inclusionAI/Ring-lite-rl-data huggingface dataset:
K2) Let $n$ be a natural number. A volleyball team consisting of $n$ women and $n$ men lines up for a game. Each team member occupies one of the positions $1,2, \ldots, 2 n$, whereby exactly positions 1 and $n+1$ are outside the court. During the game, all team members rotate, switching from position $i$ to position $i+1$ (or from $2 n$ to 1). How many possibilities are there for the starting line-up so that there are always at least $n-1$ women on the pitch, no matter how often the team rotates?

Note: Two starting line-ups are different if at least one team member occupies a different position.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n!)^{2}\cdot2^{n}}
A 250 m long train is running at a speed of 55 km/hr. It crossed a platform of a certain length in 35.99712023038157 seconds. What is the length of the platform?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Let  $a,b, c$  denote the real numbers such that  $1 \le  a, b, c\le  2$ . 
Consider  $T = (a - b)^{2018} + (b - c)^{2018} + (c - a)^{2018}$ .
Determine the largest possible value of  $T$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Joe has written 5 questions of different difficulties for a test, numbered 1 through 5. He wants to make sure that problem \(i\) is harder than problem \(j\) whenever \(i - j \geq 3\). In how many ways can he order the problems for his test?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
The parameter of a square is equal to the perimeter of a rectangle of some length and breadth 10 cm. The circumference of a semicircle whose diameter is equal to the side of the square is 21.99 cm (rounded to two decimal places). What is the length of the rectangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.1}
Given the line $l$: $y=2x$, a point $P$ on it forms a circle $C$: $(x-8)^{2}+(y-1)^{2}=2$. The tangents $l_{1}$, $l_{2}$ from point $P$ to the circle are symmetric about line $l$. Find the distance from point $P$ to the center of circle $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{5}}
I5.2 If $1-\frac{4}{x}+\frac{4}{x^{2}}=0, b=\frac{a}{x}$, find $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of different Euclidean divisions possible for a given pair of elements in the Gaussian integers, $\mathbb{Z}[i]$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
David has a store where he sells cell phones. When he takes inventory at the end of the day, he notices that he has 10 Samsung cell phones but he started the day with 14 Samsung cell phones. Then he notices that he has 5 iPhones left, but started with 8. His staff then told him that in the afternoon they realized that 2 Samsung cell phones were damaged and 1 iPhone had a manufacturing defect in the screen, so they threw these out during the day. 

During the day, there are two types of discounts offered on each phone: a 10% discount on Samsung cell phones and a 15% discount on iPhones. The tax rates applied on the final price were different for Samsung and iPhones at 12% and 10%, respectively.

Calculate the total revenue generated from the sales, considering discounts and taxes, if the retail price of a Samsung cell phone is $800, and an iPhone is $1,000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$3482.80}
Evaluate the integral \( \oint_C \frac{dz}{z-2} \) where \( C \) is the square with vertices at \( \pm 2 \pm 2i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi i}
How many four-digit numbers are there in which no two adjacent digits are the same?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6561}
Find all prime numbers $p$ such that the number of distinct positive factors of $p^2 + 2543$ is less than 16.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the genus of the complete 4-partite graph $K_{4,2,2,2}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the order of the image of the homomorphism $\varphi: \mathbb{Z}^2 \to S_{15}$ defined 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}
An ant is 40 ft away from its nest, which is a large wall. Due to rain, its scent trail is erased, and it can travel a maximum of 280 ft more before starving. Determine the shortest path the ant can take to return to the nest, considering the nest as a point on a wall and the ant's starting point as a point on a circle with a radius of 40 ft.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
If an integer is divisible by $6$ and the sum of its last two digits is $15$, then what is the product of its last two digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
A pedestrian traffic light allows pedestrians to cross the street for one minute and prohibits crossing for two minutes. Find the average waiting time for a pedestrian who approaches the intersection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Determine the cardinality of the set $S$, which consists of all infinite increasing sequences of binary strings. An infinite increasing sequence of binary strings is a series of finite binary strings $\{w_1, w_2, \dots\}$ where each string $w_i$ is a prefix of the next, i.e., $w_i$ is a substring at the beginning of $w_{i+1}$. For example, the sequence "101", "$10100$", "$101001$", "$1010010111$", \dots belongs to $S$. What is the cardinality of $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Given natural numbers \( a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} \) such that \( a_{1} + a_{2} + a_{3} = b_{1} + b_{2} + b_{3} \), \( a_{1}a_{2}a_{3} = b_{1}b_{2}b_{3} \), and \( a_{1}a_{2} + a_{1}a_{3} + a_{2}a_{3} \neq b_{1}b_{2} + b_{1}b_{3} + b_{2}b_{3} \). Find the minimal value of the largest number among \( a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let \(\mathcal{F}\) be the family of all nonempty finite subsets of \(\mathbb{N} \cup \{0\}\). Find all positive real numbers \(a\) for which the series

