inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the length of the normal chord of the parabola \(y^2 = 8x\) that subtends a right angle at the vertex of the parabola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12\sqrt{3}}
Find the invariant factors of the module \( N = \langle a,b \rangle / \langle 5a-2b \rangle \), where \( \langle a,b \rangle \) and \( \langle 5a-2b \rangle \) are \( \mathbb{Z} \)-modules.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the minimum value of the function \( a = \frac{{(1+t^2)}^3}{t^4} \) using basic inequalities, without employing calculus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27}{4}}
Find the number of ring homomorphisms from the ring \( \mathbb{Z}_{12} \) to the ring \( \mathbb{Z}_{24} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For what values of $c \in [-1,1]$ does the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) given by \[ f(x) = \begin{cases} \sin \left( \frac{1}{x} \right), & \text{if } x \neq 0; \\ c, & \text{if } x = 0 \end{cases} \] have an antiderivative?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the condition on \(\alpha\) such that the function \(|x|^{-\alpha}\) is in the \(L^1\) space with respect to the measure \(\mu\) on the set \(\{x \in \mathbb{R}^n : |x| \ge 1\}\), given that \(\mu\) is a polynomially bounded measure satisfying \(\mu(|x| \le A) \le C(1+A^N)\) for constants \(C, N > 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > N}
Evaluate the integral \( \int_0^\pi e^{2\cos(\theta)}\cos(2\sin(\theta) + \theta)\,d\theta \) using complex analysis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Six lines are drawn in the plane. Determine the maximum number of points, through which at least $3$ lines pass.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4
}
Evaluate the following limit:
$$L=\lim_{x\to 0} \frac{(1+x)^{1/x}-e+\frac{ex}{2}}{x^2}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11e}{24}}
Given $x+4y=4$ ($y > 0$), $0 < t < z$, then the minimum value of $\dfrac {4z^{2}}{|x|}+ \dfrac {|xz^{2}|}{y}+ \dfrac {12}{t(z-t)}$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Given that the diameter of the base of a cone is 4, and the slant height (generatrix) is 6, find the central angle of the sector in the lateral surface development (net).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
Determine whether the series \( \sum_{n=1}^{\infty}\left(\cos\frac{1}{n}\right)^{k(n)} \) converges, where \( k(n)=\frac{1}{\sin^a\left(\frac{1}{n}\right)} \) and \( a \in \mathbb{R} \). Use the approximations \( \cos\frac{1}{n} \sim 1-\frac{1}{2n^2} \) and \( \sin^a\left(\frac{1}{n}\right) \sim \frac{1}{n^a} \) as \( n \to +\infty \) to analyze the convergence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > 2}
Given a sample of size 66 with a frequency distribution as follows: $(11.5, 15.5]$: $2$, $(15.5, 19.5]$: $4$, $(19.5, 23.5]$: $9$, $(23.5, 27.5]$: $18$, $(27.5, 31.5]$: $11$, $(31.5, 35.5]$: $12$, $[35.5, 39.5)$: $7$, $[39.5, 43.5)$: $3$, estimate the probability that the data falls in [31.5, 43.5).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Given a cyclic quadrilateral $ABCD$ inscribed in circle $O$, where $AC=\sqrt{3}BD$ and $\angle ADC=2\angle BAD$, and given that $AB\cdot CD+BC\cdot AD=4\sqrt{3}$, calculate the radius of circle $O$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit: \[ \lim_{x \to \infty}\left(\frac {\int_{x^{2}}^{2x}t^{4}e^{t^{2}}dt}{e^{x}-1-x - \frac{x^2}{2}- \frac{x^3}{6}-\frac{x^4}{24}}\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\infty}
Determine the absolute maximum value of \( f(z) = \left| \frac{5z^4 + 4}{z^2 + 1} \right| \) on the closed disk defined by \( |z - 2i| \leq 0.5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1063}{28}}
Determine the number of equally spaced interpolation nodes required to approximate the function \( f(x) = e^x \) on the interval \([-4, 0]\) using Lagrange interpolation, such that the interpolation error is less than 0.05.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given the set
$$
S=\left\{A=\left(a_{1}, a_{2}, \cdots, a_{8}\right) \mid a_{i}=0 \text{ or } 1, i=1,2, \cdots, 8\right\},
$$
for any two elements \(A\) and \(B\) in \(S\), define the distance between \(A\) and \(B\) as
$$
d(A, B)=\sum_{i=1}^{8}\left|a_{i}-b_{i}\right|. 
$$

Determine the maximum number of elements that can be chosen from \(S\) such that the distance between any two of them is at least 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Calculate the product of the elements $(1,1,1,0,1)$ and $(0,1,0,1,0)$ in the Galois Field $GF(2^5)$. What is the result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,1,1,0,0)}
Let \( A \) be the set of any 20 points on the circumference of a circle. Joining any two points in \( A \) produces one chord of this circle. Suppose every three such chords are not concurrent. Find the number of regions within the circle which are divided by all these chords.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5036}
Evaluate the limit: \[ \lim_{n\to \infty} \left(\frac{(2n!)}{n!^2}\right)^{\frac{1}{4n}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Evaluate the series: $$\sum_{n=1}^\infty \arccos\sqrt{\frac{n}{n+1}}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In a certain kingdom, there were 32 knights. Some of them were vassals to others (a vassal can only have one liege, and the liege is always wealthier than the vassal). A knight with at least four vassals held the title of baron. What is the maximum number of barons that could exist under these conditions?

