inclusionAI/Ring-lite-rl-data huggingface dataset:
If $(1+x+x^{2})^{6}=a\_{0}+a\_{1}x+a\_{2}x^{2}+...+a\_{12}x^{12}$, then $a\_{2}+a\_{4}+...+a\_{12}=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{364}
Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$ and $f(x+ \frac{5}{2})=-\frac{1}{f(x)}$, and when $x \in [-\frac{5}{2}, 0]$, $f(x)=x(x+ \frac{5}{2})$, find $f(2016)=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Determine the number of ways to arrange 6 red roses and 4 white roses in a garland such that no two white roses are adjacent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the real values of $n$ for which the matrix \( M = \begin{bmatrix} n & 1 & 1 \\ 1 & n & 1 \\ 1 & 1 & -n \end{bmatrix} \) is negative definite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Given a circle $(x^2 + y^2 - 4x - 4y = 0)$, there are at least three distinct points on the circle such that the distance from each point to the line $l: y = kx$ is $\sqrt{2}$. Determine the range of the slope $k$ of the line $l$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2 - \sqrt{3}, 2 + \sqrt{3}]}
Evaluate the integral $$J=\int_0^{\infty} \frac{\operatorname{artanh}\left(\frac{1}{\sqrt{1+x^2}}\right)}{\sqrt{1+x^2}} d x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi^2}{4}\)}
Let $V$ be a finite-dimensional complex vector space and $G \subset GL(V)$ be a finite group. Consider the quotient variety $V/G$. Determine the codimension of the singular locus in $V/G$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
a and b are two multiples of a certain number, and q is the set of consecutive integers between a and b, inclusive. q contains 9 multiples of that number and there are 17 multiples of 7 in q. What is that certain number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Mark does a gig every other day for 2 weeks. For each gig, he plays a certain number of songs. 2 of the songs are 5 minutes long and the last song is twice that long. He played for 280 minutes. How many songs does he play at each gig?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
There are 100 chairs arranged in a circle. If \( n \) people are sitting on these chairs, such that any new person sitting down will always sit on a chair adjacent to one of the \( n \) people, what is the minimum value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
已知数列 $\left\{a_{n}\right\}$ 满足
$$
a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right]\left(n \in \mathbf{Z}_{+}\right),
$$
其中， $[x]$ 表示不超过实数 $x$ 的最大整数。设 $C$ 为实数，且对于任意的正整数 $n$ ，均有
$$
\sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C
$$
则 $C$ 的最小值为 $\qquad$原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{289}
11. There are 11 students who have signed up for the volunteer tour guide activity at the museum. The activity runs from 9 AM to 5 PM, with a public mini-lecture every hour. Each session requires 1 student to provide tour guide services to visitors. To avoid overworking the students, the museum will not schedule the same person for two consecutive sessions, and the first and last sessions must be handled by two different students. How many scheduling arrangements are possible for the museum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100000010}
Zoey read $15$ books, one at a time. The first book took her $1$ day to read, the second book took her $2$ days to read, the third book took her $3$ days to read, and so on, with each book taking her $1$ more day to read than the previous book. Zoey finished the first book on a Monday, and the second on a Wednesday. On what day the week did she finish her $15$th book?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Monday}
Two circles touch each other externally at point \( C \). A line is tangent to the first circle at point \( A \) and to the second circle at point \( B \). The line \( AC \) intersects the second circle at point \( D \), distinct from \( C \). Find \( BC \), given that \( AC = 9 \) and \( CD = 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
x and y are two towns. Ganesh covers the distance from x to y at an average speed of 43 km/hr. However, he covers the distance from y to x at an average speed of 34 km/hr. What is his average speed during the whole journey?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38}
