inclusionAI/Ring-lite-rl-data huggingface dataset:
Given the chemical reaction where compounds $A$ and $B$ combine to form compound $X$:

$$A + B \xrightarrow{k} X$$

If $2 \, \rm{g}$ of $A$ and $1 \, \rm{g}$ of $B$ yield $3 \, \rm{g}$ of $X$, the differential equation describing the reaction is:

$$\frac{dx}{dt} = k \left(a - \frac{2}{3}x\right)\left(b - \frac{1}{3}x\right)$$

where $a$ and $b$ are the initial amounts of $A$ and $B$ (in grams), $x(t)$ is the amount of $X$ at time $t$ in minutes, and $k$ is the reaction rate (per minute per gram). Assume that $x(0) = 0$. Express the solution in the form:

$$\ln \left(\frac{a - \frac{2}{3}x(t)}{b - \frac{1}{3}x(t)} \cdot \frac{b}{a}\right) = ct$$

where $c$ is a constant depending on $k$, $a$, and $b$. If $a = 15 \, \rm{g}$, $b = 20 \, \rm{g}$, and $15 \, \rm{g}$ of $X$ is formed after $10$ minutes, find the amount of $X$ after $20$ minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19.5}
Given the hyperbola $H: \frac{x^2}{9} - \frac{y^2}{16} = 1$, a tangent is drawn at any point $P$ on $H$. This tangent intersects another hyperbola $K: \frac{x^2}{m^2} - \frac{y^2}{n^2} = k$ at points $A$ and $B$. If $P$ is always the midpoint of $AB$, find the floor of the sum of the possible eccentricities of $K$, where $k = \pm 1$. \( \text{Provide your answer as an integer.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A trapezoid has side lengths $3, 5, 7,$ and $11$. The sum of all the possible areas of the trapezoid can be written in the form of $r_1 \sqrt{n_1} + r_2 \sqrt{n_2} + r_3$, where $r_1, r_2,$ and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of a prime. What is the greatest integer less than or equal to
\[r_1 + r_2 + r_3 + n_1 + n_2?\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
Given 5 observations: 4, 3, 5, 2, 6 from a \(\text{binomial}(10, p)\) random variable, find an unbiased estimate of \((1+p)^{10}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{124}{5}}
How many ways can we form a group with an odd number of members (plural) from $99$ people? Express your answer in the form $a^b + c$, where $a, b$, and $c$ are integers and $a$ is prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{98}-99}
Let $S$ be the set of all polynomials of the form $z^3+az^2+bz+c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $\left\lvert z \right\rvert = 20$ or $\left\lvert z \right\rvert = 13$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
540

}
$\triangle ABC$ satisfies $\angle A=60^{\circ}$. Call its circumcenter and orthocenter $O, H$, respectively. Let $M$ be a point on the segment $BH$, then choose a point $N$ on the line $CH$ such that $H$ lies between $C, N$, and $\overline{BM}=\overline{CN}$. Find all possible value of
\[\frac{\overline{MH}+\overline{NH}}{\overline{OH}}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\sqrt{3}$}
Evaluate the integral \( \int_0^{2\pi}\frac{\cos^2 3\theta\, d\theta}{5-4\cos 2\theta} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{8}}
Given in $\triangle ABC$, $3\sin ^{2}B+7\sin ^{2}C=2\sin A\sin B\sin C+2\sin ^{2}A$, find $\sin \left(A+\dfrac{\pi}{4} \right)=$\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{- \dfrac{\sqrt{10}}{10}}
For any natural number \(n \ge 3\), find the maximum number of odd entries in a matrix \(B\in\mathcal{M}_{n}\left(\mathbb{N}\right)\) such that \(\det(B)=1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2 - n + 1}
Find the lower bound for the expression \(-\frac{1}{4}\sqrt{(1-2a+x)^2-8(-3-4a-3x-2ax)}-\frac{1}{4}(1-2a+x)\) given that \(x \leq 0\) and \(a > 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Find the number of ordered tuples $\left(C,A,M,B\right)$ of non-negative integers such that\[C!+C!+A!+M!=B!\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
7
}
Consider the family of holomorphic functions $\mathcal F = \{f \text{ holomorphic in } \Bbb D \text{ with } f(\Bbb D)\subset\Bbb D, f\left(\frac{1}{2}\right)=f'\left(\frac{1}{2}\right)=0\}$, where $\Bbb D$ is the unit disc. Compute $\sup_{f\in\mathcal F}|f(0)|$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Evaluate the limit \( \lim_{x \to 1^-} \sqrt{1-x} \sum_{n=0}^{\infty} x^{n^3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a pyramid \(S-ABC\) with a height of \(SO = 3\) and a square base with side length 6, a perpendicular is drawn from point A to the opposite face \(SBC\), with the foot of the perpendicular being \(O'\). On the line segment \(AO'\), a point \(P\) is chosen such that \(\frac{AP}{PO'} = 8\). Determine the area of the cross-section through point \(P\) that is parallel to the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{9}}