(In the kingdom, the law stated: "the vassal of my vassal is not my vassal.")
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A class went on a camping trip and walked along the road from the school to the base, then immediately returned to the school by the same route. If the distance traveled each day increased by 1 kilometer, and it took 4 days to go and 3 days to return, calculate the distance from the school to the base. Express your answer as a whole number in kilometers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Let $A$ be the $n\times n$ matrix whose entry in the $i$-th row and $j$-th column is \[\frac1{\min(i,j)}\] for $1\le i,j\le n.$ Compute $\det(A).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[{\dfrac{(-1)^{n-1}}{n!(n-1)!}}\]}
Given that $\alpha$ and $\beta$ are both acute angles, $sin\alpha=\frac{3}{5}$, $cos(\alpha+\beta)=\frac{5}{13}$, then the value of $\cos \beta$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{56}{65}}
All the prime numbers are written in order,  $p_1 = 2, p_2 = 3, p_3 = 5, ...$ Find all pairs of positive integers  $a$  and  $b$  with  $a - b \geq 2$ , such that  $p_a - p_b$  divides  $2(a-b)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ (4, 2) }
Find the maximum dimension of a subspace of endomorphisms of a 4-dimensional vector space over the complex numbers, where all elements of the subspace commute and at least one element has three distinct eigenvalues.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A unit of blood expires after $10!=10\cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January $1$. On what day does his unit of blood expire? Provide the answer by converting the month to its corresponding number (e.g., January is 1, February is 2, etc.) and the day to a number, then give the sum of these two numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
A rook has traveled across an \( n \times n \) board, visiting each cell exactly once, where each of its moves was exactly one cell. The cells are numbered from 1 to \( n^2 \) in the order of the rook's path. Let \( M \) be the maximum difference between the numbers of adjacent (by side) cells. What is the smallest possible value of \( M \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n - 1}
A circular coin with a circumference of about 5 cm starts rolling from a point on the boundary of a quadrilateral with a circumference of 20 cm. After making one complete rotation and returning to the starting point, the center of the coin will trace a closed curve. The length of this curve is ____ cm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Determine the minimal degree of a smooth curve in $\mathbb{CP}^3$ that is not contained in any cubic surface.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let $R$ be a square region and $n \ge 4$ an integer. A point $X$ in the interior of $R$ is called n-ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2320}
Evaluate the contour integral \( \frac{1}{2\pi i}\int_{\gamma} \frac{1}{z^2(e^{z}-1)}dz \) where the contour \(\gamma\) is defined by \(z=\frac{1}{2}e^{2\pi i t}\) for \(0\leq t\leq1\), and \(z=0\) is a pole of order 3. Determine the value of the integral without directly applying the Cauchy Integral Formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
Using the random walk approach, calculate the expected time for a mouse to reach one of the three exits in a maze, where each exit is an absorbing state.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Example 8. Find the integral $\int \frac{d x}{\sqrt{3 x^{2}-6 x+9}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{\sqrt{3}}\ln|x-1+\sqrt{x^{2}-2x+3}|+C}
In the tetrahedron \(EFGH\), the side lengths are given as \(EF = GH = 7\), \(EG = FH = 10\), and \(EH = FG = 11\). The points \(K, L, M, N\) are the incenters of the triangles \(EFG\), \(EFH\), \(EGH\), and \(FGH\) respectively. Calculate the volume of the tetrahedron \(KLMN\). Round your answer to the nearest hundredth if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.09}
A regular tetrahedron has all edges of equal length. There are two spheres that each touch every edge of the tetrahedron. What is the ratio of their volumes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{27}}
How many distinct ways can the letters of the word COMBINATION be arranged such that none of the patterns CAN, BIN, or NIB appear?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4541040}
Find the limit: $$\lim_{x\to \infty} \frac{(x+3)^x}{\sum_{n=3}^{x+2}n^x}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e - 1}
The 6th problem of the 3rd China Northern Hope Star Mathematics Summer Camp:
Given non-negative real numbers $a, b, c, x, y, z$ satisfying $a+b+c=x+y+z=1$.
