inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the simple interest on a certain amount at 25/4% per annum for the period from 4th Feb 2005 to 18th April 2005, if the interest earned is Rs. 25. What is the principal amount?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
A group of 6 students decided to make [i]study groups[/i] and [i]service activity groups[/i] according to the following principle:

Each group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members.

Supposing there are at least one group and no three students belong to the same study group and service activity group, find the minimum number of groups.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the limit of the following expression as x approaches 0: $$\lim_{x \rightarrow 0} \frac{e^{2x}-e^{-2x}-2x}{x^2}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that the domain of the function $f(x)$ is $\mathbb{R}$ and $$\frac{f'(x)}{2} - f(x) > 2,$$ if $f(0) = -1$, then the solution set of the inequality $$\frac{f(x) + 2}{e^{2x}} > 1$$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, +\infty)}
$A=\{1,2,3, \cdots, 15\}, B=\{1,2,3,4,5\}, f$ 是 $A$ 到 $B$ 的映射，若满足 $f(x)=f(y)$ ，则称有序数对 $(x, y)$ 为"好对"，那么"好对"的个数的最小值是多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
In a group of order 21, there is a conjugacy class of order 3. Determine the order of an element in this conjugacy class.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Calculate the volume of the tetrahedron with vertices at $A_{1}(14, 4, 5)$, $A_{2}(-5, -3, 2)$, $A_{3}(-2, -6, -3)$, and $A_{4}(-2, 2, -1)$. The height is dropped from vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{338}{3}}
Edward starts in his house,which is at(0,0)and needs to go point(6,4),which is coordinate for his school. However,there is a
 park that shaped as a square and has coordinates \( \left( {2,1}\right) ,\left( {2,3}\right) ,\left( {4,1}\right) \) ,and(4,3). There is no road for him to walk inside the park but
 there is a road for him to walk around the perimeter of the square. How many different shortest road routes are there from
 Edward's house to his school?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
110
}
Points $A$, $B$, and $C$ lie on a circle $\Omega$ such that $A$ and $C$ are diametrically opposite each other. A line $\ell$ tangent to the incircle of $\triangle ABC$ at $T$ intersects $\Omega$ at points $X$ and $Y$. Suppose that $AB=30$, $BC=40$, and $XY=48$. Compute $TX \cdot TY$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{460}
A number when divided by 1423 leaves 215 as the remainder. When the same number is divided by 109, and then subtracted by the square of their quotient, what will the remainder be?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106}
Determine the total number of spanning trees in the undirected labelled graph $Q_3$, which is a 3-dimensional hypercube.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{384}
Find the value of $k$ for which the integral \( \int_{0}^{\infty} \frac{x^k}{2+4x+3x^2+5x^3+3x^4+4x^5+2x^6} \, dx \) is minimized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The series \(\sum a_n x^n\) and \(\sum b_n x^n\) have radii of convergence of 1 and 2, respectively. Determine the radius of convergence \(R\) of the series \(\sum a_n b_n x^n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the least positive integer $n$ such that the decimal representation of the binomial coefficient $\dbinom{2n}{n}$ ends in four zero digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{313}
Calculate the definite integral:

$$
\int_{\arcsin (1 / \sqrt{37})}^{\pi / 4} \frac{6 \tan x \, dx}{3 \sin 2x + 5 \cos^2 x}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{6} \ln \frac{6 e}{11}}
Solve the system of equations:

\[ 
x^3 - y = 6, 
\] 

\[ 
y^3 - z = 6, 
\] 

\[ 
z^3 - x = 6. 
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2,2,2)}
Calculate the expected number of coin tosses needed to achieve either $n$ consecutive heads or $n$ consecutive tails, excluding the first toss.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n - 2}
Find the derivative of the matrix product $\mathbf{A}\mathbf{B}\mathbf{c}$ with respect to the matrix $\mathbf{B}$, where $\mathbf{A}$ is an $n\times m$ matrix, $\mathbf{B}$ is an $m\times k$ matrix, and $\mathbf{c}$ is a $k\times 1$ vector.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbf{c}^\top \otimes \mathbf{A}}
In the convex quadrilateral \(ABCD\), point \(K\) is the midpoint of side \(BC\), and \(S_{ABCD} = 2S_{AKD}\).

