inclusionAI/Ring-lite-rl-data huggingface dataset:
Select 5 different letters from the word "equation" to arrange in a row, where "qu" must be together and in the same order. Find the number of different arrangements. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{480}
Each cell of a $10 \times 10$ table contains a natural number. Then, a cell is shaded if the number in that cell is less than one of its neighbors but greater than another neighbor. (Two cells are considered neighbors if they share a side.) As a result, only two cells remain unshaded. What is the minimal possible sum of the numbers in these two unshaded cells?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Evaluate the triple integral \( \iiint_B (x^2+y^2+z^2)^2 \, dV \) where \( B = \{ (x,y,z) \in \mathbb{R}^3 : x^2+y^2+z^2 \leq 1, z \leq 0 \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\pi}{7}}
Outside the right-angled triangle $ABC$, squares $ACDE$ and $BCFG$ are constructed on its legs $AC$ and $BC$. The extension of the median $CM$ of the triangle $ABC$ intersects the line $DF$ at point $N$. Find the segment $CN$ if $AC = 4$ and $BC = 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{\sqrt{17}}}
In triangle \( \triangle ABC \), the sides are given as \( AB = 7 \), \( BC = 8 \), and \( AC = 9 \). A circle is drawn such that it is tangent to side \( AC \) and the extensions of sides \( AB \) and \( BC \). Let \( D \) be the center of this circle. Calculate \( BD^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{224}
Freshmen go for a long walk in the suburbs after the start of school. They arrive at point \( A \) 6 minutes later than the originally planned time of 10:10, and they arrive at point \( C \) 6 minutes earlier than the originally planned time of 13:10. There is exactly one point \( B \) between \( A \) and \( C \) that is reached according to the original planned time. What is the time of arrival at point \( B \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11:40}
Find all positive real solutions to the equation \( \left(\dfrac{\ln (x)}{x}\right)^2=\pi^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{W(\pi)}{\pi}}
A $39$-tuple of real numbers $(x_1, x_2, \ldots, x_{39})$ satisfies the conditions:
\[2\sum_{i=1}^{39} \sin(x_i) = \sum_{i=1}^{39} \cos(x_i) = -34.\] 
Determine the ratio between the maximum of $\cos(x_1)$ and the maximum of $\sin(x_1)$ over all tuples $(x_1, x_2, \ldots, x_{39})$ satisfying these conditions. If this ratio is $\frac{a}{b}$ for coprime positive integers $a$ and $b$, find $a + b$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Given the probability density function of a continuous random variable $X$:
\[ f_{X}(x) = \begin{cases}\frac{x+1}{a+1} & -1 \leq x < a \\
1 & x=a \\
\frac{x-1}{a-1} & a < x \leq 1 \\
0 & \text{otherwise}\end{cases}\]
Calculate the expected value $E(X)$. Consider the implications of the probability being 1 at $x=a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{a}{3}}
Evaluate the double integral \( \iint _R \frac{y^2}{x} \, dx \, dy \) over the region \( R \) in the \( xy \)-plane bounded by the curves \( y = x^2 \), \( y = 4x^2 \), \( xy = 1 \), and \( xy = 5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
For a positive integer \( n \) ,define a function \( {f}_{n}\left( x\right) \) at an interval \( \left\lbrack  {0,n + 1}\right\rbrack \) as
 \( {f}_{n}\left( x\right)  = {\left( \mathop{\sum }\limits_{{i = 1}}^{n}\left| x - i\right| \right) }^{2} - \mathop{\sum }\limits_{{i = 1}}^{n}{\left( x - i\right) }^{2}. \)
 Let \( {a}_{n} \) be the minimum value of \( {f}_{n}\left( x\right) \) . Find the value of
 \( \mathop{\sum }\limits_{{n = 1}}^{{11}}{\left( -1\right) }^{n + 1}{a}_{n} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
450
}
Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to $32$? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{568}
$\mathrm{Natural~}a,b,c\mathrm{are}>100\mathrm{and}(a,b,c)=1.c|a+b,a|b+c\text{Find minimal }b$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
 10099}
Calculate the work done by a particle moving along the spiral curve given by $x = a\cos t$, $y = a\sin t$, $z = bt$ for $0 \le t \le 4\pi$, against the force field $\mathbf{F}(x,y,z) = xy\,\mathbf{i} + yz\,\mathbf{j} + xz\,\mathbf{k}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^2 b \pi}
Compute the exterior derivative \(d\beta\) of the two-form \(\beta = z(y^2-x^2)\,dx\,dy - x^2y\,dx\,dz - xy^2\,dy\,dz\) in three-dimensional real space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the function \( F(x) = \lim_{y \to 0} f(x,y) \) where \( f: \mathbb{R}^2 \to \mathbb{R} \) is defined as follows:
\[
f(x,y) = \begin{cases} \dfrac{e^{-x^2}-e^{-y^2}}{x^2+y^2} &\text{if } (x,y) \neq (0,0), \\
 k &\text{if } (x,y) = (0,0). \end{cases}
\]
