inclusionAI/Ring-lite-rl-data huggingface dataset:
How many lattice points $(v, w, x, y, z)$ does a 5-sphere centered on the origin, with radius 3, contain on its surface or in its interior?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1343}
【Example 5】 There are $n$ young people and $r$ elderly people, $n>2r$, to be arranged in a row. It is required that each elderly person has a young person on both sides to support them (but each pair of young people only supports one elderly person). How many different arrangements are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n!\cdot(n-r)!}{(n-2r)!}}
Define $x \star y=\frac{\sqrt{x^{2}+3 x y+y^{2}-2 x-2 y+4}}{x y+4}$. Compute $$((\cdots((2007 \star 2006) \star 2005) \star \cdots) \star 1)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{15}}{9}}
In the $xOy$ plane, consider two lines $A_{1}B_{1}$ and $A_{2}B_{2}$ passing through a point $P$ on the $x$-axis, where the points $A_{1}$, $B_{1}$, $A_{2}$, and $B_{2}$ lie on the parabola $\Gamma: y^{2}=x$. The lines $A_{1}B_{2}$ and $A_{2}B_{1}$ intersect the $x$-axis at points $S$ and $T$, respectively. Determine whether the magnitude of $\overrightarrow{OS} \cdot \overrightarrow{OT}$ is greater than, less than, or equal to $|OP|^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Equal}
Circle $A$ has radius $100$. Circle $B$ has an integer radius $r<100$ and remains internally tangent to circle $A$ as it rolls once around the circumference of circle $A$. The two circles have the same points of tangency at the beginning and end of circle $B$'s trip. How many possible values can $r$ have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given that the divisor function $d(\phi(n)) = 2017$, where $d()$ counts the number of divisors and $\phi()$ is the totient function, find the smallest natural number $n$. This problem is from the 2017 Regional Mathematical Olympiad (RMO) in Maharashtra and Goa, India.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2017}}
Find the coefficient of \(x^{46}\) in the generating function \(\frac{1}{1 - x^3 - x^4 - x^{20}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3224}
Given the rate of population growth \( \frac{dP}{dt} = \frac{0.2P}{4(4-\frac{P}{400})} \), and an initial population of 3 at time \( t = 0 \), find the time \( t \) when the population \( P \) will reach 1200.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{419.47}
Compute the prime factorization of \(1007021035035021007001\). (You should write your answer in the form \(p_{1}^{e_{1}} p_{2}^{e_{2}} \ldots p_{k}^{e_{k}}\), where \(p_{1}, \ldots, p_{k}\) are distinct prime numbers and \(e_{1}, \ldots, e_{k}\) are positive integers.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7^7 \cdot 11^7 \cdot 13^7}
Determine the number of field extensions between the fields $\mathbb{Q}(\zeta^3)$ and $\mathbb{Q}(\zeta)$, where $\zeta$ is a primitive 12th-root of unity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of distinct cycle subgraphs of the complete bipartite graph $K_{3,3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
The number of real roots of the equation $\frac{x}{100} = \sin x$ is:  
(32nd United States of America Mathematical Olympiad, 1981)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
Let $h$ and $k$ be functions of $u$ and $v$ such that $h(2, 1) = 1$ and $k(2, 1) = 3$. At the point $(2, 1)$, the differentials are given by $dh = 3du - 2dv$ and $dk = -du + 4dv$. The function $w$ is defined implicitly by the equation $w^3h - wk + u + v = 0$, with the condition $w(2, 1) = 2$. Find the value of $\frac{\partial w}{\partial v}$ at the point $(2, 1)$. \(\boxed{\text{Answer}}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{23}{9}}
Approximately calculate the value of the function \( f(x) = \sqrt{x^{2} + x + 3} \) at \( x = 1.97 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.975}
Let $\omega_{1}, \omega_{2}, \ldots, \omega_{100}$ be the roots of $\frac{x^{101}-1}{x-1}$ (in some order). Consider the set $S=\left\{\omega_{1}^{1}, \omega_{2}^{2}, \omega_{3}^{3}, \ldots, \omega_{100}^{100}\right\}$. Let $M$ be the maximum possible number of unique values in $S$, and let $N$ be the minimum possible number of unique values in $S$. Find $M-N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