已知集合 A=\{1,2,\cdots,10\}. 定义函数 f:A\rightarrow A，满足：(1) 对任意 x,y\in A，若 x\leq y，则 f(x)\leq f(y)；(2) f(3)=4. 这样的函数 f 的个数为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17160}
Evaluate the integral: \[ \int_0^2 \left(\sum_{n=0}^\infty \frac{(x-1)^{2n+1}}{2n+1}\right) dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $g(x),g:\mathbb{R}\to \mathbb{R}$ be a non-constant continuous function satisfying the equation $$\left(e^{2x}-1\right)g(x)=\left(e^x-1\right)g(2x).$$ If $g'(0)=2$, find the value of $$\lim_{x\to \infty}\left(\frac{g(x)}{x}\right)^{2/x}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^2}
An ant starts at one vertex of a regular dodecagon and wants to reach the opposite vertex by moving to adjacent vertices. Let \( p_n \) represent the number of such paths with length \( n \). Compute the sum \( p_1 + p_2 + p_3 + \ldots + p_{12} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{548}
Consider a sample $X_1, X_2, \ldots, X_{100}$ from a Poisson distribution with parameter $\lambda = 3.5$. Calculate the approximate probability that the total sum $T = \sum_{i=1}^{100} X_i$ exceeds 360.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.287}
Evaluate the integral \( I = \oint_{|z|=1} \frac{\cos(z)}{z(e^{z}-1)} \, dz \) along the unit circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi i}
Determine the structure of the quotient group $G/G'$ for the group $G$ defined by the presentation $\langle a, b \mid a^6 = 1, b^2 = a^3, bab^{-1} = a^{-1} \rangle$, where the derived subgroup is $G' = \{e, a^2, a^4\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}_4}
Compute the residue of the function \( \frac{1}{e^{\frac{1}{z}}-1} \) at the point \( z=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
Determine the number of homomorphisms $\psi: D_4 \longrightarrow \mathbb{Z}_4$, where $D_4$ is the dihedral group of order 8.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In a connected network with 2008 computers, no two cycles share a common vertex. A hacker and an administrator play a game on this network. On the 1st move, the hacker selects and hacks one computer. On the 2nd move, the administrator selects and protects another computer. For every 2k+1th move (k > 0), the hacker hacks a computer that is directly connected to a previously hacked computer and is not protected. For every 2k+2th move, the administrator protects a computer that is directly connected to a previously protected computer and is not hacked. The game ends when neither can make a move. What is the maximum number of computers the hacker can guarantee to hack by the end of the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1004}
Determine the values of \( \alpha \) for which the limit \( \lim_{n\rightarrow \infty}\int_{1}^{n}\frac{\sin{x^2}}{x^{\alpha}}dx \) is finite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > -1}
Call an ordered triple of sets $(A, B, C)$ nice if $|A \cap B| = |B \cap C| = |C \cap A| = 2$ and $|A \cap B \cap C| = 0$. How many ordered triples of subsets of $\{1, 2, · · · , 9\}$ are nice?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{483840}
Determine the bases for which there exists an integer N such that there are no N-digit prime numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the smallest constant \( M \) such that the inequality \[ \frac{x_{1}}{x_{1}+x_{2}}+\frac{x_{2}}{x_{2}+x_{1}}+\frac{x_{3}}{x_{3}+x_{4}}+\cdots+\frac{x_{2009}}{x_{2009}+x_{1}} < M \] holds for all positive real numbers \( x_{1}, x_{2}, \ldots, x_{2009} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1005}
Determine the fundamental group of the space \((S^1 \times S^1)/((S^1 \times \{x\})\), where \(x\) is a point in \(S^1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Calculate the work done by the vector field \( \vec{G}=(x^2y+z^2)\vec{i}+(xy^2-z)\vec{j}+(yz+x)\vec{k} \) in moving a particle along the curve formed by the intersection of the plane \( x+y+z=1 \) and the cylinder \( x^2+y^2=4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12\pi}