Determine the value of \( k \) such that \( F(x) \) is continuous at \( x = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Find the functional derivative of the function \( g(x) = x(1) - 1 \) with respect to \( x \), where \( x \) is a function and both \( x \) and \( g(x) \) are differentiable and continuous at all points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\delta(t - 1)}
For \( n \in \mathbb{N} \), consider non-negative valued functions \( f \) on \( \{1,2, \cdots , n\} \) satisfying \( f(i) \geqslant f(j) \) for \( i>j \) and \( \sum_{i=1}^{n} (i+ f(i))=2023. \) Choose \( n \) such that \( \sum_{i=1}^{n} f(i) \) is maximized. How many such functions exist in that case?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the interval of convergence for the series \(\sum_{n=1}^{\infty}\frac{((n+1)x)^n}{n^{n+1}}\). Specifically, evaluate the convergence at the endpoints of the interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 1)}
Given the ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a > b > 0)$, let the left vertex be $M$, the top vertex be $N$, and the right focus be $F$. If $\overrightarrow{NM} \cdot \overrightarrow{NF}=0$, then find the eccentricity of the ellipse. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5} - 1}{2}}
John is a hack author. He writes a book every certain period of time. He has been writing for 20 years and has earned an average of $30,000 per book. He has made $3,600,000 writing. How often does he write a book?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $PQRS$ be an isosceles trapezoid with bases $PQ=120$ and $RS=25$. Suppose $PR=QS=y$ and a circle with center on $\overline{PQ}$ is tangent to segments $\overline{PR}$ and $\overline{QS}$. If $n$ is the smallest possible value of $y$, then $n^2$ equals what?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2850}
2B. Find $x$ for which the function $y=\left(x-x_{1}\right)^{2}+\left(x-x_{2}\right)^{2}+\ldots+\left(x-x_{1996}\right)^{2}$ takes the smallest value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{x_{1}+x_{2}+\ldots+x_{1996}}{1996}}
Evaluate the integral \( \iiint_E z \,{\rm d}v \), where \( E \) is the region between the spheres \( x^2 + y^2 + z^2 = 1 \) and \( x^2 + y^2 + z^2 = 3 \) in the first octant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Determine the number of 4-tuples \((a,b,c,d) \in \mathbb{N}^{4}\) such that:

\[\mathrm{lcm}(a,b,c) = \mathrm{lcm}(b,c,d)=\mathrm{lcm}(c,d,a)= \mathrm{lcm}(d,a,b)=2^{5}3^{4}5^{3}7^{2}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42723153}
Evaluate the integral \( \int_{-\pi}^{\pi}\ln(4e^2+e^2+4e^2\cos(\theta))d\theta \) when \( a = 2e \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi(1 + \ln 2)}
There are 7 boys and 13 girls standing in a line. Let \( S \) be the number of positions where boys and girls are adjacent to each other. Calculate the average value of \( S \) for all possible arrangements of these 20 people. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the largest real $C$ such that for all pairwise distinct positive real $a_{1}, a_{2}, \ldots, a_{2019}$ the following inequality holds $$\frac{a_{1}}{\left|a_{2}-a_{3}\right|}+\frac{a_{2}}{\left|a_{3}-a_{4}\right|}+\ldots+\frac{a_{2018}}{\left|a_{2019}-a_{1}\right|}+\frac{a_{2019}}{\left|a_{1}-a_{2}\right|}>C$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1010}
Consider a convex polygon with $4n-1$ sides, where $n$ is a positive integer greater than 1. The diagonals of this polygon are divided into $k$ subsets \( S_{1}, S_{2}, \cdots, S_{k} \) such that for every pair of different indices $i$ and $j$, there exist diagonals in subset $S_{i}$ that intersect with diagonals in subset $S_{j}$ at an interior point. What is the maximum possible value of $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n - 1}
6 barrels of water have an average capacity of 12 gallons. The capacities of the barrels must be between 8 and 16 gallons, inclusive. What is the maximum possible capacity of the smallest barrel if the median is 14 gallons and the capacity of the largest barrel is a prime number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let the tangent line at point A $(x_0, y_1)$ of the curve $y=(ax-1)e^x$ be $l_1$, and the tangent line at point B $(x_0, y_2)$ of the curve $y=(1-x)e^{-x}$ be $l_2$. If there exists $x_0 \in [0, \frac{3}{2}]$ such that $l_1 \perp l_2$, then the range of the real number $a$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 \leq a \leq \frac{3}{2}}
Calculate the sum \( S = \sum_{k=1}^{2015} (-1)^{\frac{k(k+1)}{2}} k \) by considering the residue of \( k \) modulo 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the magnitude of the product of all complex numbers  $c$  such that the recurrence defined by  $x_1 = 1$ ,  $x_2 = c^2 - 4c + 7$ , and  $x_{n+1} = (c^2 - 2c)^2 x_n x_{n-1} + 2x_n - x_{n-1}$  also satisfies  $x_{1006} = 2011$ .