Given that the odd function $f(x)$ is a monotonically decreasing function on the interval $[-1,0]$, and $\alpha, \beta$ are acute angles of a triangle, determine the relationship between $f(\cos \alpha)$ and $f(\sin \beta)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(\cos \alpha)>f(\sin \beta)}
Determine the order of the image of the homomorphism \( \varphi: \mathbb{Z}^2 \to S_{15} \) given by \( \varphi(1,0)=(2,7,3)(3,11,5)(12,13) \) and \( \varphi(0,1)=(14,15)(1,4,6,8,9,10) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Evaluate the flux of the vector field \( F(x,y,z)=(z^2-x,-xy,3z) \) across the surface \( S \) of the solid bounded by the equations \( z=4-y^2, x=0, x=3 \), and \( z=0 \), with the normal vector pointing outward. Compute \( \iint_S F \cdot n \,ds \) without using the Divergence Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
What is the largest number of white and black chips that can be placed on a chessboard so that on each horizontal and each vertical, the number of white chips is exactly twice the number of black chips?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
The sine function $y = \sin x$ has an arc length from $0$ to $2\pi s$ given by:

$$ L = \int_0^{2\pi s} \sqrt{1 + \cos^2 x}\, dx $$

where $s \in \mathbb{Q}$ and $s > 1$. If the period of the sine function is changed from $2\pi$ to $\frac{4\pi}{3}$, determine the new amplitude of $\sin x$ (currently $|A|=1$) such that the arc length from $0$ to $2\pi s$ remains unchanged.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
In triangle $ABC$, medians $\overline{AM}$ and $\overline{BN}$ are perpendicular. If $AM = 15$ and $BN = 20$, and the height from $C$ to line $AB$ is $12$, find the length of side $AB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{50}{3}}
Determine the number of ordered pairs \((\mathcal{A}, \mathcal{B})\) of nonempty sets such that \(\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}\), \(\mathcal{A} \cap \mathcal{B} = \emptyset\), the number of elements in \(\mathcal{A}\) is not an element of \(\mathcal{A}\), and the number of elements in \(\mathcal{B}\) is not an element of \(\mathcal{B}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{772}
How many ordered pairs of sets \((A, B)\) have the properties:

1. \( A \subseteq \{1, 2, 3, 4, 5, 6\} \)
2. \( B \subseteq \{2, 3, 4, 5, 6, 7, 8\} \)
3. \( A \cap B \) has exactly \(3\) elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{720}
You are allowed to throw a standard die 10 times initially. Each time you roll a 6, you are allowed to throw 5 more times. Let N be the total number of throws. Calculate the expected value of \( E[N^2] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11100}
Given the price of Product A was set at 70 yuan per piece in the first year, with an annual sales volume of 118,000 pieces, starting from the second year, the price per piece increased by $$\frac {70\cdot x\%}{1-x\%}$$ yuan due to a management fee, and the annual sales volume decreased by $10^4x$ pieces, calculate the maximum value of x such that the management fee collected in the second year is not less than 1.4 million yuan.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The length of a rectangle is increased by $ 10\%$ and the width is decreased by $ 10\%$. What percent of the old area is the new area?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
Given the linear function \( y = ax + b \) and the hyperbolic function \( y = \frac{k}{x} \) (where \( k > 0 \)) intersect at points \( A \) and \( B \), with \( O \) being the origin. If the triangle \( \triangle OAB \) is an equilateral triangle with an area of \( \frac{2\sqrt{3}}{3} \), find the value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
