inclusionAI/Ring-lite-rl-data huggingface dataset:
8. Suppose $A=\{1,2, \ldots, 20\}$. Call $B$ a visionary set of $A$ if $B \subseteq A, B$ contains at least one even integer, and $|B| \in B$, where $|B|$ is the cardinality of set $B$. How many visionary sets does $A$ have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{19}-256}
已知椭圆 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ 的右焦点为 $F(c, 0)$, 存在经过点 $F$的一条直线 $l$ 交粗圆于 $A, B$ 两点, 使得 $O A \perp O B$, 求该椭圆的离心率的取值范围。该取值范围两端点的和形如\frac{a\sqrt{b}+c}{d}。请求出a+b+c+d的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
What is the asymptotic formula for the geodesic distance between equidistant points on a unit sphere as a function of the number of points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4/\sqrt{n}}
Given vectors $\overrightarrow{O A} \perp \overrightarrow{O B}$, and $|\overrightarrow{O A}|=|\overrightarrow{O B}|=24$. Find the minimum value of $|t \overrightarrow{A B}-\overrightarrow{A O}|+\left|\frac{5}{12} \overrightarrow{B O}-(1-t) \overrightarrow{B A}\right|$ for $t \in[0,1]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Considering all numbers of the form $n = \lfloor \frac{k^3}{2012} \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $k$ ranges from $1$ to $2012$, how many of these $n$’s are distinct?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1995}
Let \(V\) be a closed subspace of \(L^2[0,1]\) and let \(f,g \in L^2[0,1]\) be given by \(f(x)=x\) and \(g(x)=x^2\). If \(V^\perp = \operatorname{Span} f\) and \(Pg\) is the orthogonal projection of \(g\) on \(V\), find the expression for \((g-Pg)(x)\) for \(x \in [0,1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}x}
Find the rightmost digit of \( \left \lfloor \frac{10^{20000}}{10^{100}+3} \right\rfloor \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A circle touches the extensions of two sides \( AB \) and \( AD \) of a square \( ABCD \) with a side length of 4 cm. From point \( C \), two tangents are drawn to this circle. If the angle between the tangents is \( 60^{\circ} \), find the radius of the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4(\sqrt{2} + 1)}
How many distinct Jordan canonical forms are possible for a $4 \times 4$ complex matrix with the minimal polynomial $(x-3)^2(x-4)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that \(a, b, c > 0\) and \(a^2 + b^2 + c^2 = 4\), and \(4(a^2 + 2) = (a^2 + b + c)^2\), find the maximum possible value of \(a + b + c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 + \sqrt{2}}
For $k=1,2,\\ldots ,2011$ we denote $S_k=\frac{1}{k}+\frac{1}{k+1}+\cdots +\frac{1}{2011}$. \nCompute the sum $S_1+S_1^2+S_2^2+\cdots +S_{2011}^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4022
}
Given the values of the differentiable real functions \( f, g, h \), and the derivatives of their pairwise products at \( x=0 \):
\[ 
f(0) = 1, \quad g(0) = 2, \quad h(0) = 3, \quad (gh)^{\prime}(0) = 4, \quad (hf)^{\prime}(0) = 5, \quad (fg)^{\prime}(0) = 6.
\]
Find the value of \( (fgh)^{\prime}(0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Evaluate the integral \( \int_0^{2 \pi } \frac{e^{e^{-i \theta }}}{e^{4 i \theta }} \, d\theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \( A \) and \( B \) be subsets of a set \( M \) and define \( S_0 := \{A,B\} \). For \( i \ge 0 \), define \( S_{i+1} \) inductively as the collection of subsets of \( M \) taking the form \( C \cup D \), \( C \cap D \), or \( M - C \) where \( C,D \in S_i \). Determine the largest possible number of elements in \( S = \bigcup_{i=0}^{\infty} S_i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Determine the number of distinct ways to color a 3x4 rectangle using two colors, such that each color appears exactly 6 times. Two colorings are considered equivalent if one can be obtained from the other by permuting the rows or by cyclically permuting the columns.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Consider a party with ten people in a two-room apartment. Initially, it is equally likely that all ten people are in room A or that nine are in room A and one is in room B. Every minute, one person is chosen uniformly at random to move to the other room. What is the probability that the group is evenly split between rooms A and B (5 people in each) as the number of minutes goes to infinity?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{63}{256}}
Determine the degree of the field extension \(\mathbb{Q}(5^{1/3}, 5^{1/9})\) over \(\mathbb{Q}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the derivative of the complex conjugate function with respect to the complex variable: \( \frac{\mathrm{d} z^*}{\mathrm{d} z} \), where \( z^* = x - iy \) and \( z = x + iy \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the maximum number of vectors in \(\mathbb{C}^d\) that can have the uniform overlap property, where each vector has unit norm and is defined up to a phase. The uniform overlap property requires that for all distinct vectors \(v_i\) and \(v_j\), the squared magnitude of their inner product is a constant \(C \neq 0\). What is this maximum number of vectors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d^2}
Determine the fundamental group of the space \(\mathbb{CP}^1\) minus a finite set of points \(\{a_1, \ldots, a_n\}\), where \(a_1, \ldots, a_n \in \mathbb{C}\). What is \(\pi_1(\mathbb{CP}^1 \setminus \{a_1, \ldots, a_n\})\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F_{n-1}}
Given that the Fourier transform of the function \( f(x) \) is \( F(k) \), find the Fourier transform of the function \( f'(x) = [f(-x)]^* \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[F(k)]^*}
Calculate the mass of the region of the cone defined by the equation \(36(x^{2}+y^{2})=z^{2}\), which is located inside the cylinder \(x^{2}+y^{2}=1\) with \(x \geqslant 0\) and \(z \geqslant 0\). The density of the region is given by \(\rho=\frac{5(x^{2}+y^{2})}{6}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Consider a random variable $X$ that follows a Gamma distribution with parameters $\alpha$ and $\beta = \alpha/\mu$. The probability density function of $X$ is given by:
\[ f_{X}(x) = \frac{(\alpha/\mu)^{\alpha}}{\Gamma(\alpha)} x^{\alpha-1} e^{-\alpha x/\mu}. \]
Determine the limit of the density function as $\alpha \to \infty$:
\[ \lim_{\alpha \to \infty} f_{X}(x). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\delta(x - \mu)}
In a first-price sealed-bid auction with two bidders, each having either a low valuation \(V_L\) or a high valuation \(V_H\), determine the minimum bid \(\underline{b_i}\) for a bidder with a low valuation \(V_L\). Assume valuations are unknown to the other bidder, ties are resolved by a coin flip, and bidders use mixed strategies over the domain \([\underline{b_i},\overline{b_i}]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit: $$ \lim_{n \to \infty} \left( \frac{\sin\left(\frac{1}{n}\right)}{n+1} + \frac{\sin\left(\frac{2}{n}\right)}{n+\frac{1}{2}} + \cdots + \frac{\sin\left(\frac{n}{n}\right)}{n+\frac{1}{n}} \right) $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \cos 1}
Evaluate the complex integral \( I = \oint_C f(z)\,dz \), where \( f(z) = \sqrt{z^2-1} \) is a holomorphic function for \( 1 < |z| < \infty \) and \( C \) is a circle of radius \( R \) centered at the origin in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi i}
Determine the limiting behavior of the sequence \( \prod_{p \leq n} \frac{p}{p-1} \) as \( n \) becomes very large, where \( p \) denotes prime numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the surface integral \( \iint_{X} y \, dS \) over the parametrized surface \( X(s,t) = (2s, s+t, 2t) \), where \( D \) is the triangular region with vertices at \((0,0)\), \((0,1)\), and \((1,1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{6}}
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{}.
Answer: \boxed{65520}
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{}.
Answer: \boxed{
9
}
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{}.
Answer: \boxed{-2\delta(x - y)}
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{}.
Answer: \boxed{k + 1}
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{}.
Answer: \boxed{\sqrt{n}}
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{}.
Answer: \boxed{F_{L+3} - 1}
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{}.
Answer: \boxed{1}
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{}.
Answer: \boxed{n^{1 - 1/q}}
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{}.
Answer: \boxed{\sqrt{e} - 1}
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{}.
Answer: \boxed{p > 1}
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{}.
Answer: \boxed{n}
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{}.
Answer: \boxed{8^{9^{8^{9}}}}
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{}.
Answer: \boxed{330}
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{}.
Answer: \boxed{-\frac{1}{2}}
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{}.
Answer: \boxed{x e^{a t}}
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{}.
Answer: \boxed{7}
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{}.
Answer: \boxed{\dfrac{5}{3}}
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{}.
Answer: \boxed{-\infty}
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{}.
Answer: \boxed{76}
Compute \( \operatorname{Ext}_{k[x] / \langle x^2 \rangle}(k,k) \) where \( k \) is a field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k}
What is the smallest non-trivial $E_8$-module?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{248}
Calculate the square root of $e$ to within an error of $10^{-4}$ using the Beuker-style integration by parts method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.6487}
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{}.
Answer: \boxed{S_p}
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{}.
Answer: \boxed{\dfrac{1}{b}}
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{}.
Answer: \boxed{4\pi i}
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{}.
Answer: \boxed{0.717}
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{}.
Answer: \boxed{15}
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{}.
Answer: \boxed{5\pi}
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{}.
Answer: \boxed{\dfrac{1}{2}}
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{}.
Answer: \boxed{670}
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{}.
Answer: \boxed{108}
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{}.
Answer: \boxed{11}
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{}.
Answer: \boxed{-\dfrac{7}{75}}
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{}.
Answer: \boxed{48}
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{}.
Answer: \boxed{orthogonal transformation}
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.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4775400}
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{}.
Answer: \boxed{y^2 = 8ax}
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{}.
Answer: \boxed{-\dfrac{2}{3}}
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{}.
Answer: \boxed{2\pi}
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{}.
Answer: \boxed{7}
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{}.
Answer: \boxed{\dfrac{1}{12}}
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{}.
Answer: \boxed{10^{20}}
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{}.
Answer: \boxed{
2003
}
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{}.
Answer: \boxed{5}
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{}.
Answer: \boxed{\emptyset}
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{}.
Answer: \boxed{\sqrt[3]{2}}
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{}.
Answer: \boxed{\pi + 2\pi n}
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{}.
Answer: \boxed{2\pi^2 I_0(a)}
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{}.
Answer: \boxed{\dfrac{7}{5}}
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{}.
Answer: \boxed{3}
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{}.
Answer: \boxed{1 + \sqrt{5}}