On the leg \(BC\) of the right triangle \(ABC\), a circle is constructed with \(BC\) as its diameter. This circle intersects the hypotenuse \(AB\) at point \(P\). Chord \(PQ\) is parallel to leg \(BC\). Line \(BQ\) intersects leg \(AC\) at point \(D\). Given that \(AC = b\) and \(DC = d\), find \(BC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{bd}}
Medians $\overline{AD}$ and $\overline{BE}$ of $\triangle ABC$ intersect at an angle of $45^\circ$. If $AD = 12$ and $BE = 16$, then calculate the area of $\triangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64\sqrt{2}}
We fill a graduated cylinder with a diameter of $2.9 \mathrm{~cm}$ with water up to a height of $4 \mathrm{~cm}$. Then, we place solid copper cubes with an edge length of $2 \mathrm{~cm}$ into the cylinder. What is the maximum number of cubes that can be submerged in the water in the cylinder?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Among the four-digit numbers formed by the digits 6, 7, 8, and 9, where each digit is different, there are $\boxed{\text{several}}$ numbers that are divisible by 11.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given the curve $y=\sin x$ ($0 \leq x \leq \pi$) and the line $y= \frac {1}{2}$, calculate the area of the enclosed shape formed by these two functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt {3}- \frac {\pi}{3}}
In a geometry problem, Isabella has drawn a pair of concentric circles. She starts drawing chords $\overline{CD}$, $\overline{DE}$, and so on, of the larger circle, each tangent to the smaller one. If $m\angle CDE=60^\circ$, how many chords will Isabella draw before she returns to her starting point at $C$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Under the new AMC $10, 12$ scoring method, $6$ points are given for each correct answer, $2.5$ points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between $0$ and $150$ can be obtained in only one way, for example, the only way to obtain a score of $146.5$ is to have $24$ correct answers and one unanswered question. Some scores can be obtained in exactly two ways, for example, a score of $104.5$ can be obtained with $17$ correct answers, $1$ unanswered question, and $7$ incorrect, and also with $12$ correct answers and $13$ unanswered questions. There are (three) scores that can be obtained in exactly three ways. If the sum of these scores is $x$, what is the value of $10x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1885}
Find the sum of the digits in base ten of the expression (10^{4n^2 + 8} + 1)^2, where n is a positive integer. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
\( p(x) \) is a polynomial of degree less than 1992 such that \( p(0) \), \( p(1) \), \( p(-1) \) are all non-zero. The 1992nd derivative of \( \frac{p(x)}{x^3 - x} \) is \( \frac{f(x)}{g(x)} \) for polynomials \( f(x) \) and \( g(x) \). Find the smallest possible degree of \( f(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3984}
The negation of the proposition "For all $x > 0$, $\sin x \geq -1$":
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\exists x > 0, \text{ such that } \sin x < -1}
Let f(x) be a monotonically decreasing odd function defined on R. Given x1+x2>0, x2+x3>0, and x3+x1>0, determine the sign of f(x1)+f(x2)+f(x3). Express your answer as a single inequality in the format f(x1)+f(x2)+f(x3) [inequality] 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x1)+f(x2)+f(x3) < 0}
What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1, k_2, \ldots, k_n$ such that
$k^2_1 + k^2_2 + \ldots + k^2_n = 2002?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
# 2.1. Condition:

In the campus, rooms are numbered consecutively $1,2,3,4 \ldots, 10,11, \ldots$ For room numbering, stickers with digits were purchased, with the digits 1, 2, and 3 being purchased in equal quantities, and the digit 5 being purchased three more than the digit 6. How many rooms are there in the campus, given that there are fewer than 100?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
