inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the abscissa of convergence for the Dirichlet series: $$\sum_{m,n\in\mathbb{Z}}'\frac{m^6+n^6}{(m^2+n^2)^s}$$ where the prime on the summation symbol indicates that the term $(0,0)$ is omitted.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the ring to which \( R[X,Y,Z]/(X-Y^3,Z-Y^9) \) is isomorphic, given that substitution for \( X \) yields \( R[x,x^{1/3},x^3] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{R[T]}
A shop owner professes to sell his articles at a certain cost price, but he uses false weights with which he cheats by some percentage while buying and by 20% while selling. His percentage profit is 50%. What is the percentage by which he cheats while buying?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
In triangle \(ABC\), side \(BC = 28\). The angle bisector \(BL\) is divided by the intersection point of the angle bisectors of the triangle in the ratio \(4:3\) from the vertex. Find the radius of the circumscribed circle around triangle \(ABC\) if the radius of the inscribed circle is 12.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Given a function $y = f(x)$, where $x \in \mathbb{R}$, it satisfies $f(-x + 2) = f(-x)$. For $x \in [-1, 1]$, $f(x) = |x|$. Find the number of intersection points between $y = f(x)$ and $y = \log_7 x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $x_1 = 4, x_2 = 6$ and define $x_n$ for $n ≥ 3$ to be the least non-prime greater than $2x_{n−1} − x_{n−2}.$ Find $x_{1000}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{501500}
Determine the number of pairwise non-isomorphic 2-connected graphs with 20 vertices that do not contain a $K_4^-$ minor.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the largest possible integer $n$ such that there exist two bi-infinite sequences $A$ and $B$ with the following properties: sequence $A$ has a period of 1995, and any segment of length $n$ from sequence $B$ is contained in $A$. However, sequence $B$ is not periodic with period 1995. The sequences can consist of arbitrary symbols.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1994}
Given \( x, y, z > 0 \) and \( x + y + z = 1 \), find the maximum value of 
$$
f(x, y, z) = \sum \frac{x(2y - z)}{1 + x + 3y}.
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/7}
A sequence is recursively defined as follows: \( t_{1} = 1 \). For \( n > 1 \), if \( n \) is even, \( t_{n} = 1 + t_{\frac{n}{2}} \); if \( n \) is odd, \( t_{n} = \frac{1}{t_{n-1}} \). Given that \( t_{n} = \frac{19}{87} \), find the sum of the digits of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
From point \( A \), a secant and a tangent are drawn to a circle with radius \( R \). Let \( B \) be the point of tangency, and \( D \) and \( C \) be the points where the secant intersects the circle, with point \( D \) lying between \( A \) and \( C \). It is known that \( BD \) is the angle bisector of angle \( B \) in triangle \( ABC \) and its length is equal to \( R \). Find the distance from point \( A \) to the center of the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{R \sqrt{7}}{2}}
Find the number of incongruent integers of order $2^{10}$ modulo $3 \cdot 2^{12}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2048}
Find the maximum value of the distance \(|PQ|\) where point \(P\) moves on the circle \(x^{2}+(y-4)^{2}=1\) and point \(Q\) moves on the ellipse \(\frac{x^{2}}{9} + y^{2} = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + 3\sqrt{3}}
