inclusionAI/Ring-lite-rl-data huggingface dataset:
A pixel is initially colored red, which corresponds to color number 5. A program recolors the pixel using the following rules: if the current color number \( n \leq 19 \), it changes to \( n+4 \); if \( n \geq 20 \), it changes to \( |129-2n| \). The program is applied 2019 times in succession. Determine the final color number of the pixel.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Evaluate the line integral \( \oint\limits_C \ \frac{x-y}{x^2+y^2}dx+\frac{x+y}{x^2+y^2}dy \) over the closed path \( C \), where \( C \) is formed by the line \( x=2 \) and the parabola \( y^2=2(x+2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Let \( \phi: G \rightarrow H \) be a homomorphism and let \( g \in G \) such that \( h = \phi(g) \). Suppose \( o(h) = 100 \) and \( g \) has finite order. What are the possible values of \( o(g) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100k}
Find the fundamental group of a cube with 12 points removed, where each point is located at the center of each face.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F_{11}}
Evaluate the integral \( \int_{-\infty}^\infty \frac{\sin x}{x-i} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{e}}
A uniform solid semi-circular disk of radius $R$ and negligible thickness rests on its diameter. It is then tipped over by some angle $\gamma$ with respect to the table. Determine the minimum angle $\gamma$ at which the disk will lose balance and tumble over. Express your answer in degrees, rounded to the nearest integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Evaluate the contour integral \( \oint_{|z|=2} \frac{f'(z)}{f(z)} \, dz \) for the function \( f(z) = \frac{2z^4 - 7z^3 - 2z - i}{(z-i)^2(3z^2 - z + 9)^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-6\pi i}
Calculate the derivative \( \frac{\partial (ABC)}{\partial B} \) where \( A \), \( B \), and \( C \) are matrices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{C^\top \otimes A}
Find a function that is integrable over any finite interval but whose limit as \( x \) approaches infinity does not exist.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin(x)}
Determine the smallest cardinal number \( \kappa \) such that the interval \([0, \omega_1)\) can be embedded into the Euclidean space \( \mathbb{R}^\kappa \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_1}
Suppose $P(x) = 2x^8 + x^6 - x^4 +1$, and that $P$ has roots $a_1$, $a_2$, $...$ , $a_8$ (a complex number $z$ is a root of the polynomial $P(x)$ if $P(z) = 0$). Find the value of
p10. 假设 $P(x) = 2x^8 + x^6 - x^4 +1$ ，并且 $P$ 有根 $a_1$ ， $a_2$ ， $...$ ， $a_8$ （一个复数 $z$ 是多项式 $P(x)$ 的根当且仅当 $P(z) = 0$ ）。求值$$(a^2_1-2)(a^2_2-2)(a^2_3-2)...(a^2_8-2).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{4}(37)^2$}
Let $S$ be the surface obtained by rotating the curve $(x-2)^2+z^2=1$ around the $z$-axis. Compute the surface integral $$\int_{S}\mathbf{F\cdot n } \, dA$$ where $$\mathbf{F}=(x+\sin(yz), y+e^{x+z}, z-x^2\cos(y)).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12\pi^2}
Evaluate the triple integral \( \int_{1}^{2}\int_{1}^{2}\int_{1}^{2}\frac{x^2}{x^2+y^2+z^2}dxdydz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Suppose we have $n$ real numbers $a_1,\ldots,a_n$ where $|a_i| \geq 1$ for all $i$. If we randomly choose $2^n$ vectors $v_1,\ldots,v_{2^n}$ such that $v_i \in \{-1,1\}$, what is the probability that at least one of the vectors satisfies the condition $-1 < \sum_{i=1}^n v_i \cdot a_i < 1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Decompose the symmetric polynomial \( \Sigma{x_1^2x_2^2x_3^2} \) into elementary symmetric polynomials.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sigma_3^2}
Find the radius of convergence of the function \( \frac{z-z^3}{\sin{\pi z}} \) at \( z=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a bank, people arrive according to a Poisson process with a rate of \( \lambda = 1 \) client per minute. Each client makes a deposit of \( Y \sim \mathrm{Unif}\{1,2\} \) thousand dollars. Calculate the probability that at time \( t=5 \) minutes, the total deposit is exactly 6 thousand dollars.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{154625}{9216} e^{-5}}
Determine the best known bound on the index of an abelian subgroup in a discrete subgroup of the group of isometries in Euclidean space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n n!}
