inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the values of $d$ for which the field $\mathbb{Q}(\sqrt{d})$ has an integral basis of the form $\alpha, \alpha'$ where $\alpha'$ is the conjugate of $\alpha$. Provide the condition on $d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d \equiv 1 \pmod{4}}
Compute the first Chern class, \( c_1(\mathcal{O}_C(1)) \), where \( C \) is the twisted cubic defined by the map \( \nu: \mathbb{P}^1_{\mathbb{C}} \to \mathbb{P}^3_{\mathbb{C}} \) given by \( \nu(s,t)=(s^3,s^2t,st^2,t^3) \). Assume \( c_1(\mathcal{O}_{\mathbb{P}^3}(1))=1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the probability \( P(T_1 < T_{-1} < T_2) \), where \( T_a = \inf\{t \geq 0 | B_t = a\} \), and \( B_t \) denotes a Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Let $x, y, z$ be nonzero real numbers such that $ x + y + z = 0$ and $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}= 1 -xyz + \frac{1}{xyz}.$$ Determine the value of the expression $$\frac{x}{(1-xy) (1-xz)}+\frac{y}{(1- yx) (1- yz)}+\frac{z}{(1- zx) (1-zy)}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3
}
Given the differential equation \( y^{\prime}=4\sqrt{y-x^2} \) and the initial condition \( y(1)=2 \), find the value of \( y(3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Determine the number of integer solutions for the equation $a^b b^c c^a = p$, where $p$ is a prime number, and exactly two of $a, b, c$ are negative.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the integral \( \int_{\Gamma} \mathrm{Log}z \, \mathrm{d}z \) where \( \Gamma(t) = e^{2i\pi t} \) for \( 0 \leq t \leq 1 \), considering that the principal argument of \( e^{2i\pi t} \) is \( 2\pi t \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
Determine the isomorphism class of the line bundle \(L\) over \(\mathbb{CP}^1\) with the transition function \(\psi_{12}([z_1, z_2]) = \frac{z_1/z_2}{|z_1/z_2|}\) on \(U_1\cap U_2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathcal{O}(1)}
Find the product of the 5th roots of \((1+\sqrt{2}i)^3.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5 + \sqrt{2}i}
In a checkered plane painted with 10 different colors, each cell is colored with one color, and all colors are present. Any two adjacent cells are colored in different colors. A pair of colors is considered "good" if there are two neighboring cells painted in these colors. What is the least number of good pairs possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Define \(\{ a_{n}\} \) as an arithmetic sequence where all terms are positive integers. If \(a_{10}-a_{1}=225\), find \[\lim _{n\rightarrow \infty }\dfrac {na^{2}_{n}+1}{\sum\limits^{n}_{k=1}\left( 1+2+3+\ldots +k\right) }\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3750}
Compute the sum of all 4-digit numbers that can be formed using the digits from the multiset \(\{1^2, 2^2, 3^2, 4^2\}\), where each digit can appear at most twice.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{566610}
Find the number of horizontal tangents to the function \( f(x) = xe^{-x^2-x^{-2}} \) for \( x \neq 0 \) and \( f(0) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the following limit of the integral: \[ \lim_{n \rightarrow \infty} \int_{0}^{\pi/4} \frac{\sin^2(\frac{nx}{2})}{n\,\sin^2(\frac{x}{2})} \,dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Find the smallest integer \( n_0 \) such that for all \( n > n_0 \), the inequality \( \sqrt{n^3+1} + \sqrt{n^3} > 10^3 \) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
Find the smallest variance that can be achieved by an unbiased estimator of the parameter \( \varphi \) given the probability density function: \[ f_{\varphi}(x)= \begin{cases} \varphi x^{\varphi - 1} &\text{if}\, 0 < x < 1, \varphi > 0\\ 0 &\text{otherwise} \end{cases} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\varphi^2}
Find the residue of the function \( \frac{\sin z}{z^2} \) at \( z = \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let \( {\left\{  {a}_{n}\right\}  }_{n = 0}^{\infty } \) be the sequence defined by the recurrence relation \( {a}_{n + 3} = 2{a}_{n + 2} - {23}{a}_{n + 1} + 3{a}_{n} \) for all \( n \geq  0 \) ,with initial