Consider $\mathbb{R}$ with the lower limit topology, and $Y=(0, 1] \subset \mathbb{R}$ with the subspace topology. Determine the closure $\overline{Y}$ in this topological space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 1]}
A circle $C: (x-3)^2+(y+1)^2=2$ has two tangent lines passing through a point $P$ on the line $l: y=3x$. If the two tangent lines are symmetric with respect to line $l$, then the distance from point $P$ to the center of circle $C$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{10}}
Calculate the monoid of natural transformations from the identity functor to itself in the category of abelian groups, \( \text{Id}_{\mathfrak{Ab}} \rightarrow \text{Id}_{\mathfrak{Ab}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
In triangle \( A D E \), point \( C \) is the midpoint of side \( A D \). Points \( B \) and \( C \) are marked on side \( A D \) such that triangles \( A B E \) and \( B C E \) have the same area, and triangle \( B D E \) is isosceles with base \( B E \). The perimeter of triangle \( A B E \) is 6 cm greater than the length of the broken line \( E B C \), and the perimeter of triangle \( A C E \) is 2 cm greater than the perimeter of triangle \( C D E \). Calculate the perimeter of triangle \( A D E \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{46}{3}}
In $\triangle ABC$, points $E$ and $F$ lie on $\overline{AC}$ and $\overline{AB}$, respectively. Denote by $P$ the intersection of $\overline{BE}$ and $\overline{CF}$. Compute the maximum possible area of $\triangle ABC$ if $PB = 14$, $PC = 4$, $PE = 7$, and $PF = 2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Evaluate the expression \(10\sqrt{5}\,\alpha\), where \(\alpha = \lim_{n\to\infty}\sum_{m = n^2}^{2n^2}\dfrac{1}{\sqrt{5n^4 + n^3 + m}}.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a group of 21 students, there are 5 excellent students, 10 good students, and 6 poorly performing students. In the upcoming exam, excellent students can only receive excellent grades. Good students can receive either good or excellent grades with equal probability. Poorly performing students can receive good, satisfactory, or unsatisfactory grades with equal probability. A student is randomly selected to take the exam. Find the probability that the student will receive a good or excellent grade (event \( A \)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{21} \approx 0.81}
Given that \( m, n, \) and \( k \) are distinct natural numbers greater than 1, and \( \log_{m} n \) is rational, with the condition:

\[ k^{\sqrt{\log_{m} n}} = m^{\sqrt{\log_{n} k}} \]

Determine the minimum possible value of the sum \( k + 5m + n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{278}
Evaluate the double integral \( \int_0^1 \int_x^{\sqrt{x}} \frac{e^y}{y} \, dy \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e - 2}
Evaluate the limit: \[ \lim_{n\rightarrow\infty} \int_{-\pi}^\pi x^2 \frac{\sin(2nx)}{\sin x} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi^3}
Determine the limit of the sequence defined by the following terms as a function of $n$ and $k$:
\[ a_1 = \text{LOG }n \]
\[ a_2 = \text{LOG }k \cdot \text{LOG }\text{LOG }n \]
\[ a_3 = \text{LOG }k \cdot \text{LOG }\text{LOG }k \cdot \text{LOG }\text{LOG }\text{LOG } n \]
\[ a_4 = \text{LOG }k \cdot \text{LOG }\text{LOG } k \cdot \text{LOG }\text{LOG }\text{LOG }k \cdot \text{LOG }\text{LOG }\text{LOG }\text{LOG }n \]
... 