Find the value of \( f'(0) \) for the function \( f(x) = \int_{0}^{x} \cos\left(\frac{1}{t}\right) dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find a basis for the vector space V over R, where addition and scalar multiplication are defined as follows: \((a, 1 + a) \oplus (b, 1 + b) = (a + b, 1 + a + b)\) and \(k \ast (a, 1 + a) = (ka, 1 + ka)\), for \(k \in R\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{(1, 2)\}}
Evaluate the Riemann-Stieltjes integral \( \int_{1}^{5} (x-1)~df(x) \) where \( f(x)= \begin{cases} x, & \text{for $1\leq x\leq 3,$}\\ 3, & \text{for $3<x<4,$}\\ x^2, & \text{for $4\leq x\leq 5.$} \end{cases} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{218}{3}}
Determine the number of standard ways to number the faces of a regular polyhedron die such that the sum of the numbers on opposite faces is constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a positive integer $n \ge 3$. Find the smallest real number $k$ such that for any positive real number except $a_1, a_2,..,a_n$,$$\sum_{i=1}^{n-1}\frac{a_i}{ s-a_i}+\frac{ka_n}{s-a_n} \ge \frac{n-1}{n-2}$$where, $s=a_1+a_2+..+a_n$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$$k_{min}=\left(\frac{n-1}{n-2}\right)^2$$
}
Find the natural numbers \( n \) for which the integral \( \int_{0}^{\infty} \cos x \cdot \cos 2x \cdot \cos 3x \cdots \cos nx \cdot \frac{\sin 2020x}{x} \, \operatorname{d}x = \frac{\pi}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
How many positive integers $N$ satisfy all of the following three conditions?
(i) $N$ is divisible by 2020 .
(ii) $N$ has at most 2020 decimal digits.
(iii) The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{508536}
In the secretary problem with 500 numbers, determine the optimal number of numbers to observe before selecting the next largest number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{184}
A circular track with diameter $500$ is externally tangent at a point $A$ to a second circular track with diameter $1700$. Two runners start at point $A$ at the same time and run at the same speed. The first runner runs clockwise along the smaller track while the second runner runs clockwise along the larger track. There is a first time after they begin running when their two positions are collinear with the point $A$. At that time, each runner will have run a distance of $\frac{m\pi}{n}$ 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{2128}
Determine the relationship between the prime factor \( P \) and the smallest factor of \( M \) when \( M = 2^P - 1 \) and \( M \) is composite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{q \equiv 1 \pmod{2P}}
For which values of \( x \) is the fourth term the largest in the expansion of \( (5 + 3x)^{10} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x \in \left( \frac{5}{8}, \frac{20}{21} \right)}
Calculate the Galois group \( \text{Gal}(L/\mathbb{Q}) \), where \( L \) is the splitting field of the polynomial \( f(x) = x^4 - 6x^2 + 4 \) over \( \mathbb{Q} \). The zeros of \( f \) are \( x_{1,2} = \pm\sqrt{3+\sqrt{5}} \) and \( x_{3,4} = \pm\sqrt{3-\sqrt{5}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{V_4}
Let $T_1, T_2 : \mathbb{R}^5 \to \mathbb{R}^3$ be linear transformations such that $\text{rank}(T_1) = 3$ and $\text{nullity}(T_2) = 3$. Let $T_3 : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation such that $T_3 \circ T_1 = T_2$. Find the rank of $T_3$. \(\text{Note: Use the rank-nullity theorem and properties of linear transformations.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
From the $8$ vertices of a cube, select any $4$ vertices. The probability that these $4$ points lie in the same plane is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{6}{35}"]}
Given nonnegative real numbers $x, y, z,$ and $t$ such that $|x-y| + |y-z| + |z-t| + |t-x| = 4$, find the minimum value of $x^2 + y^2 + z^2 + t^2$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $H$ and $K$ be subgroups of a group $G$. If every left coset of $H$ is contained in some right coset of $K$, and every right coset of $K$ is contained in some left coset of $H$, determine the relationship between $H$ and $K$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H = K}