 conditions \( {a}_{0} = {20},{a}_{1} = 0 \) ,and \( {a}_{2} = {23} \) . Let \( {b}_{n} = {a}_{n}^{3} \) for all \( n \geq  0 \) . There exists a unique positive integer \( k \) and const
 \( {c}_{0},\ldots ,{c}_{k - 1} \) with \( {c}_{0} \neq  0 \) and \( {c}_{k - 1} \neq  0 \) such that for all sufficiently large \( n \) ,we have the recurrence relation
 \( {b}_{n + k} = \mathop{\sum }\limits_{{t = 0}}^{{k - 1}}{c}_{t}{b}_{n + t} \) . Find \( k + \sqrt{\left| {c}_{k - 1}\right| } + \sqrt{\left| {c}_{0}\right| } \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
262

}
Evaluate the limit: \[ \lim_{x\to0}(x\Gamma(x))^\frac{1}{x} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-\gamma}}
Two individuals, A and B, start at opposite poles of a sphere with radius \( r \). They move randomly along either a latitude or a longitude, but only one individual moves at a time, with a step length of 1 unit. What is the expected time for them to meet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi^2 r^2}
Given vectors $\overrightarrow{a}, \overrightarrow{b}$ satisfy $\overrightarrow{a}=(1, \sqrt {3}), |\overrightarrow{b}|=1, |\overrightarrow{a}+ \overrightarrow{b}|= \sqrt {3}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{2\\pi}{3}"]}
Determine the smallest odd integer \( k \) such that for every cubic polynomial \( f \) with integer coefficients, if there exist \( k \) integers \( n \) such that \(|f(n)|\) is a prime number, then \( f \) is irreducible in \( \mathbb{Z}[n] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let $a$, $b$, $c$ and $d$ be real numbers such that no two of them are equal, \n\n\\[\\frac{a}{b}+\\frac{b}{c}+\\frac{c}{d}+\\frac{d}{a}=4\\] and $ac=bd$. Find the maximum possible value of \n\n\\[\\frac{a}{c}+\\frac{b}{d}+\\frac{c}{a}+\\frac{d}{b}.\\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
-12
}
Let \( f: [0,1] \to \mathbb{R} \) be defined by \( f(x) = 0 \) if \( x \) is rational, and \( f(x) = 9^n \) if \( x \) is irrational, where \( n \) is the number of zeros immediately after the decimal point in the decimal representation of \( x \). Compute the Lebesgue integral \( \int_0^1 f(x) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the center $Z(G)$ of the group $G$ generated by the elements $\sigma$ and $\tau$, where $\sigma$ has order $5$, $\tau$ has order $4$, and $\tau\sigma\tau^{-1} = \sigma^2$. The group $G$ is of order $20$, and every element $g$ in $G$ can be uniquely expressed as $g = \sigma^k\tau^m$, where $0 \le k < 5$ and $0 \le m < 4$. Determine $Z(G)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{e\}}
Determine whether the function \(f(x) = (x^2-1) \log\left(\frac{1+x}{1-x}\right)\), defined on \((-1,1)\), can be continuously extended at \(x = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \( k \) be an integer greater than 1, and let \( a \) be a root of the equation \( x^{2} - kx + 1 = 0 \). For any natural number \( n \) greater than 10, the units digit of \( a^{2^{n}} + a^{-2^{n}} \) is always 7. Determine the units digit of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Evaluate the contour integral \( I = \int_0^\infty \frac{dx}{(x+1)^2(x+2)} \) given that there are two first-order poles at \( x = -1 \) and \( x = -2 \) on the real axis, and their residues cancel each other out.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \ln 2}
Find a polynomial $p(x)$ in $\mathbb{Z}_4[x]$ such that $p(x)$ is a unit and $\deg p(x) > 1$. Provide an example of such a polynomial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + 2x^2}
Determine the number of group homomorphisms from $A_5$ to $S_5$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
Calculate the mass of the tile in the first quadrant defined by the inequalities \(1 < xy < 3\) and \(1 < x^2 - y^2 < 3\) using the density function \(p(x, y) = (x^4 + y^4)(x^2 + y^2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Evaluate the degree of the field extension \([L:\mathbb{Q}]\), where \(L\) is the splitting field of the polynomial \(x^4 - 2\) over \(\mathbb{Q}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given the iteration formula \(x_{n+1}=x_n-(\cos x_n)(\sin x_n)+R\cos^2x_n\), where \(R\) is a positive constant, determine the derivative of the function \(f(x)\) that is being approximated using Newton's method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sec^2 x}