Express the limit in terms of big-Oh notation, if applicable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(1)}
Sandy likes to eat waffles for breakfast. To make them, she centers a circle of waffle batter with a radius of $3\text{ cm}$ at the origin of the coordinate plane. Her waffle iron imprints non-overlapping unit-square holes centered at each lattice point. How many of these holes are contained entirely within the area of the waffle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
The base of the pyramid \( S A B C D \) is a quadrilateral \( A B C D \). The edge \( S D \) is the height of the pyramid. Find the volume of the pyramid, given that \( |A B| = |B C| = \sqrt{5} \), \( |A D| = |D C| = \sqrt{2} \), \( |A C| = 2 \), and \( |S A| + |S B| = 2 + \sqrt{5} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
Let's suppose that $x^2$ is inversely proportional to $y^4$. If $x = 6$ when $y = 2$, determine the value of $x^2$ when $y = 4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{4}}
Determine the period of the cycle of numbers that are coprime to 9 when you repeatedly take the digit sum of their powers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given a trapezoid $ABCD$ and a point $M$ on the lateral side $AB$ such that $DM \perp AB$. It turns out that $MC = CD$. Find the length of the upper base $BC$ if $AD = d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{d}{2}}
The little sprite wrote down consecutive natural numbers starting from 1 in a row:

From the 30th digit, 2021 (the four digits marked in red) first appears. The little sprite continues to write, and from the $\qquad$th digit, 2022 first appears.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{496}
Find the units digit of the sum \(2^{2^2} + 4^{4^4} + 6^{6^6} + \cdots + 2016^{2016^{2016}} + 2018^{2018^{2018}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Evaluate the integral \( \int_0^{90} f(x) \, dx \), where the function \( f(x) \) is defined as follows:
\[
f(x) = \begin{cases} 
(x-3k)^2 & \text{if } 3k-1 \le x < 3k+1 \\
1 & \text{if } 3k+1 \le x < 3k+2 
\end{cases}
\]
for \( k \in \mathbb{Z} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Calculate: \( 7 \frac{4480}{8333} \div \frac{21934}{25909} \div 1 \frac{18556}{35255} = \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{35}{6}}
Determine the number of times the argument of the complex function \( f = \ln[z(z+1)] \) increases when traversing a closed contour that encircles the branch cut \([-1,0]\) on the real axis in the counter-clockwise direction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
设函数 f(x)=\df{1-2x}{(1-x+x^2)^2}，则 \left[\df{f^{(2022)}(0)}{2022!}\right]=__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2023}
Evaluate the polynomial \( P_{2010}(x) \), which satisfies \( P_{2010}(n) = 1^{2010} + 2^{2010} + \cdots + n^{2010} \) for positive integers \( n \), at \( x = -\frac{1}{2} \). Find the value of \( P_{2010}\left(-\frac{1}{2}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit \( \lim_{a\to1^+}\int_0^{\pi/2}\frac{dx}{(1+\cos2x)^a} \). Determine if the integral converges and find its value if it does.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the integral \( I_C = \frac{1}{2\pi}\int_{0}^{2\pi}\left(\frac{d}{d\theta}\phi(\theta)\right) d\theta \), where \( \phi(\theta) = \arctan\left[\frac{3\cos(\theta)}{4(\cos(\theta)+\sin(\theta))}\right] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Find the inverse Laplace transform of \( \frac{2ks}{(s^2 + k^2)^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t \sin(kt)}
Two boxes of candies have a total of 176 pieces. If 16 pieces are taken out from the second box and put into the first box, the number of pieces in the first box is 31 more than m times the number of pieces in the second box (m is an integer greater than 1). Then, determine the number of pieces originally in the first box.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{131}
On a purchase of $110, a store offered a payment plan consisting of a certain down payment and 12 monthly payments of $10 each. The customer paid 9.090909090909092% of the purchase price in interest by using this plan. What was the down payment amount?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$0}
How many elements of order p are there in GL(2,Zp)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2 - 1}
Find the maximum value of \(x_{1}^{2}+x_{2}^{2} + x_{3}^{2}\) subject to the constraints \(x_{1}-x_{2}\leq0\), \(x_{2}-x_{3}\leq0\), \((x_{1} - x_{2})^{2} + (x_{2} - x_{3})^{2} = \delta\), and \(x_{1} + x_{2} + x_{3} = 0\), where \(\delta\) is a positive constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\delta}
Find a differentiable function $g: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(n) = O(g(n))$ as $n \to \infty$, where $f(n) = \exp\left(2\log\log\omega(n) + O(1)\right)$ and $\omega(n)$ is the number of prime divisors of $n$. Provide your answer in terms of a function $g(n)$. \( \omega(n) \) is the number of distinct prime factors of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\log \log n)^2}