Evaluate the series: \[ \sum_{k=1}^{\infty} (-1)^{k+1}\frac{\pi^{6k}}{(6k)!} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Let $F$ be a field of order $32$. Determine the number of non-zero solutions $(a,b) \in F \times F$ to the equation $x^2 + xy + y^2 = 0$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In an alphabet with $n > 1$ letters, a word is defined as any finite sequence of letters where any two consecutive letters are different. A word is called "good" if it is not possible to delete all but four letters from it to obtain a sequence of the form $a a b b$, where $a$ and $b$ are different letters. Determine the maximum possible number of letters in a "good" word.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n + 1}
Find the sum of the infinite series \(\sum_{n=1}^{\infty} \frac{f(n)}{n(n+1)}\), where \(f(n)\) is the number of ones in the binary representation of the integer \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \ln 2}
Find the spectrum of a bounded self-adjoint operator $S: H \rightarrow H$ on a Hilbert space $H$ that satisfies the equation $g(S) = 0$, where $g(z) = z^4 + z^3 + z^2 + z + 1$. Provide your answer as a set of values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
A unit square is divided into rectangles, each colored either yellow or blue. For a blue rectangle, the number written inside it is the rectangle's width divided by its height. For a yellow rectangle, the number is the rectangle's height divided by its width. Let $x$ be the sum of all these numbers. If the total area of the blue rectangles equals the total area of the yellow rectangles, find the smallest possible value of $x$. \( \text{(Express your answer as a single number.)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{2}}
Evaluate the integral \( \int_\gamma \frac{dz}{(z-\frac{1}{2}-i)(z-1-\frac{3i}{2})(z-1-\frac{i}{2})(z-\frac{3}{2}-i)} \), where \( \gamma \) is the polygon with vertices \( 0, 2, 2+2i, 2i, 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
How many positive three-digit integers $\\underline{a}\\underline{b}\\underline{c}$ can represent a valid date in $2013$, where either $a$ corresponds to a month and $\\underline{b}\\underline{c}$  corresponds to the day in that month, or $\\underline{a}\\underline{b}$ corresponds to a month and $c$ corresponds to the day?  For example, 202 is a valid representation for February 2nd, and 121 could represent either January 21st or December 1st.  \n\n(Note:  During the actual test they had to write the number of days in each month so don feel bad if you have to google that .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
273
}
Solve for \( t \) in the equation:
\[ 5000 = 21t - 111.37e^{-\frac{12t}{6.5625}} + 1207.12 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180.61}
Let \( N \) be the set \( \{ 1,2,\ldots ,{2018}\} \) . For each subset \( A \) of \( N \) with exactly 1009 elements,define
 \( f\left( A\right)  = \mathop{\sum }\limits_{{i \in  A}}i\mathop{\sum }\limits_{{j \in  N,j \notin  A}}j. \)
 If \( \mathbb{E}\left\lbrack  {f\left( A\right) }\right\rbrack \) is the expected value of \( f\left( A\right) \) as \( A \) ranges over all the possible subsets of \( N \) with exactly 1009 elements, find
 remainder when the sum of the distinct prime factors of \( \mathbb{E}\left\lbrack  {f\left( A\right) }\right\rbrack \) is divided by 1000 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
441
}
Compute the limit \(\lim_{n\to \infty} \frac{\ln^2n}{n}\sum_{k=2}^{n-2}\frac{1}{\ln k\ln(n-k)}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the range of values for $a$ such that the system of inequalities \( \left\{\begin{array}{l}x^{2}-x+a-a^{2}<0 \\ x+2a>1\end{array}\right. \) has exactly two integer solutions for $x$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 2]}
Determine the number of non-decreasing tuples of integers \((a_1, a_2, \dots, a_{16})\) such that \(0 \leq a_i \leq 16\) for all \(i\), and the sum of all \(a_i\) is even.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300546630}
Find the volume of the region in \( \mathbb{R}^3 \) described by the inequalities \( x^2 + y^2 + z^2 \le 1 \) and \( |y| \le x^2 + y^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{16}{9}}
Determine the set of values of $a > 0$ for which the function \( f(x) = \frac{\sin([x])}{a} + \frac{\cos([x])}{a} \), where \([ \cdot ]\) denotes the fractional part of \(x\), can attain its maximum value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \infty)}