Given a hyperbola with equation x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0), let the left and right foci be F1 and F2, respectively. A circle centered at F2 with radius |F1F2| intersects the hyperbola in the first and second quadrants at points A and B, respectively. If 3|F1B| = |F2A|, find the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
The random variable $X$ follows a normal distribution $N(1,4)$. Given that $P(X \geqslant 2) = 0.2$, calculate the probability that $0 \leqslant X \leqslant 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.3}
If m is an integer such that a certain number squared by 2m equals 2 ^ ( 18 - m ), and m equals 6, what is that certain number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that the radius of convergence of the power series \( \sum_{n=1}^{\infty}c_{n}z^{n} \) is finite and nonzero, find the radius of convergence of the series \( \sum_{n=1}^{\infty}c_{n}z^{n^{2}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Bill buys a stock for $100. On the first day, the stock decreases by $25\%$, on the second day it increases by $35\%$ from its value at the end of the first day, and on the third day, it decreases again by $15\%$. What is the overall percentage change in the stock's value over the three days?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-13.9375\%}
Determine the minimum number of charts required to cover a torus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The sum of 99 consecutive natural numbers equals $abcd$, where $a$, $b$, $c$, and $d$ are all prime numbers. What is the minimum value of $a+b+c+d$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Given that vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are perpendicular and $|\overrightarrow{OA}| = |\overrightarrow{OB}| = 24$. If $t \in [0,1]$, the minimum value of $|t \overrightarrow{AB} - \overrightarrow{AO}| + \left|\frac{5}{12} \overrightarrow{BO} - (1-t) \overrightarrow{BA}\right|$ is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
A doctor told Mikael to take a pill every 75 minutes. He took his first pill at 11:05. At what time did he take his fourth pill?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14:50}
What is the probability of never losing an infinite game where you start with a single four-sided die numbered 1 to 4? In each round, rolling a 1 loses the die, a 2 keeps the die, a 3 or 4 gains an additional die. You lose the game when you have no dice left.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
A function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have
\[3f\left(x\right) + f\left(\frac{1}{x}\right) = 6x + \sin x + 3\]
Let $S$ denote the sum of all of the values of $x$ for which $f(x) = 1001$. Compute the integer nearest to $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{445}
Compute
$$\sum^{\infty}_{m=1} \sum^{\infty}_{n=1} \frac{m\cos^2(n) + n \sin^2(m)}{3^{m+n}(m + n)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{8}$}
Let $S$ be the set of integers of the form $2^{x}+2^{y}+2^{z}$, where $x, y, z$ are pairwise distinct non-negative integers. Determine the 100th smallest element of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{577}
Determine whether the endomorphism $\phi: f(x) \mapsto x^2 f(x)$ has an infinite norm in the space $C(\mathbb{R})$ of continuous functions on $\mathbb{R}$, equipped with the norm $\|f\| = \sup_{x \in \mathbb{R}} |f(x)|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let real numbers \(x\) and \(y\) be such that \(x-y\), \(x^2-y^2\), and \(x^3-y^3\) are all prime numbers. Find the value of \(x-y\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The diagonals of a convex quadrilateral \(ABCD\) intersect at point \(E\). Given that \(AB = AD\) and \(CA\) is the bisector of angle \(C\), with \(\angle BAD = 140^\circ\) and \(\angle BEA = 110^\circ\), find angle \(CDB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Calculate the area enclosed between the curves \( r = \frac{1}{\phi} \) and \( r = \frac{1}{\sin\phi} \) for \( \phi \in (0, \frac{\pi}{2}) \) using polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{\pi}}