Let $a_{1}, a_{2}, \ldots, a_{2018}$ be non-negative real numbers whose sum is 1. Determine the maximum possible value of

$$
S=\sum_{i \neq j, i \mid j} a_{i} a_{j}
$$

where $i \mid j$ denotes that $i$ divides $j$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{11}}
Evaluate the limit: $$\lim_{x\to0} {\frac{\frac{1}{1+x^3} + \frac{1}{3}\log{\left(1+3x^3\right)}-1}{2\sin{\left(3x^2\right)}-3\arctan{\left(2x^2\right)}}}$$ without using L'Hôpital's rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
At least how many days must a festival last if 11 theatrical groups participate, and each group must see at least one performance of every other group during their days off?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the degree of the polynomial \( p(x) \) that satisfies the equation \( (x)p(x - 1) = (x - 15) p(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
The number of the members of a club is more than 20 and less than 50. When 5 people sit at a table, other people are exactly divided into 6-people groups or 7-people groups. If the members are divided into groups of a certain number of people, 7 people will be left. How many people are in this new group size?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the bases $b$ for which the recursive sequence defined by $a_0 = 1$ and $a_{n+1} = \log_{b} (1 + a_n)$ results in the series $\sum_{n=0}^\infty a_n$ having a finite sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b > e}
In an isosceles triangle, the angle at the base is $\alpha$. The altitude drawn to the base is greater than the radius of the inscribed circle by $m$. Find the radius of the circumscribed circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{m}{4 \sin^2 \frac{\alpha}{2}}}
In Möbius geometry, at how many points do two distinct lines intersect?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find \( \lim_{\varepsilon\rightarrow 0^+} f_\varepsilon \) in \( D'(\mathbb{R}) \), where \( f_\varepsilon(x)=\frac{\sin{\varepsilon x}}{x} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \delta(x)}
Determine the radius of convergence for the Taylor series expansion of \( f(z) = \frac{1}{1+e^z} \) around \( z = \pi i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the exact y-coordinate of a circle with radius \( \frac{1}{4} \) that makes it tangent to the function \( y = e^{-x^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
Evaluate the limit: \[ \lim _{p\to \infty }\left(p\ln \left(e\left(1+\frac{1}{p}\right)^{1-p}\right)\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Evaluate the double integral \( \iint_R 8(x^2+y^2)(x^2-y^2) \, dA \), where \( R \) is the region in the first quadrant of the \( xy \)-plane bounded by the circles \( x^2+y^2=1 \), \( x^2+y^2=4 \), and the lines \( y=x+1 \) and \( y=x-1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the gradient of the function \( f(x) = \frac{1}{2}\|G(x)\|_2^2 \) in terms of \( G(x) \) and \( G'(x) \), where \( G(x): \mathbb{R}^n \rightarrow \mathbb{R}^n \) is a twice continuously differentiable function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{G'(x)^\top G(x)}
Find all natural numbers \( m \) such that \( 5^{m-2} - 2 \equiv 0 \pmod{m} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the number of iterations required to reach an error of $10^{-6}$ when approximating Dottie's number using the iterative formula $\epsilon^{(k)}\approx(-\sin D)^k\epsilon$, where $D$ is Dottie's number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Evaluate the limit: $$\lim_{x\to\infty}\left(\sum_{n=1}^{\infty}\left(\frac{x}{n}\right)^n \right)^{1/x}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\frac{1}{e}}}
A car starts its trip in town A and travels towards town B. During the first 4 hours, the car averages 75 mph. The car then slows down to an average speed of 60 mph for the next 3 hours and finally averages 50 mph for the remaining 2 hours before reaching town B. Upon arrival, the car stops for a 30 minute break, then heads towards town C, covering a distance with varying average speeds of 40 mph for the first hour, 55 mph for the second hour and 65 mph for the next 1.5 hours. The car stops again in town C for a 45 minute break and finally goes back to town A with a consistent average speed of 70 mph. Considering the entire trip from town A to town B, town B to town C, and town C back to town A, determine the total time it takes for the complete journey including breaks.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24.79}
From point \( K \) on side \( AC \) of triangle \( ABC \), perpendiculars \( KL_1 \) and \( KM_1 \) are dropped to sides \( AB \) and \( BC \) respectively. From point \( L_1 \), perpendicular \( L_1 L_2 \) is dropped to \( BC \), and from point \( M_1 \), perpendicular \( M_1 M_2 \) is dropped to \( AB \).