Find the length of the median \(KE\) of triangle \(AKD\), if \(AB = a\) and \(CD = b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a + b}{2}}
In the context of the theorem on Formal Functions (Hartshorne, III.11.1), consider a projective morphism $f: X \to Y$ between noetherian schemes, with a coherent sheaf $\mathcal{F}$ on $X$ and a point $y \in Y$. The natural isomorphism $R^i f_\ast (\mathcal{F})_y^\hat{} \to \varprojlim H^i(X_n, \mathcal{F}_n)$ holds for all $i \geq 0$. The sequence $0 \rightarrow \mathcal{I}^n \mathcal{F} \rightarrow \mathcal{F} \rightarrow \mathcal{F}_n \rightarrow 0$ defines $\mathcal{F}_n$. Determine the starting point of the inverse limit: does it start at $n = 1$ or $n = 0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Let \( n \) be a positive integer. Chef Kao has \( n \) different flavors of ice cream. He wants to serve one small cup and one large cup for
 each flavor. He arranges the \( {2n} \) ice cream cups into two rows of \( n \) cups on a tray. He wants the tray to be colorful,so he
 arranges the ice cream cups with the following conditions:
 - each row contains all ice cream flavors, and
 - each column has different sizes of ice cream cup.


 Determine the number of ways that Chef Kao can arrange cups of ice cream with the above conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!(n+1)!}
Find the interval containing the solution to the equation $2^{x-1}+x=5$. Express your answer as an interval in the form (a, b).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 3)}
Let \( g : \mathbb{R}^{3} \to \mathbb{R}^{3} \) be a function defined by \( g(x,y,z) = (3y+4z, 2x-3z, x+3y) \). Define the set \( S = \{(x,y,z) \in \mathbb{R}^{3} : 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1 \} \).

If \[ \int\int\int_{g(S)} (2x+y-2z) \, dx \, dy \, dz = \alpha \int\int\int_{S} z \, dx \, dy \, dz, \] find the value of \( \alpha \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Determine the set of real values of \(\alpha\) for which the improper integral \(\int_0^\infty x^\alpha \cos x\, dx\) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 0)}
Find the third derivative of the given function.

\[ 
y=\frac{\ln (x-1)}{\sqrt{x-1}}, \quad y^{\prime \prime \prime}=?
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{46 - 15 \ln(x-1)}{8 (x-1)^{7/2}}}
Determine the line integral of the vector field $\vec{F}(x,y,z) = (3y, 5x, \frac{2x+3}{x^2 + y^2})$ over the intersection curve of the cone $x^2 + y^2 - z^2 = 0$ and the cylinder $(x-1)^2 + y^2 = 4$, using a parameterization without applying Stoke's theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8π}
Farmer John owns 2013 cows. Some cows are enemies of each other, and Farmer John wishes to divide them into as few groups as possible such that each cow has at most 3 enemies in her group. Each cow has at most 61 enemies. Compute the smallest integer $G$ such that, no matter which enemies they have, the cows can always be divided into at most $G$ such groups.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
The increasing interval of the function $f(x) = (x - 1)^2 - 2$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, +\infty)}
Given a polynomial equation \( z^{4} + a z^{3} + b z^{2} + c z + d = 0 \) with real coefficients, where all roots lie on the unit circle in the complex plane, find the sum of the reciprocals of these roots. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-a}
The sector of a circle has a certain perimeter and a central angle of 180 degrees. Its radius is 28.000000000000004 cm. What is the perimeter of the sector?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{143.96459430079216}
Find the limit:

$$\lim_{x \to -\infty} \frac{\sqrt[n]{x+1} + \sqrt[n]{x+2} - 2\sqrt[n]{x-3}}{\sqrt[n]{x-1} + \sqrt[n]{x-2} - 2\sqrt[n]{x+3}} \quad \text{for} \quad n \in \mathbb{N}, \quad n > 2, \quad \text{odd}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:  
$abc=70,$
$cde=71,$  
$efg=72.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
The set \( S \) is given by \( S = \{1, 2, 3, 4, 5, 6\} \). A non-empty subset \( T \) of \( S \) has the property that it contains no pair of integers that share a common factor other than 1. How many distinct possibilities are there for \( T \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Find the equation of the projection of the surface $x^2+y^2+z^2-yz=1$ onto the $xoy$ plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4x^2 + 3y^2 = 4}
Find the distributional limit \( \lim_{\varepsilon\rightarrow 0^+}f_{\varepsilon} \) in \( D'(\mathbb{R}) \), where \( f_\varepsilon(x) \) is defined as follows:

\[
f_\varepsilon(x) = \begin{cases} 
\frac{1}{\varepsilon^3}, & x \in (0,\varepsilon) \cup (2\varepsilon,3\varepsilon), \\
\frac{-2}{\varepsilon^3}, & x \in (\varepsilon,2\varepsilon), \\
0, & x \notin (0,3\varepsilon).
\end{cases}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\delta''}
Find the equation of the line joining the points of contact of the tangents from the line \(2x-3y = c\) to the curve \(x^2+2y^2=2\). Given that \(c = \pm \sqrt{17}\), determine the equation of the line joining these points of contact.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3x + 4y = 0}
Evaluate the following integral:

$$\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{611\pi}}
If $x$ is a real number and $|x-4|+|x-3|<a$ where $a>0$, then:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > 1}
Evaluate the limit $$\lim_{x\rightarrow 0} \frac{e - (1+x)^{\frac{1}{x}}}{\tan x}$$ using L'Hopital's rule, without relying on series expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e/2}
Determine the value of
\[2023 + \frac{1}{2} \left( 2022 + \frac{1}{2} \left( 2021 + \dots + \frac{1}{2} \left( 4 + \frac{1}{2} \cdot (3 + 1) \right) \right) \dotsb \right).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4044}
Given the function
\[ f(x) = 256x^9 - 576x^7 + 432x^5 - 120x^3 + 9x \]
find the range of the function \( f(x) \) over the interval \( x \in [-1, 1] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 1]}
Evaluate the line integral \( \int_C (\ln y) e^{-x} \,dx - \dfrac{e^{-x}}{y}\,dy + z\,dz \) where \( C \) is the curve parametrized by \( r(t)=(t-1)i+e^{t^4}j+(t^2+1)k \) for \( 0\leq t\leq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
The Fibonacci numbers are defined recursively by $F_{0}=0, F_{1}=1$, and $F_{i}=F_{i-1}+F_{i-2}$ for $i \geq 2$. Given 15 wooden blocks of weights $F_{2}, F_{3}, \ldots, F_{16}$, compute the number of ways to paint each block either red or blue such that the total weight of the red blocks equals the total weight of the blue blocks.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Given that $\sin(\theta + 3\pi) = -\frac{2}{3}$, find the value of $\frac{\tan(-5\pi - \theta) \cdot \cos(\theta - 2\pi) \cdot \sin(-3\pi - \theta)}{\tan(\frac{7\pi}{2} + \theta) \cdot \sin(-4\pi + \theta) \cdot \cot(-\theta - \frac{\pi}{2})} + 2 \tan(6\pi - \theta) \cdot \cos(-\pi + \theta)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
In the arithmetic sequence $\{a_{n}\}$, $\frac{a_{1010}}{a_{1009}} < -1$. If its first $n$ terms' sum $S_{n}$ has a maximum value, then the maximum positive integer value of $n$ that makes $S_{n} > 0$ is $\_\_\_\_\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2018}
Antonio plays a game where he continually flips a fair coin to see the sequence of heads ($H$) and tails ($T$) that he flips. Antonio wins the game if he sees on four consecutive flips the sequence $TTHT$ before he sees the sequence $HTTH$. The probability that Antonio wins the game is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Given $f(x)= \frac{2^{x}}{2(\ln 2-1)x}$, find $f'(1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
3. [30] Let $A B C$ be an acute triangle with incenter $I$ and circumcenter $O$. Assume that $\angle O I A=90^{\circ}$. Given that $A I=97$ and $B C=144$, compute the area of $\triangle A B C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14040}
In $\triangle ABC$, if $\overrightarrow{AB} \cdot \overrightarrow{AC} = \overrightarrow{AB} \cdot \overrightarrow{CB} = 4$, then the length of edge $AB$ is _____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Evaluate the complex integral \( K = \frac{i}{4-\pi} \int \frac{1}{z\cos(z)}\,dz \) over the circle centered at the origin with radius 4. Determine the value of \( K \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given 6 points in space such that no 4 points are coplanar, find the smallest positive integer \( n \) such that whenever 10 line segments are drawn connecting these points, it is always possible to find \( n \) distinct pairs of triangles, each pair sharing a common edge.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $M$ and $N$ be two closed subspaces of a Hilbert space $H$ with $N \subset M$. Given that $P_M$ and $P_N$ denote the orthogonal projections onto $M$ and $N$, respectively, determine the subspace of $H$ on which the operator $P_M - P_N$ is an orthogonal projection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{M \ominus N}
Shapovalov A.V.