Determine the number of dimensions in which the statement "a projectivity that fixes any three points is the identity" is true.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the point on the surface defined by the equations \(x_1x_3-x_2x_4=1\) and \(x_2x_3+x_1x_4=0\) that is furthest from the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 0, 1, 0)}
In a contest with n questions, each question is worth a specific positive number of points, and no partial credit is awarded. By redistributing the question scores, any desired contestant ranking can be achieved. What is the maximum possible number of contestants, m?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Compute the intersection multiplicity of the curves $V(f)$ and $V(g)$ at the point $p = [1:0:-1]$, where $f = x^2 + y^2 - z^2$ and $g = (x-z)^2 - y^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given a real \(2\times2\) matrix \(A\) such that \(A^6 = I\), where \(I\) is the identity matrix, determine the number of distinct possibilities for the characteristic polynomial of \(A\). The characteristic polynomial must divide the annihilating polynomial \(x^6-1 = (x-1)(x^2+x+1)(x+1)(x^2-x+1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the largest integer $k$ such that $\Bbb{Z}^{k}$ can be embedded in $\mathcal{GL}(n,\Bbb{Z})$. What is this largest value of $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - 1}
Find the minimum value of \(|A \cap B \cap C|\) given that \(n(A) + n(B) + n(C) = n(A \cup B \cup C)\), \(|A| = |B| = 100\), and \(n(S) = 2^{|S|}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{97}
Find the sum of all values of $a$ such that there exist positive integers $a$ and $b$ satisfying the equation $(a - b)\sqrt{ab} = 2016$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{209}
Given the matrix \( A = \begin{bmatrix} 1 & -1 & -5 & 1 & 4\\ -1 & 2 & 8 & -3 & -4\\ 3 & -1 & -9 & 0 & 4 \\ 2 & 2 & 2 & -5 & -10\\ 0 & -3 & -9 & 5 & 13 \end{bmatrix} \), define the subspace \( W_2 \) as \( W_2 = \{ Y \in M_{5 \times 5} \mid YA = 0 \} \). If the rank of matrix \( A \) is 3, determine the dimension of \( W_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a country, there are 28 cities and between some cities there are two-way flights. In
 every city there is exactly one airport and this airport is either small or medium or big. F
 every route which contains more than two cities, doesn't contain a city twice and ends
 where it begins; has all types of airports. What is the maximum number of flights in this
 country?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
53
}
Let \( f(x,y) \) be defined as follows:
\[
f(x,y) =
\begin{cases}
\frac{xy(x^2-y^2)}{x^2+y^2}, & \text{if } (x,y) \ne (0,0) \\
0, & \text{if } (x,y) = (0,0)
\end{cases}
\]
Find the value of \( \frac{\partial^2 f}{\partial x \partial y} \) at the point \((0,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the dimension of $H^0(T,\mathcal{L})$, where $T$ is a complex torus of dimension $2$ and $\mathcal{L}$ is a line bundle that provides $T$ with a $(m,n)$-polarization.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{mn}
Let $N$ denote the number of ordered pairs of sets $(A, B)$ such that $A \cup B$ is a size-$999$ subset of $\{1,2,\dots,1997\}$ and $(A \cap B) \cap \{1,2\} = \{1\}$. If $m$ and $k$ are integers such that $3^m5^k$ divides $N$, compute the the largest possible value of $m+k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1005
}
Determine the upper boundary of the function \( f_n(x) = \frac{\sin nx}{\sin x} \) for \( n \in \mathbb{Z} \), where \( x \) is not a rational multiple of \( \pi \). Express your answer in terms of a function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|n|}
Consider the Cauchy problem for the partial differential equation:

$$\frac{\partial u}{\partial t}+ \bigg(\frac{\partial u}{\partial x}\bigg)^2=1 \quad x\in \mathbb{R},t>0$$

with the initial condition:

$$u(x,0)=-x^2 \quad \quad x \in \mathbb{R}$$

Determine the value of $t^*$ such that $|u(x,t)| \to \infty$ as $t \to t^*$. \(t^*\) is the time at which the solution becomes unbounded.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Calculate the limit: $$\lim \limits_{n \to \infty}\left(1-e^{\left(\frac{t+ \log^2(n)}{t-\log(n)}\right)}\right)^n$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-e^{-t}}}
Determine the minimum number of times two fair six-sided dice must be rolled to ensure a 95% probability that the proportion of times they show the same face is between \( \frac{5}{36} \) and \( \frac{7}{36} \) using the central limit theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{692}
David is traveling to Budapest from Paris without a cellphone and he needs to use a public payphone. He only has two coins with him. There are three pay-phones - one that never works, one that works half of the time, and one that always works. The first phone that David tries does not work. Assuming that he does not use the same phone again, what is the probability that the second phone that he uses will work?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{2}{3}$}
Given the probability density function \( \frac{1}{\pi a_0^3}e^{-2r/a_0} \), find the most probable value of \( r \) for the wavefunction of the lowest energy state of hydrogen, \( \psi(r) = \sqrt{\frac{1}{\pi a_0^3}}e^{-r/a_0} \), where \( r \) is the radial distance between the electron and nucleus and \( a_0 \) is the Bohr radius.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a_0}
Determine the value of \( n > 1000 \) that maximizes the probability \( P(x=10) = \frac{\binom{1000}{10}\binom{n-1000}{140}}{\binom{n}{150}} \), where \( n \) is the total number of identical balls in a box. Initially, 1000 balls are marked and returned to the box, and then 150 balls are randomly selected. Find the value of \( n \) that makes this probability largest.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15000}
Find the probability that a random variable \( \xi \sim \text{Beta}(1,2) \) is less than another independent random variable \( \eta \sim \text{Exp}(1) \). Calculate \( P(\xi < \eta) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{e}}
On a 3D chess board of size \(N \times N \times N\), determine the minimum number of rooks required to ensure that every empty cube on the board can be reached by a rook in a single move.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N}
Let $f: \mathbb{D} \to \mathbb{R}$ be defined as \( f(x) = \frac{x^2 + 2x + a}{x^2 + 4x + 3a} \), where \( \mathbb{D} \) is the domain of \( f \). Determine the range of the parameter \( a \) such that \( f \) is a surjective mapping.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1)}
Determine the number of natural numbers \( n \) such that there is no natural number \( M \) for which \( n^2 \) can be expressed as a sum of distinct squares of natural numbers \( a_i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Consider the set $\mathcal F$ of all holomorphic functions $f$ in the unit disc $\Bbb D$ such that $f(\Bbb D)\subset\Bbb D$, and satisfying $f\left(\frac{1}{2}\right)=f'\left(\frac{1}{2}\right)=0$. Determine the supremum of $|f(0)|$ over all functions $f$ in $\mathcal F$. What is this supremum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Determine the center of the algebra $PB(H)P$, where $P$ is a projection on a Hilbert space $H$. Express your answer in terms of $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{C}P}
Evaluate the integral \( \int_{-\infty}^\infty |f\ast f'|^2(x)\,dx \) given that \( \mathcal{F}[f](\omega)=\frac{1}{1+|\omega|^3} \) using Plancherel’s theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{9\pi}}
Determine the largest exponent \(d\) such that the volume of the set \(X(r)\), consisting of matrices \(A \in M(n \times m)\) with \(n \leq m\), where the largest singular value of \(A\) is at most 1 and the smallest singular value is less than \(r\), divided by \(r^d\) remains uniformly bounded as \(r\) approaches 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m - n + 1}