As shown in the diagram, in the plane Cartesian coordinate system, the ellipse $\Gamma$: $\frac{x^{2}}{2} + y^{2} = 1$ has left and right foci $F_{1}$ and $F_{2}$, respectively. Let $P$ be a point on $\Gamma$ in the first quadrant, and let the extended lines $P F_{1}$ and $P F_{2}$ intersect $\Gamma$ at points $Q_{1}\left(x_{1}, y_{1}\right)$ and $Q_{2}\left(x_{2}, y_{2}\right)$, respectively. Find the maximum value of $y_{1} - y_{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 \sqrt{2}}{3}}
Yann writes down the first $n$ consecutive positive integers, $1,2,3,4, \ldots, n-1, n$. He removes four different integers $p, q, r, s$ from the list. At least three of $p, q, r, s$ are consecutive and $100<p<q<r<s$. The average of the integers remaining in the list is 89.5625. What is the number of possible values of $s$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
In triangle $XYZ$, which is equilateral with a side length $s$, lines $\overline{LM}$, $\overline{NO}$, and $\overline{PQ}$ are parallel to $\overline{YZ}$, and $XL = LN = NP = QY$. Determine the ratio of the area of trapezoid $PQYZ$ to the area of triangle $XYZ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{16}}
In the vertices of a regular 100-gon, 100 chips numbered $1, 2, \ldots, 100$ are placed in exactly that order in a clockwise direction. During each move, it is allowed to swap two chips placed at adjacent vertices if the numbers on these chips differ by no more than $k$. What is the smallest $k$ such that, in a series of such moves, every chip is shifted one position clockwise relative to its initial position?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
The number 175 can be written as the sum of the squares of 4 different positive integers. What is the sum of these 4 integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Let $\mathcal{S}$ be a sphere with radius $2.$ There are $8$ congruent spheres whose centers are at the vertices of a cube, each has radius $x,$ each is externally tangent to $3$ of the other $7$ spheres with radius $x,$ and each is internally tangent to $\mathcal{S}.$ There is a sphere with radius $y$ that is the smallest sphere internally tangent to $\mathcal{S}$ and externally tangent to $4$ spheres with radius $x.$ There is a sphere with radius $z$ centered at the center of $\mathcal{S}$ that is externally tangent to all $8$ of the spheres with radius $x.$ Find $18x + 5y + 4z.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Given that the sequence $\{a\_n\}$ satisfies $a\_1$, $a\_2-a\_1$, $a\_3-a\_2$, $…$, $a\_n-a_{n-1}$, $…$, is an arithmetic sequence with the first term being $1$ and the common ratio being $2$, find the value of $a\_n$ equal to _____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n-1}
Through the vertex \(B\) of triangle \(ABC\), a line \(\ell\) is drawn parallel to side \(AC\). The angle bisector of angle \(A\) intersects \(\ell\) at point \(A'\), and the angle bisector of angle \(C\) intersects \(\ell\) at point \(C'\). It is known that the area of triangle \(ABA'\) is \(50\%\) greater and the area of triangle \(CBC'\) is \(25\%\) greater than the area of triangle \(ABC\). The bisectors intersect at point \(O\). What fraction of the area of triangle \(AOC\) is the area of triangle \(ABC\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{15}}
There are $n$ integers $a$ such that $0 \leq a < 91$ and $a$ is a solution to the congruence $x^3 + 8x^2 - x + 83 \equiv 0 \pmod{91}$. What is $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
If $ABCDE$ is a regular pentagon and $X$ is a point in its interior such that $\triangle CDX$ is equilateral, compute $\angle AXE$ in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
If the positive sequence $\{a_n\}$ satisfies $\log a_{n+1} - \log a_n = 1$, and $a_{2001} + a_{2002} + a_{2003} + \ldots + a_{2010} = 2015$, then the value of $a_{2011} + a_{2012} + a_{2013} + \ldots + a_{2020}$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015 \times 10^{10}}
Given $A=\{y|y=|x+1|, x\in[-2,4]\}$, $B=[2,5)$, then $\complement_{A}B=$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0,2)\cup\{5\}}
Compute the integer $m > 3$ for which
\[\log_{10} (m - 3)! + \log_{10} (m - 1)! + 3 = 2 \log_{10} m!.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Al told Bob that he was thinking of $2011$ distinct positive integers. He also told Bob the sum of those $2011$ distinct positive integers. From this information, Bob was able to determine all $2011$ integers. How many possible sums could Al have told Bob?