In a set of 25 elements, determine the minimum number of subsets of size 5 needed to ensure that every possible subset of size 3 is included in at least one of these subsets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
7.5. Find the volume of a regular triangular pyramid, the midpoint of the height of which is at a distance of 2 and $\sqrt{6}$ from the lateral face and the lateral edge, respectively. Round your answer to two decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{334.63}
Joe, Susie's brother, collected all 94 trading cards scattered in his room and decided to sort them by their types - magic, rare, and common. He had 33 magic cards, 28 rare cards, and 33 common cards. If he can only place a maximum of 8 magic cards, 10 rare cards, and 12 common cards per box, how many boxes were fully filled?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Consider a power series $f(z)$ with radius of convergence $R > 0$. If $f(z)$ has infinitely many distinct zeros that converge to a point $z_0$ within the radius of convergence, what can we conclude about $f(z)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(z) = 0}
There are eight envelopes numbered 1 to 8. Find the number of ways in which 4 identical red buttons and 4 identical blue buttons can be put in the envelopes such that each envelope contains exactly one button, and the sum of the numbers on the envelopes containing the red buttons is more than the sum of the numbers on the envelopes containing the blue buttons.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Mary is baking a cake. The recipe calls for some cups of sugar, 6 cups of flour, and 9 cups of salt. She already put in 12 cups of flour. She needs to add 2 more cups of sugar than cups of salt now. How many cups of sugar does the recipe call for?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A perpendicular dropped from the vertex $C$ of parallelogram $A B C D$ to the line $C D$ intersects at point $F$ a perpendicular dropped from vertex $A$ to the diagonal $B D$. A perpendicular dropped from point $B$ to the line $A B$ intersects at point $E$ the perpendicular bisector of segment $A C$. In what ratio does segment $E F$ divide side $B C$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:2}
A jug is filled with 5 liters of water and a bucket is filled with 4 jugs. A barrel can hold 3 buckets and a gallon is equal to 3.79 liters. If you have 1 barrel and 2 buckets of water, and you want to find out the total amount of water in gallons, how many gallons of water do you have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26.39}
Evaluate the integral \( \int_\gamma x \cdot n(x) \, ds(x) \), where \( x = (x, y) \in \mathbb{R}^2 \), \( n(x) \) is the unit outward normal to the ellipse \( \gamma \) given by \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12\pi}
Four fair six-sided dice are rolled. What is the probability that they can be divided into two pairs which sum to the same value? For example, a roll of $(1,4,6,3)$ can be divided into $(1,6)$ and $(4,3)$, each of which sum to $7$, but a roll of $(1,1,5,2)$ cannot be divided into two pairs that sum to the same value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{7}{27}$}
Calculate the surface integral of the portion of the paraboloid \(x^2+y^2=2ay\) that is cut by the plane \(y=a\). Express your answer in terms of \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\pi a^2}{3} \left(3\sqrt{3} - 1\right)}
Determine the values of $n$ for which the equation $x^2 - (2n+1)xy + y^2 + x = 0$ has no solution in natural numbers. Note that for $n=1$, the equation has a trivial solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 2}
Compute the sixth derivative of the function \( F(x) = \int_1^x \sin^3(1-t)\, dt \) at \( x_0 = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
If each point \((x,y)\) in the plane, where both coordinates are rational numbers, represents a tree, how far can you see if you are standing at the point \((0,0)\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Leticia has a  $9\times 9$  board. She says that two squares are *friends* is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has  $4$  friends on the board. Leticia will paint every square one of three colors: green, blue or red. In each square a number will be written based on the following rules:

 - If the square is green, write the number of red friends plus twice the number of blue friends.

