number of steps to reach the origin starting from state $x$. Find $e(1)$. (Hint: First consider the expected return time starting at the origin and express $e(1)$ in terms of this.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the number of connected graphs that can be formed with 6 labeled vertices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the infinite series \(S\) represented by \(2 - 1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} - \frac{1}{81} + \frac{1}{243} - \cdots\). Find the sum \(S\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For the four-digit number \(\overline{abcd}\) where \(1 \leqslant a \leqslant 9\) and \(0 \leqslant b, c, d \leqslant 9\), if \(a > b, b < c, c > d\), then \(\overline{abcd}\) is called a \(P\)-type number. If \(a < b, b > c, c < d\), then \(\overline{abcd}\) is called a \(Q\)-type number. Let \(N(P)\) and \(N(Q)\) represent the number of \(P\)-type and \(Q\)-type numbers respectively. Find the value of \(N(P) - N(Q)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The library initially has 250 books. Over the course of three weeks, the library experiences the following transactions:

Week 1:
- Tuesday: 120 books are taken out to be read by children.
- Wednesday: 35 books are returned.
- Thursday: 15 books are withdrawn from the library.
- Friday: 42 books are taken out by adults.

Week 2:
- Monday: 60% of the books taken out by children last week are returned.
- Tuesday: 80% of the books taken out by adults last week are returned.
- Wednesday: 75 books are withdrawn from the library.
- Thursday: 40 books are taken out by teenagers.
- Friday: An inventory is conducted.

Week 3:
- Monday: 50% of the books taken out by teenagers last week are returned.
- Tuesday: 20 books are taken out by senior citizens.
- Wednesday: 95% of the books taken out by children in week 1 that had not yet returned are returned.
- Thursday: 10 more books are withdrawn from the library.
- Friday: 55 books are taken out by a local book club.

Taking into account all these transactions, how many books are in the library by the end of the third week?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABC$ be a triangle with $AB=7$, $BC=9$, and $CA=4$. Let $D$ be the point such that $AB \parallel CD$ and $CA \parallel BD$. Let $R$ be a point within triangle $BCD$. Lines $\ell$ and $m$ going through $R$ are parallel to $CA$ and $AB$ respectively. Line $\ell$ meets $AB$ and $BC$ at $P$ and $P^\prime$ respectively, and $m$ meets $CA$ and $BC$ at $Q$ and $Q^\prime$ respectively. If $S$ denotes the largest possible sum of the areas of triangle $BPP^\prime$, $RP^\prime Q^\prime$, and $CQQ^\prime$, determine the value of $S^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the line integral

$$
\int_{L} \operatorname{arctg} \frac{y}{x} d l
$$

where \( L \) is part of the Archimedean spiral \( \varrho = \varphi \) (for \( 0 \leq \varphi \leq \pi / 2 \)).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If for any $x\in R$, $2x+2\leqslant ax^{2}+bx+c\leqslant 2x^{2}-2x+4$ always holds, then the maximum value of $ab$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways are there for Nick to travel from \((0,0)\) to \((16,16)\) in the coordinate plane by moving one unit in the positive \(x\) or \(y\) direction at a time, such that Nick changes direction an odd number of times?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence for the series \( \sum_{n=0}^{\infty} a_n (x-3)^n \), where the series is given by \( 1 - x^2 + x^4 - x^6 + \cdots \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Provide an example of a function $f(x)$ whose derivative $\frac{df}{dx}$ is periodic, but the function $f(x)$ itself is not periodic.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p$ be a prime number and $\mathbf{W}\subseteq \mathbb{F}_p[x]$ be the smallest set satisfying the following :
(a)
$x+1\in \mathbf{W}$ and $x^{p-2}+x^{p-3}+\cdots +x^2+2x+1\in \mathbf{W}$
(b) For $\gamma_1,\gamma_2$ in $\mathbf{W}$, we also have $\gamma(x)\in \mathbf{W}$, where $\gamma(x)$ is the remainder $(\gamma_1\circ \gamma_2)(x)\pmod {x^p-x}$.
How many polynomials are in $\mathbf{W}?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the conditional Shannon entropy $H[R_{513}, C_{513} | R_1, R_2, \ldots, R_{512}]$ for a scenario where 1024 rooks are randomly and uniquely placed on a 1024 × 1024 chessboard without any rook attacking another. Here, $R_i$ represents the row number and $C_i$ the column number of the i-th rook.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the derivative of the expression \( \operatorname{trace}(XWW^T X^T) \) with respect to the matrix \( W \), where \( X \) and \( W \) are \( n \times n \) real matrices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the area of the quadrilateral formed by the intersection of two circles with radii \(a\) and \(b\), intersecting at an angle \(\theta\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x\to0^-} (x^{-x^x+1})$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit: $$\lim_{x \to 0} \frac{e^{\sin(x)} - e^{\tan (x)}}{e^{\sin (2x)}-e^{\tan (2x)}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Galois group of the splitting field of the polynomial $t^4 - 3t^2 + 4$ over $\mathbb{Q}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the frequency distribution histogram of a sample, there are a total of $m(m\geqslant 3)$ rectangles, and the sum of the areas of the first $3$ groups of rectangles is equal to $\frac{1}{4}$ of the sum of the areas of the remaining $m-3$ rectangles. The sample size is $120$. If the areas of the first $3$ groups of rectangles, $S_1, S_2, S_3$, form an arithmetic sequence and $S_1=\frac{1}{20}$, then the frequency of the third group is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Eleven points are marked from left to right on a straight line. The sum of all the distances between the first point and the other points is 2018 cm. The sum of all the distances between the second point and the other points, including the first one, is 2000 cm. Find the distance between the first and second points. Express your answer in centimeters (cm) as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cofinality of the ordinal $\omega+1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the convergence and determine the value, if it exists, of the integral \( \int_{0}^{+\infty }\frac{x \log(x)}{(x^{2}+1)^{2}}dx. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a complex circular zero-mean Gaussian random process (or vector) \( \textbf{x} \) where \( \textbf{x} \sim \mathcal{CN}(0,\sigma^2) \), find the expectation \( \mathbb{E}\{\textbf{x}^n\} \) for \( n \in \mathbb{N}^{*+} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \( \alpha \in \mathbb{R} \) such that the integral \( \int_{\mathbb{S}^{n}} \frac{d\theta}{|\theta-\theta_{0}|^{\alpha}} \) converges, where \( \theta_{0} \in \mathbb{S}^{n} \) is fixed.
Please reason step by step, and put your final answer within \boxed{}.