Inside a rectangular sheet of paper, $n$ rectangular holes with sides parallel to the edges of the sheet have been cut out. What is the smallest number of rectangular pieces into which this perforated sheet can be guaranteed to be cut? (The holes do not overlap and do not touch.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3n+1}
Calculate the probability that $X + Y \geq 2$, where $X$ is a binomial random variable with parameters $n = 10$ and $p = 0.5$, and $Y$ is an exponential random variable with rate $\lambda = \frac{1}{5}$. Assume $X$ and $Y$ are independent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.9979}
Find the distance between the local maximum and local minimum of a cubic polynomial \( P(x) \) of degree 3, given that \( P(-1) = 10 \), \( P(1) = -6 \), there is a maximum at \( x = -1 \), and the derivative \( P'(x) \) has a minimum at \( x = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{65}}
If $f(x) = (m-2)x^2 + mx + 4$ ($x \in \mathbb{R}$) is an even function, find the interval where $f(x)$ is monotonically decreasing.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, +\infty)}
Vasya was driving from Sosnovka to Petrovka. Along the way, he saw a sign indicating "70 km to Petrovka". After traveling another 20 km, Vasya saw a sign indicating "130 km to Sosnovka". What is the distance (in kilometers) from Sosnovka to Petrovka? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Find the limit of the following: $$\lim\limits_{n \to \infty}\frac{3^{-n^2}}{\sum\limits_{k=n+2}^{\infty} 3^{-k^2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{infinity}
Let \( f \in (\mathbb{R}^{3})^* \) be a linear functional defined as \( f(x_1, x_2, x_3) = x_1 - 3x_2 + x_3 \). Consider a basis \( \mathcal{A} = \{\alpha_1, \alpha_2, \alpha_3\} \) for the linear space \( \mathbb{R}^3 \) and its dual basis \( \mathcal{A}^* = \{g_1, g_2, g_3\} \subset (\mathbb{R}^{3})^* \), where \( g_1(x_1, x_2, x_3) = x_1 - x_3 \) and \( g_2(x_1, x_2, x_3) = x_1 - 2x_2 \). If the coefficient of \( f \) in the dual basis representation is 1 for \( g_3 \), determine the vector \( \alpha_3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 1, 2)}
A classroom has the shape of a regular hexagon with a side length of 3 m. A snore-o-meter is installed in each corner, measuring the number of sleeping students within a distance of up to 3 m. How many sleeping students are there in total in the classroom if the sum of the snore-o-meter readings is 7?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given that the area of a triangle with vertices $(x_1, x_1^2)$, $(x_2, x_2^2)$, and $(x_3, x_3^2)$ is 2 square units, and the area of another triangle with vertices $(y_1, y_1^2)$, $(y_2, y_2^2)$, and $(y_3, y_3^2)$ is 10 square units, calculate the value of the determinant:
$$
\begin{vmatrix}
(1+x_1y_1)^2 & (1+x_2y_1)^2 & (1+x_3y_1)^2 \\
(1+x_1y_2)^2 & (1+x_2y_2)^2 & (1+x_3y_2)^2 \\
(1+x_1y_3)^2 & (1+x_2y_3)^2 & (1+x_3y_3)^2 \\
\end{vmatrix}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