It turns out that triangles \( BL_1M_1 \) and \( BL_2M_2 \) are similar (point \( L_1 \) in the first triangle corresponds to point \( M_2 \) in the second). Additionally, \( BL_2 = 6 \) and \( L_2M_1 = 4 \). Find \( L_1L_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
You start with a single piece of chalk of length 1. Every second, you choose a piece of chalk that you have uniformly at random and break it in half. You continue this until you have 8 pieces of chalk. What is the probability that they all have length $\frac{1}{8}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{63}}
How many lines pass through point A (1, 2) and have intercepts on the two coordinate axes with equal absolute values?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let \(\triangle ABC\) have side lengths \(AB=30\), \(BC=32\), and \(AC=34\). Point \(X\) lies in the interior of \(\overline{BC}\), and points \(I_1\) and \(I_2\) are the incenters of \(\triangle ABX\) and \(\triangle ACX\), respectively. Find the minimum possible area of \(\triangle AI_1I_2\) as \( X\) varies along \(\overline{BC}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{126}
Given a function $f(x)$ which is an odd function defined on $(-\infty,0) \cup (0,+\infty)$ and is monotonically decreasing on $(0,+\infty)$, with the condition $f(4)=0$. If $f(x-3) \leqslant 0$, then the range of $x$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1,3) \cup [7,+\infty)}
Let $x$ be a real number, $P=e^x+e^{-x}$, $Q=(\sin x+\cos x)^2$. Determine the relationship between $P$ and $Q$, expressing your answer as an inequality or equality relationship between $P$ and $Q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{P \geq Q}
Find the positive integer $k$ such that the function \( f(19, k) = 20 \), where \( a_{n} = 4 - 3n \) for all integers \( n \geq 1 \) and \( f(x, y) = x + \sum_{i=1}^{\infty}\left [\left(\frac{\prod_{j=1}^{i}a_{j}}{3^i\cdot i!} \right )x^{a_{i+1}}y^{i}\right] \) for all real \( x \) and \( y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1141}
Determine the cardinality of the subset of the interval [0,1] consisting of infinite decimal expansions that use only the digits 2 and 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
Two numbers are chosen from the interval $[0,1]$. The probability that the sum of their squares is greater than $l$ is $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \frac{\pi}{4}}
A domino is a $2 \times 1$ or $1 \times 2$ tile. Determine in how many ways exactly $n^{2}$ dominoes can be placed without overlapping on a $2 n \times 2 n$ chessboard so that every $2 \times 2$ square contains at least two uncovered unit squares which lie in the same row or column. Please provide the value when n = 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In a cyclic quadrilateral $PQRS$, the sides are given as $PQ = RS = x$, $QR = 2x$, and $PS = 1$. Determine the value of $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find the sum of all integers \( n \) such that \( n^2+2n+2 \) divides \( n^3 + 4n^2 + 4n - 14 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-11}
With one mighty blow, Maria cracked open the pinata, and countless candies spilled all over the floor. There were 50 red candies, 35 less than three times as many yellow candies as red candies, and a third as many blue candies as twice the number of yellow candies. Additionally, there were 20 green candies, which was twice the number of purple candies. Finally, 10 silver candies rolled to the side, completing the assortment.

If Carlos ate all of the yellow candies and half of the green candies, how many candies remained?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{156}
Evaluate the limit: $$\lim_{T\to \infty}\frac{\int_0^T\sin^2(s) \exp(-s)\int_0^s \sin(\sin a)\exp(a)\,\mathrm{d}a\,\mathrm{d}s}{T}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Compute the number of nonempty subsets $S \subseteq \{-10, -9, -8, \ldots, 8, 9, 10\}$ that satisfy $|S| + \min(S) \cdot \max(S) = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{335}