 - If the square is red, write the number of blue friends plus twice the number of green friends.

 - If the square is blue, write the number of green friends plus twice the number of red friends.

Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{486}
Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random
 to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in
 appearance?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( \frac{147}{1024} \)}
Three players, A, B, and C, are playing a game where the highest score wins. A's score follows a Poisson distribution with parameter 2, B's score follows a Poisson distribution with parameter 3, and C's score follows a Poisson distribution with parameter 4. The scores are independent. Given that the scores obtained were 1, 2, and 3, what is the conditional probability that player C won?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{40}{81}}
Given that the real numbers $x$ and $y$ satisfy the equation $4x^{2}-2 \sqrt {3}xy+4y^{2}=13$, determine the range of values for $x^{2}+4y^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[10-4 \sqrt {3},10+4 \sqrt {3}]}
A permutation of a finite set \( S \) is a one-to-one function from
 \( S \) to \( S \) . Given a permutation \( f \) of the set \( \{ 1,2,\ldots ,{100}\} \) ,
 define the displacement of \( f \) to be the sum
 \( \mathop{\sum }\limits_{{i = 1}}^{{100}}\left| {f\left( i\right)  - i}\right| . \)
 How many permutations of \( \{ 1,2,\ldots ,{100}\} \) have
 displacement 4 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
5047
}
What is $\log_{7}{2400}$ rounded to the nearest integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Consider a $4 \times 4$ grid of squares. Aziraphale and Crowley play a game on this grid, alternating turns, with Aziraphale going first. On Aziraphales turn, he may color any uncolored square red, and on Crowleys turn, he may color any uncolored square blue. The game ends when all the squares are colored, and Aziraphales score is the area of the largest closed region that is entirely red. If Aziraphale wishes to maximize his score, Crowley wishes to minimize it, and both players play optimally, what will Aziraphales score be?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\[ 6 \]}
How many digits could possibly be the last digit of a perfect square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
6
}
Let the function $f(x)= \frac {1-a}{2}x^{2}+ax-\ln x$ where $(a > 1)$. If for any $a\in(3,4)$ and any $x_{1}, x_{2}\in[1,2]$, it always holds that $\frac {a^{2}-1}{2}m+\ln 2 > |f(x_{1})-f(x_{2})|$, then the range of values for the real number $m$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m\geqslant \frac {1}{15}}
In the parallelogram \(ABCD\), point \(E\) is on side \(AB\) such that \(\frac{\mathrm{AE}}{\mathrm{EB}}=\frac{1}{4}\), and point \(F\) is on side \(DC\). Let \(G\) be the intersection of lines \(AF\) and \(DE\), and \(H\) be the intersection of lines \(CE\) and \(BF\). If the area of parallelogram \(ABCD\) is 1 and the area of triangle \(BHC\) is \(\frac{1}{8}\), determine the area of triangle \(ADG\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{92}}
Evaluate the limit \( \lim_{{x\to\infty}}\frac{\sin^2\left(\sqrt{x+1}-\sqrt{x}\right)}{1-\cos^2\left(\frac{1}{x}\right)} \) using the hint that \( \lim_{{x\to\infty}}\frac{\sin(x)}{x} = 0 \). Simplify the expression using trigonometric identities.