In triangle $ABC$, the area is $5$ and $BC = 10$. Points $E$ and $F$ are the midpoints of sides $AC$ and $AB$, respectively. Lines $BE$ and $CF$ intersect at point $G$. Given that quadrilateral $AEGF$ is cyclic, find the value of $AB^2 + AC^2$. \(\text{(Proposed by Ray Li)}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Given a cube with edge length \( a \). Let \( N \) be a point on the diagonal of a side face, and \( M \) be a point on a circle located in the plane of the base, with the center at the center of the base and radius \( \frac{5}{12}a \). Find the minimum value of \( |MN| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \frac{\sqrt{34}}{24}}
For each positive integer $n$ (written with no leading zeros), let $t(n)$ equal the number formed by reversing the digits of $n$. For example, $t(461) = 164$ and $t(560) = 65$. Determine the number of three-digit positive integers $m$ for which $m + t(t(m))$ is odd.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Calculate the value of the expression: $3 - 7 + 11 - 15 + 19 - \cdots - 59 + 63 - 67 + 71$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-36}
A train 700 m long can cross an electric pole in 40 sec. What is the speed of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17.5}
Find the maximum value of $T$ with $x, y, z \in \Bbb{R_+}$
$$T=\frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{3}{16}\)}
Let $SABC$ is pyramid, such that $SA\le 4$, $SB\ge 7$, $SC\ge 9$, $AB=5$, $BC\le 6$ and $AC\le 8$.
Find max value capacity(volume) of the pyramid $SABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$8\sqrt{6}$}
Suppose the random variable Y has a moment generating function given by \( M(t) = \frac{e^{ct}}{1-dt^2} \) for \(-1 < t < 1\). Given that the mean and variance of Y are 5 and 4, respectively, find the value of \( c + d \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Point $I$ is the center of the inscribed circle of triangle $ABC$, $M$ is the midpoint of the side $AC$, and $W$ is the midpoint of the arc $AB$ of the circumcircle, not containing $C$. It turns out that $\angle AIM = 90^\circ$. In what ratio does point $I$ divide the segment $CW$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 : 1}
Example. Evaluate the definite integral

$$
\int_{0}^{2} \frac{4 \sqrt{2-x}-\sqrt{2+x}}{(\sqrt{x+2}+4 \sqrt{2-x})(x+2)^{2}} d x
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\ln5}{16}}
There is a repeating decimal $0.2\dot{5}8\dot{7}$. What is the sum of the digits in the 1st, 99th, 199th, and 299th decimal places?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Suppose that $k \geq 2$ is a positive integer. An in-shuffle is performed on a list with $2 k$ items to produce a new list of $2 k$ items in the following way: - The first $k$ items from the original are placed in the odd positions of the new list in the same order as they appeared in the original list. - The remaining $k$ items from the original are placed in the even positions of the new list, in the same order as they appeared in the original list. For example, an in-shuffle performed on the list $P Q R S T U$ gives the new list $P S Q T R U$. A second in-shuffle now gives the list $P T S R Q U$. Ping has a list of the 66 integers from 1 to 66, arranged in increasing order. He performs 1000 in-shuffles on this list, recording the new list each time. In how many of these 1001 lists is the number 47 in the 24th position?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83}