\[ 
\sum_{A \in \mathcal{F}} \frac{1}{\sum_{k \in A} a^{k}} 
\]

is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > 2}
Determine the irreducible representations of the tensor product $\varphi_2 \otimes \varphi_3$ for the $SU(2)$ group, where $\varphi_n$ denotes the standard irreducible representation with highest weight $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\varphi_1 \oplus \varphi_3 \oplus \varphi_5}
An elementary school teacher is making Halloween goodie bags for her class. She wants the bags to be personalized, so she surveys her students asking whether they'd like a vampire-themed bag or a pumpkin-themed bag. Of her 25 students, 11 indicate they want the vampire-themed bag and 14 indicate they want the pumpkin-themed bag. The store the teacher shops at sells packs of 5 of each theme at a price of $3 per package, as well as individual bags of each theme at a price of $1 each. Additionally, the store offers a "Buy 3 packages, Get 1 package Free" discount to encourage bulk purchase. What is the minimum amount of money the teacher can spend on the bags if she buys every student the theme they requested while taking advantage of the discount offer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$13$}
Evaluate the integral \( \iint_D (3x^3 + 3x^2y + x + y) \, dA \) over the region \( D \) bounded by the curves \( y = x^3 - 4 \), \( y = x^3 + 4 \), \( x + y = 6 \), and \( x + y = 0 \), using a suitable change of variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
A father wants to distribute his estate to his children: first giving 1000 yuan and $\frac{1}{10}$ of the remaining estate to the eldest child, then 2000 yuan and $\frac{1}{10}$ of the remaining estate to the second child, next 3000 yuan and $\frac{1}{10}$ of the remaining estate to the third child, and so on. It turns out that each child received the same amount of property. How many children does the father have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the number of matrices in the group $SL_3(\mathbb{Z}_2)$ by using the method of counting bases.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
A wall contains three switches $A$, $B$, and $C$, each of which powers a light when flipped on. Every $20$ seconds, switch $A$ is turned on and then immediately turned off again. The same occurs for switch $B$ every $21$ seconds and switch $C$ every $22$ seconds. At time $t = 0$, all three switches are simultaneously on. Let $t = T > 0$ be the earliest time that all three switches are once again simultaneously on. Compute the number of times $t > 0$ before $T$ when at least two switches are simultaneously on.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
In a contest with \( n \) questions taken by \( m \) contestants, each question is worth a certain positive number of points, and no partial credits are given. After grading, it was observed that by reassigning the scores of the questions, any desired ranking of the contestants could be achieved. What is the largest possible value of \( m \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Evaluate the integral \( \int_{0}^{2\pi} e^{\sin(t)} \sin(\cos(t) + 5t) \, dt \) using complex analysis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{60}}
设正整数$a, b, c, d$满足$a>b>c>d$，且
$$ab+bc+ca+d^{2}\mid(a+b)(b+c)(c+a)$$
求$ab+bc+ca+d^{2}$的素因子个数的最小值。（素因子的个数计重数，如12的素因子个数为3）。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the greatest common factor (GCF) of the polynomials \( P(x) = x^{50} - 2x^2 - 1 \) and \( Q(x) = x^{48} - 3x^2 - 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 + 1}
Calculate the limit $$\lim_{n\rightarrow\infty} \sum_{k=1}^{n}\cos\left( \frac{2\pi k}{2n+1} \right).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2}}
Given four points \( A, B, C, D \) on the surface of a sphere with a radius of 2, and that \( AB = CD = 2 \), determine the maximum volume of the tetrahedron \( ABCD \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4 \sqrt{3}}{3}}
Given ${(2x+\frac{a}{x})}^{5}$, the coefficient of $x$ in its expansion is $80$. Find the value of $a$. Express your answer as a numerical value, including both positive and negative possibilities.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm 1}
Let  $S$  be the set of all lattice points  $(x, y)$  in the plane satisfying  $|x|+|y|\le 10$ . Let  $P_1,P_2,\ldots,P_{2013}$  be a sequence of 2013 (not necessarily distinct) points such that for every point  $Q$  in  $S$ , there exists at least one index  $i$  such that  $1\le i\le 2013$  and  $P_i = Q$ . Suppose that the minimum possible value of  $|P_1P_2|+|P_2P_3|+\cdots+|P_{2012}P_{2013}|$  can be expressed in the form  $a+b\sqrt{c}$ , where  $a,b,c$  are positive integers and  $c$  is not divisible by the square of any prime. Find  $a+b+c$ . (A *lattice point* is a point with all integer coordinates.)
<details><summary>Clarifications</summary>