Danielle Bellatrix Robinson is organizing a poker tournament with 9 people. The tournament will have 4 rounds, and in each round the 9 players are split into 3 groups of 3. During the tournament, each player plays every other player exactly once. How many different ways can Danielle divide the 9 people into three groups in each round to satisfy these requirements?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20160}
Given the ellipse $\frac{x^{2}}{49}+ \frac{y^{2}}{24}=1$, if the angle between the lines connecting a point $P$ on the ellipse and the two foci $F\_1$ and $F\_2$ is a right angle, then $|PF\_1| \cdot |PF\_2| =$ _____ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Evaluate the integral: $$\int_{-\infty }^{\infty } \frac{-1+e^{-i k x}}{x^2} \, dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi |k|}
If both $a$ and $b$ in the fraction $\frac{a+b}{a^2+b^2}$ are enlarged by a factor of $2$, then calculate the new value of the fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Each side of the square grid is 15 toothpicks long. Calculate the total number of toothpicks used to construct the square grid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{480}
Given that \( x \) and \( y \) are positive numbers, determine the minimum value of \(\left(x+\frac{1}{y}\right)^{2}+\left(y+\frac{1}{2x}\right)^{2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 + 2 \sqrt{2}}
Let $a$ be the greatest natural number such that $4^{2016} + 4^a + 4^{37}$ is the square of a natural number. Determine $a \pmod{37}$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Suppose $A B C$ is a triangle with incircle $\omega$, and $\omega$ is tangent to $\overline{B C}$ and $\overline{C A}$ at $D$ and $E$ respectively. The bisectors of $\angle A$ and $\angle B$ intersect line $D E$ at $F$ and $G$ respectively, such that $B F=1$ and $F G=G A=6$. Compute the radius of $\omega$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 \sqrt{5}}{5}}
Lauren sent 65 pieces of mail on Monday. On Tuesday, she sent twice as many pieces as she sent on Monday, minus 5 pieces. On Wednesday, she sent 5 fewer pieces than twice the amount she sent on Tuesday, and on Thursday, she sent the square root of the number of pieces sent on Wednesday multiplied by 10. On Friday, Lauren sent the average number of pieces she sent on the previous days plus 15 pieces. How many pieces of mail did Lauren send in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{755}
If the maximum value of the function $f(x)=a^{x} (a > 0, a \neq 1)$ on $[-2,1]$ is $4$, and the minimum value is $m$, what is the value of $m$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Evaluate the integral \( \int\limits_{\mathbb{R}^n} -\Delta \Phi(x) \, dx \), where \( \Phi(x) = \frac{1}{\alpha(n)}((\lvert x\rvert^2 + 1)^{\frac{-n-2}{2}}\lvert x\rvert^2 + (\lvert x\rvert^2 + 1)^{-n/2}) \) and \( x \in \mathbb{R}^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
An isosceles triangle with a base of $\sqrt{2}$ has medians intersecting at a right angle. What is the area of this triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.5}
Evaluate the limit: $$\lim_{n\to \infty}{\left(\prod_{k=1}^{n}\left(1+\frac{2k-1}{2n}\right)\right)^{\frac{1}{n}}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{e}}
A line $l$ passing through the origin intersects the circle $x^2+y^2=1$ at points $P$ and $Q$. Point $A$ is the intersection of the circle and the negative semi-axis of the $x$-axis. The circle with $AQ$ as the diameter intersects line $l$ at point $N$, which is different from point $Q$. The product of the slopes of lines $AN$ and $AP$ is equal to 1. Determine the equation of line $l$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm \sqrt{3} x}
Let $x_0, x_1, \ldots$ be a sequence of real numbers such that $x_n = \frac{1 + x_{n -1}}{x_{n - 2}}$ for $n \geq 2$. Find the number of ordered pairs of positive integers $(x_0, x_1)$ such that the sequence gives $x_{2018} = \frac{1}{1000}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128}
Compute the sum of  $x^2+y^2$  over all four ordered pairs  $(x,y)$  of real numbers satisfying  $x=y^2-20$  and  $y=x^2+x-21$ .