Find the maximum value of $\left(a-x^{2}\right)\left(b-y^{2}\right)\left(c-z^{2}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{16}}
Let $x_{1}$, $x_{2}$, $\\ldots$, $x_{n}$ be $n$ real numbers in $\\left(\\frac{1}{4},\\frac{2}{3}\\right)$. Find the minimal value of the expression: \\[ \\log_{\\frac 32x_{1}}\\left(\\frac{1}{2}-\\frac{1}{36x_{2}^{2}}\\right)+\\log_{\\frac 32x_{2}}\\left(\\frac{1}{2}-\\frac{1}{36x_{3}^{2}}\\right)+\\cdots+ \\log_{\\frac 32x_{n}}\\left(\\frac{1}{2}-\\frac{1}{36x_{1}^{2}}\\right). \\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
The minimal value of the expression is \( 2n \).}
Consider an ergodic Markov chain \( \{Y_t\} \) with state space \( \{0, \dots, m\} \). Given that \( E(Y_{t+1}-Y_t)=\alpha \) for \( Y_t\in [1,m-2] \), \( E(Y_{t+1}-Y_t)>\alpha \) for \( Y_t=m-1 \), and \( E(Y_{t+1}-Y_t)=\gamma \) for \( Y_t=0 \), where \( \alpha \) is a small negative number, \( \gamma>\alpha \), and \( m \) is a large positive constant, determine the behavior of \( \lim_{t\to\infty}E(Y_t)/m \) as \( m\rightarrow\infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Set 
\[A_n=\sum_{k=1}^n \frac{k^6}{2^k}.\]
Find \(\lim_{n\to\infty} A_n.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9366}
Simplify the following limit:

$$\lim _{ n\rightarrow \infty }{ \left( \sin ^{ 4 }{ x } +\frac { 1 }{ 4 } \sin ^{ 4 }{ 2x+\cdots+\frac { 1 }{ { 4 }^{ n } } \sin ^{ 4 }{ 2nx } } \right) } $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin^2 x}
Find the closed-form expression for the limit of the series \( \sum_{m=M+1}^{\infty}\frac{(-1)^{m+1}}{2m+1}\sin((2m+1)\frac{\pi}{2}x) \) as \( M \to \infty \), given that \( \sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{2k-1}\sin((2k-1)x)=\frac{1}{2}\ln\tan(\frac{\pi}{4}+\frac{x}{2}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
$n$ coins are arranged in a circle. If two neighboring coins both show heads or both show tails, you can flip both coins. How many distinct arrangements of the coins are there such that no arrangement can be transformed into another by performing these operations?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Express \(\cos(x^2)\) in terms of \(\cos(x)\) and \(\sin(x)\), if possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\cos(x^2)}
Al has $2019$ red jelly beans. Bob has $2018$ green jelly beans. Carl has $x$ blue jelly beans. The minimum number of jelly beans that must be drawn in order to guarantee $2$ jelly beans of each color is $4041$. Compute $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2020}
If 60% of (x - y) = 30% of (x + y + z), and 40% of (y - z) = 20% of (y + x - z), then what percent of x is (y - z)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100\%}
14:There are N permutations $(a_{1}, a_{2}, ... , a_{30})$ of 1, 2, ... , 30 such that for $m \in \left\{{2, 3, 5}\right\}$, m divides $a_{n+m} - a_{n}$ for all integers n with $1 \leq n < n+m \leq 30$. Find the remainder when N is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{440}
Given that 10 spots for the finals of the 2009 National High School Mathematics Competition are to be distributed to four different schools in a certain district, with the requirement that one school gets 1 spot, another gets 2 spots, a third gets 3 spots, and the last one gets 4 spots, calculate the total number of different distribution schemes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Evaluate the limit \( \lim_{n \to \infty} \frac{\prod_{k=0}^{n-1} \binom{n-1}{k} \cdot \prod_{k=0}^{n+1} \binom{n+1}{k}}{\left(\prod_{k=0}^{n} \binom{n}{k}\right)^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Alex, a self-described nerd and an avid comic book collector, has been attending Comic-Con annually since 1995. To celebrate his love for comics, each year at Comic-Con, Alex buys a set of comic books equal to the number of years he's been attending. In 2023, how many comic books did Alex purchase in total at Comic-Con?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{435}
Since the implementation of quality education in high schools in a certain province, student clubs have developed rapidly. Five freshmen from a certain high school plan to join one of the four clubs: "Chunhui Literature Society," "Dancers Roller Skating Club," "Basketball Home," and "Go Garden." If each club has at least one student join, each student joins at least one club and only one club, and student A does not join "Go Garden," then the number of different participation methods is ______. (Fill in the number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Consider finitely many points in the plane with no three points on a line. All these points can be coloured red or green such that any triangle with vertices of the same colour contains at least one point of the other colour in its interior.