The measures of the interior angles of a convex polygon of $n$ sides are in arithmetic progression. If the common difference is $5^{\circ}$ and the largest angle is $160^{\circ}$, then $n$ equals:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$9$}
If the eccentricity of the hyperbola $x^{2}+ky^{2}=1$ is $2$, then the value of the real number $k$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{3}}
Find the derivative of the expression $a^{T}Xb$ with respect to the vector $b$, where $a$ and $b$ are $d$-dimensional vectors and $X$ is a $d \times d$ matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X^\top a}
Two differentiable functions $f(x)$ and $g(x)$ are defined on the interval $[0, 2]$. Given that $f''(x) - g''(x) = 0$, $f'(A) = 2$, $g'(A) = 4$, $f(B) = 3$, and $g(B) = 9$, find the value of $f\left(\frac{3}{2}\right) - g\left(\frac{3}{2}\right)$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5}
If the inequality $mx^2 + 4mx - 4 < 0$ holds for all real numbers $x$, then the range of the real number $m$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1 < m < 0}
We choose 100 points in the coordinate plane. Let  $N$  be the number of triples  $(A,B,C)$  of distinct chosen points such that  $A$  and  $B$  have the same  $y$ -coordinate, and  $B$  and  $C$  have the same  $x$ -coordinate. Find the greatest value that  $N$  can attain considering all possible ways to choose the points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8100}
Crisp All, a basketball player, is *dropping dimes* and nickels on a number line. Crisp drops a dime on every positive multiple of  $10$ , and a nickel on every multiple of  $5$  that is not a multiple of  $10$ . Crisp then starts at  $0$ . Every second, he has a  $\frac{2}{3}$  chance of jumping from his current location  $x$  to  $x+3$ , and a  $\frac{1}{3}$  chance of jumping from his current location  $x$  to  $x+7$ . When Crisp jumps on either a dime or a nickel, he stops jumping. What is the probability that Crisp *stops on a dime*?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{20}{31}}
For how many different values of integer $n$, one can find $n$ different lines in the plane such that each line intersects with exacly $2004$ of other lines?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12
}
Evaluate the limit: \[ \lim_{n \to \infty} n^2 \int_0^1 \frac{1}{(1+x^2)^n} \, dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In measuring the sides of a rectangle, one side is taken 6% in excess, and the other in deficit. The error percent in the area calculated from these measurements is 0.7%. What is the percentage in deficit for the other side?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\%}
Given that \(X_1\) and \(X_2\) are independent and identically distributed discrete random variables, the moment generating function of \(X_1 + X_2\) is given by:
\[ M(t) = 0.01e^{-2t} + 0.15e^{-t} + 0.5925 + 0.225e^{t} + 0.0225e^{2t} \]
Find the probability \(P[X < 0]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.1}
Given a polynomial \( P(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \ldots + a_1 x + a_0 \) where each coefficient \( a_i \) belongs to the interval \([100,101]\), what is the smallest natural number \( n \) such that this polynomial can have a real root?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
In triangle \(ABC\), where \(AB = 9\), \(BC = 10\), and \(CA = 17\), let \(\Omega\) be its circumcircle. Points \(A_1\), \(B_1\), and \(C_1\) are diametrically opposite to \(A\), \(B\), and \(C\) on \(\Omega\), respectively. Calculate the area of the convex hexagon with vertices \(A, B, C, A_1, B_1, C_1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1155}{4}}
Let \( r_{1}, \ldots, r_{n} \) be the distinct real zeros of the equation
\[ x^{8} - 14x^{4} - 8x^{3} - x^{2} + 1 = 0. \]
Evaluate \( r_{1}^{2} + \cdots + r_{n}^{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Express the total installed capacity of 16,780,000 kilowatts in scientific notation in the format a × 10^n, where 1 ≤ a < 10 and n is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.678 × 10^7}
Calculate the fundamental group of the space $S^3 - S^1$. Provide your answer as a well-defined mathematical group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
考虑所有满足以下两个条件的实数序
列 $x_{0}, x_{1}, \cdots, x_{100}$ :
(1) $x_{0}=0$;
（2）对任意整数 $i(1 \leqslant i \leqslant 100)$ 有
$1 \leqslant x_{i}-x_{i-1} \leqslant 2$.