*2021 CCA Math Bonanza Lightning Round #3.4*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{164}
If all $x_{k}$ ($k=1,2,3,4,5$) are positive reals, and {$a_{1},a_{2},a_{3},a_{4},a_{5}$} = {$1,2,3,4,5$}, find the maximum of
$$\frac{(\sqrt{s_{1}x_{1}}+\sqrt{s_{2}x_{2}}+\sqrt{s_{3}x_{3}}+\sqrt{s_{4}x_{4}}+\sqrt{s_{5}x_{5}})^{2}}{a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}+a_{4}x_{4}+a_{5}x_{5}}$$

($s_{k}=a_{1}+a_{2}+...+a_{k}$)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{117}{4}$}
In the Cartesian coordinate plane \(xOy\), given that points \(A\) and \(B\) lie on the hyperbola \(C: 2x^{2} + 4x - y^{2} = 0\), and \(\triangle ABC\) is an isosceles right triangle with \(O\) as the right-angle vertex. Find the total number of such \(\triangle OAB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given three positive real numbers $a, b, c$ satisfying $abc + a + c = b$, find the maximum value of the expression:
\[ P = \frac{2}{a^2 + 1} - \frac{2}{b^2 + 1} + \frac{3}{c^2 + 1}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{10}{3}}
Example 5 (2005 National High School Mathematics Competition Question) Define the function
$$f(k)=\left\{\begin{array}{l}
0, \text { if } k \text { is a perfect square } \\
{\left[\frac{1}{\{\sqrt{k}\}}\right], \text { if } k \text { is not a perfect square }}
\end{array} \text {, find } \sum_{k=1}^{240} f(k)\right. \text {. }$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{768}
Evaluate the contour integral \(\int_{|z|=1}\sin(1/z)\cos(1/z)\,dz\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
Suppose $T$ is a $C_0$-semigroup with $\omega \leq 0$ and $M \geq 0$ such that $\| T(t) \| \leq M e^{\omega t}$. Let $A$ be the generator for $T$ and let $\xi_1, ..., \xi_n > \omega$ be constants. Obtain a bound for $\| (\xi_1 \cdots \xi_n)^{-1} (\xi_1^{-1} - A)^{-1} \cdots (\xi_n^{-1} - A)^{-1} \|$ in terms of $M$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{M^n}
There are a total of 4 lamps in the classroom. Each is controlled by a separate switch. The total number of ways to turn on the lights in the classroom is $\_\_\_\_\_\_\_\_\_\_\_$ (Answer in digits).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
If there is a positive integer \( m \) such that the factorial of \( m \) has exactly \( n \) trailing zeros, then the positive integer \( n \) is called a "factorial trailing number." How many positive integers less than 1992 are non-"factorial trailing numbers"?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{396}
Consider the nth order Taylor polynomial for cos x centered at 0 denoted T(n) (x,0). How large must we take n to guarantee that the error |cos x-T(n) (x,0) |is at most 10^-3 for x in [-pi/2,pi/2)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the number of squares in the sequence given by $a_0 = 91$ and $a_{n+1} = 10a_n + (-1)^n$ for $n \ge 0.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the range of the function \( f \) defined for integers \( k \) by

\[ 
f(k) = (k)_{3} + (2k)_{5} + (3k)_{7} - 6k 
\]

where \( (k)_{2n+1} \) denotes the multiple of \( 2n+1 \) closest to \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-6, 6]}
Given a triangle with vertices A, B, and C, where random points (X, Y) are uniformly distributed along the line segments AB and AC, find the covariance of X and Y. The line segments AB and AC are defined as follows: AB consists of points (x, 0) for 0 ≤ x ≤ 1, and AC consists of points (0, y) for 0 ≤ y ≤ 1. Consider the distribution of XY first.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{16}}
Given the geometric series $6 - \frac{12}{5} + \frac{36}{25} - \dots$, calculate the limiting sum of this series as the number of terms increases without bound.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{30}{7}}
Find the Cauchy Principal Value of the integral \( \int_0^\infty \frac{x^{-p}}{x-1} \, dx \) for \(|p|<1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \cot(\pi p)}
In order to obtain steel for a specific purpose, the golden section method was used to determine the optimal addition amount of a specific chemical element. After several experiments, a good point on the optimal range $[1000, m]$ is in the ratio of 1618, find $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
Given a sample \(X_1, X_2, \ldots, X_n\) from a random variable \(X\) with probability density function (pdf) \(f(x|\theta) = \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{1}{2}(x-\theta)^2\right)\), find the 1-dimensional sufficient statistic for the parameter \(\theta\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sum_{i=1}^n X_i}
Given three iid Unif(0,1) random variables a, b, and c, what is the probability that c is the maximum if a+b < 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}