Determine the minimum value of $n$ for which the probability $P(0.39 \leq \overline{Y} \leq 0.41)$ is at least $0.95$, where $\overline{Y}$ is the sample mean of $n$ independent and identically distributed Bernoulli random variables with success probability $p = 0.4$. Assume the mean and variance of a Bernoulli variable are $np$ and $np(1-p)$, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9220}
Find the direction of a light beam that should be projected onto the plane \(\pi: x-2y+4z=0\) such that its reflection has the direction \((-2,1,2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-50, 37, 10)}
Evaluate the limit: $$\lim_{n \to \infty} \frac{(2 n)! e^n (n)^n}{n! (2 n)^{2 n}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Determine a closed formula for the nth value of x such that \( 15^x \equiv x \pmod{1000} \) for \( x \geq 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000n - 625}
Find the argument of the complex number $z$ that satisfies the equation $|z| = |z - 2|z||$. Express your answer in radians.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the system of equations \(a x + b y = 1\) and \(x^{2} + y^{2} = 50\), determine the number of real pairs \((a, b)\) such that there is at least one solution, and all solutions \((x, y)\) are integer solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Simplify the ordinal expression \(((\omega + 1) + (\omega + 1))^\omega\) and express it in Cantor Normal Form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\omega^{\omega}}
Let a board game has 10 cards: 3 skull cards, 5 coin cards and 2 blank cards. We put these 10 cards downward and shuffle
 them and take cards one by one from the top. Once \( 3\mathrm{{skull}} \) cards or coin cards appears we stop. What is the possibility of it
 stops because there appears 3 skull cards?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
5/28
}
Determine the highest degree of the polynomial for which the quadrature formula \( \int_{-1}^1 |x| f(x) \, dx \approx \frac{1}{2}[f(x_0) + f(x_1)] \) is exact, given that \( x_0 \) and \( x_1 \) are the quadrature points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find an entire function that is bounded on the imaginary axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^z}
Given the function \( H(s) = \frac{\zeta(s)}{\phi(q)} \sum_{\chi \mod{q}} L(s,\chi) = \sum_{n=1}^{\infty} \frac{h(n)}{n^s} \), determine the smallest integer \( n \) as a function of \( q \) such that \( h(n) \neq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{q + 1}
What is the least positive integer \( m \) such that the following is true?
 Given \( m \) integers between 1 and 2023,inclusive,there must exist two of them \( a,b \) such that \( 1 < \frac{a}{b} \leq  2 \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
12
}
Determine the value of $d$ for which the generalized Cantor set, constructed by retaining intervals of size $d<1$ at both ends and removing an interval of size $1-2d$ at each stage, has Lebesgue measure zero. Specifically, find the value of $d$ that satisfies the equation:

$$ \sum_{n=1}^{\infty} 2^{n-1} (1-2d^n) = 1. $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Evaluate the double integral of the floor function over the rectangular region: \[ \int_{1}^{3} \int_{2}^{5} \lfloor x+y \rfloor \, dy \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Let \( E \) be the set of all continuous functions \( f: [0,1] \to \mathbb{R} \) such that \( f(x) + f(y) \ge |x-y| \) for all \( x, y \in [0,1] \). Find the minimum value of \( \int_0^1 f(x) \, dx \) for \( f \in E \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Find the smallest positive integer \( k \) such that for any coloring of the squares of a \( 2n \times k \) chessboard with \( n \) colors, there exist 2 columns and 2 rows where the 4 squares at their intersections all have the same color.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2 + 1}