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In a regular tetrahedron \(ABCD\) with edge length 2, a point \(P\) is located on edge \(AB\) such that \(AP < 1\). A cut is made through point \(P\) perpendicular to edge \(AB\) and extends through the faces of the tetrahedron. The cut stops when the length of the cut on face \(ABD\) is \(PM = 1\) and the length of the cut on face \(ABC\) is \(PN = \frac{2}{3}\). Determine the length of segment \(MN\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the maximum value of the real number \( t \) such that the inequality \[ \sum_{i=1}^n a_i \sum_{i=1}^n \frac{1}{a_i} \ge n^2 + t \cdot \frac{\displaystyle\sum_{1\le i<j \le n} (a_i - a_j)^2 }{\left( \displaystyle\sum_{i=1}^n a_i\right)^2} \] holds for all positive numbers \( a_i \) where \( n \ge 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the integral \( K = \int_0^4 \frac{\ln (4x-x^2)}{\sqrt{4x-x^2}} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Form an eight-digit number, using only the digits $1,2,3,4$ each twice, so that: there is one digit between the $1$'s, there are two digits between the $2$'s, there are three digits between the $3$'s, and there are four digits between the $4$'s.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23421314}
What is the sum of the infinite series $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{2(n+1)(2n+1)}$$? Attempts using the telescoping method have not led to a simplified form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{4}-\frac{1}{2}\ln 2\)}
Determine the values of the parameter \( \alpha \in \mathbb{R} \) for which the function \( f: (0,1) \rightarrow \mathbb{R} \) defined by \( f(x) = \frac{x-1}{x^{\alpha} \ln(x)} \) is integrable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1)}
Evaluate the integral \( J = \int^{2\pi}_{0} e^{\cos x} \cos(2x - \sin x)\, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Determine the kernel of the ring homomorphism $R[T] \to R[w]$ defined by $T \mapsto w$, where $w = u/v$ with $u, v \in R$ and $R$ is an integrally closed domain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(vT - u)}
Calculate the surface area of the portion of the sphere defined by \(x^2 + y^2 + z^2 = a^2\) that lies inside the circular cylinder given by \(x^2 + y^2 = ay\). Assume \(a > 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2a^2(\pi - 2)}
In a dark room, there are four pairs of socks in disarray on a shelf, consisting of two different sizes and two different colors. What is the minimum number of socks that need to be moved from the shelf to a suitcase to ensure that there are two pairs of different sizes and colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let $G = H \times K$ be a nonabelian group where $o(H) = p^2$ and $o(K) = p^3$, with $p$ being a prime number. Determine the order of the commutator subgroup $G'$. \( \text{(Express your answer as a power of } p \text{.)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p}
求最小的实数 \lambda，使得对任意满足 \ds{i=1}{20}ix_i=0 的实数 x_1, x_2, \cdots, x_{20}，都有\left|\ds{i=1}{20}i^2x_i\right|\leq\lambda\displaystyle{\max_{1\leq i\leq 20}|x_i|}.（张端阳供题）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{840}
A random variable $X$ has the cumulative distribution function $$F(x)=\begin{cases} 0 & x<1\\\frac{x^2-2x+2}{2}&x\in[1,2)\\1&x\geq2\end{cases}.$$ Calculate the variance of $X$. \( \boxed{\text{Variance}} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{36}}
Given a regular tetrahedron \( P-ABC \) with a base edge length of 1 and a height of \( \sqrt{2} \), find the area of the shape obtained by intersecting the sphere centered at the incenter of the tetrahedron with radius \( 2r \) with the base triangle \( \triangle ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4} + \frac{\pi}{24}}
In the figure, lines \(AB\) and \(AC\) are tangents to circle \(\odot O\) at points \(B\) and \(C\), respectively. Point \(P\) is a moving point on arc \(\overparen{BC}\). Line \(PD\) is perpendicular to \(AB\) at point \(D\), and line \(PE\) is perpendicular to \(AC\) at point \(E\). Connect \(DE\) and \(BC\). Given that the radius of \(\odot O\) is 5 and the length of chord \(BC\) is 8, find the maximum area of \(\triangle PDE\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{48}{25}}