求最大的正整数 $k \leqslant 100$ ，使得对任意这样的序列 $x_{0}, x_{1}, \cdots, x_{100}$, 均有
$$
x_{k}+x_{k+1}+\cdots+x_{100} \geqslant x_{0}+x_{1}+\cdots+x_{k-1}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Evaluate the limit \(\lim_{n\to\infty}\frac{n!}{x^n}\) using the Stolz-Cesaro theorem, where \(x\) is a real number such that \(|x|>1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the cardinality of the set $F$ of all entire functions $f$ such that $f(x) = |x^k|$ for all $x \in (-1, 1)$, where $K$ is an odd integer $\geq 1$. What is the cardinality of $F$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $f(n)=\displaystyle\sum_{k=2}^\infty \dfrac{1}{k^n\cdot k!}.$ Calculate $\displaystyle\sum_{n=2}^\infty f(n)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\[ 3 - e \]}
Determine the maximum length of a roof that can be obtained by stacking an unlimited number of identical rectangular bricks with some offset so that they do not fall.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
There is a graph with 30 vertices. If any of 26 of its vertices with their outgoiing edges are deleted, then the remained graph is a connected graph with 4 vertices.\nWhat is the smallest number of the edges in the initial graph with 30 vertices?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
405}
Determine the number of possible values for the tuple \((a, b, c, d)\) of real numbers that satisfy the following equations: \(abc = d\), \(bcd = a\), \(cda = b\), and \(dab = c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
The necessary and sufficient condition for the lines $ax+2y+1=0$ and $3x+(a-1)y+1=0$ to be parallel is "$a=$ ______".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Find the number of elements in the intersection of $A_7$ and the normalizer $N_G(H)$, where $G = S_7$ and $H = \langle (1\ 2\ 3\ 4\ 5\ 6\ 7) \rangle$. Given that $|N_G(H)| = 42$ and $|A_7| = 2520$, use the internal product formula to determine $|A_7 \cap N_G(H)|$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Evaluate the limit: $$\lim_{n\to\infty}\frac{\int_1^{n^{1/n}} y^{n^2}(\arctan y) ny^{n-1}\,dy}{n^n}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Determine the set of complex values of $z$ for which the series \( \sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\dfrac{(-z)^{n+2nk}}{1+z^{2n}} \) is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
For the sequence defined by \(x_1=\frac{1}{2}\) and \(x_{n+1}=x_n-x_n^2\), determine if there exists a value of \(\alpha\) such that \(\lim\limits_{n\to\infty}n^\alpha \cdot \frac{1}{x_n} \neq 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Along the diagonals $AC$ and $B_{1}D_{1}$ of the base of a cube $ABCDEFGH$ with edge length $a$, two ants, Gosha and Lesha, are crawling. They started moving simultaneously from points $A$ and $B_{1}$ respectively at constant speeds, with Lesha's speed being three times the speed of Gosha, and finished when Lesha reached point $D_{1}$. What was the minimum distance between Gosha and Lesha during their movement?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \sqrt{\frac{6}{5}}}