What is the maximal possible number of points with this property?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A number is *interesting*if it is a  $6$ -digit integer that contains no zeros, its first  $3$  digits are strictly increasing, and its last  $3$  digits are non-increasing. What is the average of all interesting numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{308253}
Find the positive real number $x$ such that $\lfloor x \rfloor \cdot x = 54$. Express $x$ as a decimal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.714285714285714}
Let $S = \{1, 2, \dots, 9\}$. Compute the number of functions $f : S \rightarrow S$ such that, for all $s \in S$, $f(f(f(s))) = s$ and $f(s) - s$ is not divisible by $3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
Solve for \( x \) in the equation \( x\left(\frac{e^x+1}{e^x-1}\right)=4 \) without using numerical methods or series expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Suppose $X_1, X_2, \ldots, X_n$ are independent and identically distributed (i.i.d.) positive random variables with expected value $E(X_i) = \mu$ and variance $Var(X_i) = \sigma^2$. Calculate the expectation of the ratio $\frac{X_1 + X_2 + \ldots + X_m}{X_1 + X_2 + \ldots + X_n}$, where $m < n$. \( \text{Express your answer in terms of } m, n, \mu, \text{ and } \sigma^2. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{m}{n}}
If one takes 6 different letters from "CN Dream" and arranges them in a row, how many different permutations are there that include the letter combination "ea" (in the same order)? Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{600}
We define a number as "funny" if it is divisible by the sum of its digits plus one. For example, since $1 + 2 + 1 \mid 12$, the number $12$ is considered funny. What is the maximum number of consecutive funny numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
There are $360$ permutations of the letters in $MMATHS.$ When ordered alphabetically, starting from $AHMMST,$ $MMATHS$ is in the $n$th permutation. What is $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{173
}
Let $A B C D E F G H$ be an equilateral octagon with $\angle A \cong \angle C \cong \angle E \cong \angle G$ and $\angle B \cong \angle D \cong \angle F \cong$ $\angle H$. If the area of $A B C D E F G H$ is three times the area of $A C E G$, then $\sin B$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{405}
Given the sets \(A\) and \(B\) defined as follows:
\[ A=\{(x, y) \mid x=m, y=-3m+2, m \in \mathbf{Z}_{+}\} \]
\[ B=\{(x, y) \mid x=n, y=a(a^{2}-n+1), n \in \mathbf{Z}_{+}\} \]
Determine the total number of integers \(a\) such that the intersection \(A \cap B\) is non-empty.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A community plans to invest 500 yuan to purchase three types of books, A, B, and C. Type A costs 30 yuan per book, type B costs 25 yuan per book, and type C costs 20 yuan per book. The community must buy at least 5 books of type A and at most 6 books of type A, and all three types of books must be purchased. Calculate the total number of purchasing schemes that satisfy these conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
What is the cardinality of the set of all sequences of real numbers that have a finite number of terms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
Given a matrix $A$ with entries of either 0 or 1, where each row contains at most $r$ ones, and each row of the matrix product $AA^T$ contains at most $z$ non-zero entries, find an upper bound on the number of ones in each column of $A$ as a function of $r$ and $z$. The bound need not be tight but should be expressible as a simple function of $r$ and $z$. Provide your answer as a function of $r$ and $z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{z}
Let $A \in \mathbb{R}^{m \times m}$ be an invertible matrix such that $A^k = I$ for some positive integer $k$. Define the linear map $T_A: \mathbb{R}^{m \times m} \to \mathbb{R}^{m \times m}$ by $T_A(X) = AXA^{-1}$. Determine the minimal polynomial of $T_A$ when restricted to the space of skew-symmetric $m \times m$ matrices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^k - 1}
In a regular quadrilateral prism \( ABCDA_1B_1C_1D_1 \) \( (A \| A_1 \| B \| B_1 \| C \| C_1 \| D \| D_1 ) \), a plane passes through the midpoints of two adjacent sides of the base \( DC \) and \( AD \), and the vertex \( B_1 \) of the upper base. Find the angle between this plane and the base plane if the perimeter of the cross-section is three times the diagonal of the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arccos \frac{3}{4}}
Given two linear functions \( f(x) \) and \( g(x) \) such that the graphs of \( y = f(x) \) and \( y = g(x) \) are parallel lines, not parallel to the coordinate axes. It is known that the graph of the function \( y = (f(x))^2 \) touches the graph of the function \( y = 11g(x) \). Find all values of \( A \) such that the graph of the function \( y = (g(x))^2 \) touches the graph of the function \( y = Af(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-11}
In the diagram, each of four identical circles touch three others. The circumference of each circle is 48. Calculate the perimeter of the shaded region formed within the central area where all four circles touch. Assume the circles are arranged symmetrically like petals of a flower.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Determine the radius of convergence of the power series for \( \frac{1}{1+x^2} \) centered at \( x=1 \) without using complex analysis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16.67\%}