[i]Author: Ray Li[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a sequence $\{a_n\}$ that satisfies $a_n-(-1)^n a_{n-1}=n$ ($n\geqslant 2, n\in \mathbb{N}^*$), and $S_n$ is the sum of the first $n$ terms of $\{a_n\}$, then $S_{40}=$_______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{440}
If there exists a real number $x$ such that the inequality $\left(e^{x}-a\right)^{2}+x^{2}-2ax+a^{2}\leqslant \dfrac{1}{2}$ holds with respect to $x$, determine the range of real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\{\dfrac{1}{2}\right\}}
设整数 $a_{1}, a_{2}, a_{3}$ 满足 $1 \leqslant a_{k} \leqslant 24(k=1,2,3)$ ，且对任意整数 $x, 2 a_{1} x^{2}+3 a_{2} x+4 a_{3}$ 是 24 的倍数。求满足条件的有序数组 $\left(a_{1}, a_{2}, a_{3}\right)$ 的个数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Given the equation of the ellipse is $\frac{x^2}{4}+\frac{y^2}{3}=1$, $F_1$ and $F_2$ are the left and right foci of the ellipse, respectively. The chord $PQ$ passes through the right focus $F_2$ of the ellipse. The maximum area of $\Delta PF_1Q$ is \_\_\_\_
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Dad bought a cylindrical birthday cake for his daughter Yuanyuan. Yuanyuan wants to cut the cake into several pieces (not less than 10 pieces) of different sizes to share with 10 friends. If the cake is cut vertically, at least how many cuts are needed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $ABCD$ be a convex quadrilateral with $AB = AD$ and $CB = CD$. The bisector of $\angle BDC$ intersects $BC$ at $L$, and $AL$ intersects $BD$ at $M$, and it is known that $BL = BM$. Determine the value of $2\angle BAD + 3\angle BCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{450}
From Sunday to Thursday, Prudence sleeps 6 hours a night. On Friday and Saturday, she sleeps for a different number of hours a night. She also takes a 1-hour nap on Saturday and Sunday. Prudence gets 200 hours of sleep in 4 weeks. How many hours does she sleep per night on Friday and Saturday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let $ABC$ be a triangle such that $AB=6,BC=5,AC=7.$ Let the tangents to the circumcircle of $ABC$ at $B$ and $C$ meet at $X.$ Let $Z$ be a point on the circumcircle of $ABC.$ Let $Y$ be the foot of the perpendicular from $X$ to $CZ.$ Let $K$ be the intersection of the circumcircle of $BCY$ with line $AB.$ Given that $Y$ is on the interior of segment $CZ$ and $YZ=3CY,$ compute $AK.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{147}{10}}
If the function $f(x)=a\sin^2x+b\tan x+1$, and $f(2)=5$, then calculate $f(\pi-2)+f(\pi)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Eight celebrities meet at a party. Each celebrity shakes hands with exactly two others. A fan makes a list of all unordered pairs of celebrities who shook hands with each other. If the order does not matter, how many different lists are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3507}
Let \( n \in \mathbf{N}^{*} \). Consider the set \( S = \{1, 2, \ldots, 2n\} \) and its \( k \) subsets \( A_{1}, A_{2}, \ldots, A_{k} \) that satisfy the following conditions:

1. For any \( i \neq j \) (with \( i, j \in \{1, 2, \ldots, k\} \)), the intersection \( A_{i} \cap A_{j} \) has exactly an odd number of elements.
2. For any \( i \) (where \( i = 1, 2, \ldots, k \)), \( i \notin A_{i} \).
3. If \( i \in A_{j} \), then \( j \in A_{i} \).

Determine the maximum value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n - 1}