- $k = 2013$ , i.e. the problem should read, ``... there exists at least one index  $i$  such that  $1\le i\le 2013$  ...''.  An earlier version of the test read  $1 \le i \le k$ .
</details>

*Anderson Wang*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{222}
Find the number of ordered triples of complex numbers $(a, b, c)$ for which $a^3 - b$, $b^3 - c$, and $c^3 - a$ are all rational numbers, and the following equation holds:

\[ a^2(a^4 + 1) + b^2(b^4 + 1) + c^2(c^4 + 1) = 2\left[(a^3)b + (b^3)c + (c^3)a\right] \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the difference between the sum of the first one hundred positive even integers and the sum of the first one hundred positive multiples of 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5050}
Evaluate the surface integral \( \iint_{S} \vec{F} \cdot d\vec{S} \) to find the flux of the vector field \( \vec{F}(x,y,z) = y\mathbf{j} - z\mathbf{k} \) across the surface \( S \). The surface \( S \) consists of the paraboloid \( y = x^2 + z^2, 0 \leq y \leq 1, \) and the disk \( x^2 + z^2 \leq 1, y=1 \). Use the positive (outward) orientation for closed surfaces.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the rank of the symmetric bilinear form \( f(A,B) = \operatorname{tr}(A) \operatorname{tr}(B) - n\operatorname{tr}(AB) \) for square matrices \( A \) and \( B \) of size \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2 - 1}
Find the index of the module $\mathbb{Z}[\sqrt{7}]$ as a submodule of $\mathbb{Z}\left[\frac{1+\sqrt{7}}{2}\right]$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the value of the integral ratio:

$$\frac{\int_0^\pi x^3\ln(\sin x)\,dx}{\int_0^\pi x^2\ln(\sqrt{2}(\sin x))\,dx}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{2}}
Compute the number of ordered quadruples of complex numbers $(a,b,c,d)$ such that \[(ax+by)^3 + (cx+dy)^3 = x^3 + y^3\] holds for all complex numbers $x, y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
If the zero of the function $f(x) = -|x - 5| + 2x - 1$ lies in the interval $(k, k + 1)$, then the integer $k = \_\_\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the infinite series \( \sum_{n=1}^{\infty}{\frac{\prod_{r=0}^{n-1}{(\pi-r})}{2^nn!}} \) and express it in the form \( \left(\frac{A}{B}\right)^{\pi}-C \), where \(A\), \(B\), and \(C\) are coprime integers. Find the value of \(A+B-C\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Two people start on opposite vertices of an octahedron. Each turn, they move to a random adjacent vertex. What is the probability that the person who moves first will land on the vertex occupied by the other person and win the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{5}}
Find the gradient of the quadratic form $z^TX^TXz$ with respect to the vector $z$, where $X$ is a matrix in $\mathbb{R}^{m \times n}$ and $z$ is a vector in $\mathbb{R}^{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 X^T X z}
In a Cartesian coordinate system, you can move from point \((a,b)\) to \((a+1,b)\), \((a,b+1)\), or \((a+1,b+1)\). However, you must ensure that no right angles are formed when drawing lines during the move. Determine the number of ways to move from \((0,0)\) to \((5,5)\) under these conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83}
令 $f(x)=[x]+[2 x]+[3 x]+[4 x]+[5 x]$, 当 $x$ 取遍 $[0,50]$ 中的实数时, $f(x)$ 可取到 $\qquad$个不同的整数值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{501}
Given a random sample $Y_1, Y_2, \ldots, Y_{10}$ from a normal distribution $Y \sim N(\mu, \sigma^2)$, consider the statistic:
\[ U^2 = \frac{1}{9} \sum_{i=1}^{10} (Y_i - \bar{Y})^2 \]
where $U^2$ is the empirical variance. Determine the distribution of the statistic \( \frac{9U^2}{\sigma^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\chi^2(9)}
Let $A_1A_2A_3\cdots A_{12}$ be a dodecagon ($12$-gon). Three frogs initially sit at $A_4,A_8,$ and $A_{12}$. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping 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{19}
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $1+ \dfrac {\tan A}{\tan B}+ \dfrac {2c}{b}=0$, then $A=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac {2\pi}{3}}
Two teachers and 4 students need to be divided into 2 groups, each consisting of 1 teacher and 2 students. Calculate the number of different arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The sequence $x_1, x_2, x_3, . . .$ is defined by $x_1 = 2022$ and $x_{n+1}= 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that $$\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!}$$ is never a multiple of $7$ for any positive integer $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{404}
42. Factorize the number $989 \cdot 1001 \cdot 1007 + 320$ into prime factors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{991\cdot997\cdot1009}
In the number $2016^{* * * *} 02 * *$, each of the six asterisks must be replaced with any of the digits $0, 2, 4, 5, 7, 9$ (digits may be repeated) so that the resulting 12-digit number is divisible by 15. How many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5184}
Let us define a staircase of height \( n \) as a figure consisting of all the cells of an \( n \times n \) square that lie not above the diagonal. How many different ways are there to partition a staircase of height \( n \) into several rectangles, with sides along the grid lines, and areas pairwise distinct?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-1}}
Let \(ABCD\) be a quadrilateral inscribed in the unit circle such that \(\angle BAD\) is 30 degrees. Let \(m\) denote the minimum value of \(CP+PQ+CQ\), where \(P\) and \(Q\) may be any points lying along rays \(AB\) and \(AD\), respectively. Determine the maximum value of \(m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the residue of the function \( f(z) = \frac{\sin z}{z^3} \) at \( z = 0 \) using the formula for residues of poles of higher order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Manya has a stack of $85=1+4+16+64$ blocks comprised of 4 layers (the $k$ th layer from the top has $4^{k-1}$ blocks). Each block rests on 4 smaller blocks, each with dimensions half those of the larger block. Laura removes blocks one at a time from this stack, removing only blocks that currently have no blocks on top of them. Find the number of ways Laura can remove precisely 5 blocks from Manya's stack (the order in which they are removed matters).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3384}
Find the negation of the proposition "∀x > 0, x + 1/x ≥ 2". Express your answer as a logical statement with the correct quantifier and inequality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{∃x > 0, x + 1/x < 2}
Given the function y = (x - 16)(x - 14)(x + 14)(x + 16), find the minimum value of y. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-900}
Given that the solution set of the inequality $ax^2 + bx + 3 > 0$ regarding $x$ is $(-1, \frac{1}{2})$, where $a$ and $b$ are constants, find the solution set of the inequality $3x^2 + bx + a < 0$. Express your answer in interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 2)}
Anastasia's tablecloth has a regular pattern. The tablecloth can be divided into 25 equal squares, with half of each of the 16 outer squares colored black. Find the percentage of Anastasia's tablecloth that is black. Express your answer as a percentage.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Given that \( T_1 \) is an \( \alpha \times 100\% \) lower confidence limit for \( \theta \) and \( T_2 \) is an \( \alpha \times 100\% \) upper confidence limit for \( \theta \), with the condition \( P(T_1 < T_2) = 1 \), find a \( (2\alpha - 1) \times 100\% \) confidence interval for \( \theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[T_1, T_2]}
Triangle $ABC$ has side lengths $AB=19, BC=20$, and $CA=21$. Points $X$ and $Y$ are selected on sides $AB$ and $AC$, respectively, such that $AY=XY$ and $XY$ is tangent to the incircle of $\triangle ABC$. If the length of segment $AX$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, compute $100 a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6710}
What is the percentage of 12,356 that equals 12.356000000000002?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.1\%}
Let  $n$  be a positive integer. There are  $n$  soldiers stationed on the  $n$ th root of unity in the complex plane. Each round, you pick a point, and all the soldiers shoot in a straight line towards that point; if their shot hits another soldier, the hit soldier dies and no longer shoots during the next round. What is the minimum number of rounds, in terms of  $n$ , required to eliminate all the soldiers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lceil \log_2(n) \rceil}
In a 500 m race, two contestants A and B have a certain speed ratio. A has a start of 300 m and wins by 100 m. What is the ratio of the speeds of A and B?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Two identical cubes each have an edge length of $a$. The diagonal of the first cube coincides with the diagonal of the second cube. A vertex of the second cube is positioned at the center of the first cube, and the second cube is rotated by $60^{\circ}$ around the diagonal relative to the first cube. Calculate the volume of the intersection of these two cubes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{a^3}{2}}
Evaluate the limit:

$$\lim_{n\to \infty} \sum_{i=1}^n \frac{1}{\frac{n}{2-1}+i}$$

where $a=1$ and $b=2$. Express your answer in terms of natural logarithms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
Evaluate the integral \( \int_0^{2\pi}\int_0^R\frac{rx}{\sqrt{R^2-x^2-y^2}}\,dx\,dy \) by converting to polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}