## Task A-1.3.

Determine how many six-digit natural numbers exist such that by removing the first two, or the last two digits, we obtain two four-digit numbers that give the same remainder when divided by 99.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8190}
Points \( K, L, M, N \) are the centers of the circles inscribed in the faces \( SAB, SAC \), \( SBC \), and \( ABC \) of the tetrahedron \( SABC \). It is known that \( AB = SC = 5 \), \( AC = SB = 7 \), and \( BC = SA = 8 \). Find the volume of the tetrahedron \( KLMN \). If necessary, round the answer to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.66}
In triangle $ABC$, $BC = 40$ and $\angle C = 45^\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ at $D$ and extend to meet an extension of $AB$ at $E$. Find the length of $DE$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
实系数方程 $x^{4}+a x^{3}+b x^{2}+c x+d=0$ 的根都不是实数，其中两个根的和为 $2+\mathrm{i}$, 另两个根的积为 $5+6 \mathrm{i}$。求 $b$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
If the negation of the proposition "There exists a real number $x_0$ such that $x + ax_0 + 1 < 0$" is true, then the range of the real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-2, 2]}
Given that the complex number $z$ and $(z+2)^2 - 8i$ are both purely imaginary, find $z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2i}
In a $2 \times 8$ squared board, you want to color each square red or blue such that on each $2 \times 2$ sub-board there are at least $3$ squares painted blue. In how many ways can this coloring be done?

**Note:** A $2 \times 2$ board is a square made up of $4$ squares that have a common vertex.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{341}
Given the complex number \( z \) satisfies \( |z|=1 \) and \( \operatorname{Re} \frac{z+1}{\bar{z}+1}=\frac{1}{3} \), find the value of \( \operatorname{Re} \frac{z}{\bar{z}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{7}{9}}
Density is defined as the ratio of mass to volume. There are two cubes. The second cube is made from a material with twice the density of the first, and the side length of the second cube is 100% greater than the side length of the first. By what percentage is the mass of the second cube greater than the mass of the first?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1500\%}
If \( k \) is an integer greater than 1 and \( a \) is a root of \( x^2 - kx + 1 = 0 \), for any natural number \( n \) greater than 10, the units digit of \( a^{2n} + a^{-2n} \) is always 7. What is the units digit of \( k \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Two right triangles, $ABC$ and $ACD$, are joined at side $AC$. Squares are drawn on four of the sides. The areas of three of the squares are 25, 49, and 64 square units. Determine the number of square units in the area of the fourth square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{138}
Given rectangle ABCD, AB=4, BC=8, points K and L are midpoints of BC and AD, respectively, and point M is the midpoint of KL. What is the area of the quadrilateral formed by the rectangle diagonals and segments KM and LM.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
For each integer $n>1$, let $F(n)$ be the number of solutions to the equation $\sin{x}=\sin{(nx)}$ on the interval $[0,\pi]$. What is $\sum_{n=2}^{2007} F(n)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016532}
In what ratio does the line $TH$ divide the side $BC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:1}
These points form a new triangle with angles \(45^{\circ}, 60^{\circ},\) and \(75^{\circ}\). Find the ratio of the areas of the original and the new triangles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3} - 1}
Let
$$p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3.$$Suppose that
\begin{align*}
p(0,0) &=p(1,0) = p( - 1,0) = p(0,1) = p(0, - 1)= p(1,1) = p(1, - 1) = p(2,2) = 0.
\end{align*}There is a point $(r,s)$ for which $p(r,s) = 0$ for all such polynomials, where $r$ and $s$ are not integers.  Find the point $(r,s).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \frac{5}{19}, \frac{16}{19} \right)}
Evaluate the line integral \( \int_c \frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy \) along the curve \( c(t) = (1+2\cos t, \sin t) \) for \( 0 \leq t \leq 2\pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Compute the integral \( \int_\Omega  \left|\,x\,\right|\,dx\,dy\,dz \) over the volume of a torus. In \(\mathbb{R}^3\), let \(C\) be the circle in the \(xy\)-plane with radius 2 and the origin as the center, i.e., \(C= \Big\{ \big(x,y,z\big) \in \mathbb{R}^3 \mid x^2+y^2=4, \ z=0\Big\}.\) Let \(\Omega\) consist of all points \((x,y,z) \in \mathbb{R}^3\) whose distance to \(C\) is at most 1. Use appropriate coordinate transformations and symmetry considerations to simplify the computation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17\pi}