*Author: Alex Zhu*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( {a}_{1},{a}_{2},\ldots ,{a}_{300} \) be nonnegative real numbers,with 
$$
\sum_{i=1}^{300} \alpha_i = 1.
$$
 Find the maximum possible value of \( \mathop{\sum }\limits_{{i \neq  j,i \mid  j}}{a}_{i}{a}_{j} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
 \(\frac{4}{9}\)

}
Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of $N$ colors. The coloring is done in such a way that any path between any two nonadjacent vertices consists of edges with at least two different colors. What is the smallest possible value of $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths \(3\) and \(4\) units. In the corner where those sides meet at a right angle, he leaves a small unplanted square \(S\) so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from \(S\) to the hypotenuse is \(2\) units. What fraction of the field is planted? The original answer is in \(\frac{m}{n}\) format, please give the value of \(m + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{292}
Find the matrix that corresponds to reflecting over the $x$-axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}}
Two ants, one starting at $(-1,1)$ and the other at $(1,1)$, walk to the right along the parabola $y=x^{2}$ such that their midpoint moves along the line $y=1$ with constant speed 1. Determine the speed of the left ant when it first reaches the line $y=\frac{1}{2}$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{3} - 3}
In a parallelogram $ABCD$ with an area of $240$, points $E$ and $H$ are the midpoints of sides $AD$ and $AB$, respectively. On side $BC$, point $G$ is such that $BG = 2GC$, and on side $CD$, point $F$ is such that $DF = 3FC$. Point $K$ is on side $AC$ such that the area of triangle $EKF$ is $33$. Determine the area of triangle $HKG$. \( \text{Provide your answer in square units.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
In a rectangle $ABCD$, the sides $AB = 3$ and $BC = 4$ are given. Point $K$ is at distances $\sqrt{10}$, 2, and 3 from points $A$, $B$, and $C$, respectively. Find the angle between lines $CK$ and $BD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arcsin \left(\frac{4}{5}\right)}
Consider a three-person game involving the following three types of fair six-sided dice. - Dice of type $A$ have faces labelled $2,2,4,4,9,9$. - Dice of type $B$ have faces labelled $1,1,6,6,8,8$. - Dice of type $C$ have faces labelled $3,3,5,5,7,7$. All three players simultaneously choose a die (more than one person can choose the same type of die, and the players don't know one another's choices) and roll it. Then the score of a player $P$ is the number of players whose roll is less than $P$ 's roll (and hence is either 0,1 , or 2 ). Assuming all three players play optimally, what is the expected score of a particular player?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{9}}
Find the smallest integer \(x\) such that \(\gcd(50 + x^2, 4x + 2) = 6\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the number of ways to insert + and - signs between the numbers 0, 1, 2, ..., 12 such that the resulting expression evaluates to a multiple of 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{816}
Given a natural number \( n \), find the number of distinct triplets of natural numbers whose sum is \( 6n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3n^2}
What is the result of the sum of three Cantor sets, denoted as \( C + C + C \), where \( C \) is the usual Cantor set obtained from the interval [0,1]? Express your answer as a set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 3]}
Determine the values of the real parameter \( \alpha > 0 \) for which the function \( \frac{1}{|x|^{\alpha} + |y|^{\alpha} +1} \) is in \( L^2(\mathbb{R}^2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > 1}
Evaluate the integral: \[ \int_{0}^{\infty}\frac{\sin\left(x^{2}\right) + \cos\left(x^{2}\right) - 1} {x^{2}}\,\mathrm{d}x. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The school invites 6 out of 10 students to participate in a study seminar, among which students A and B cannot attend at the same time. Calculate the number of different ways to invite the students.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{140}
Given the numbers \(\log _{\sqrt{x+34}}(2x+23), \log _{(x+4)^{2}}(x+34), \log _{\sqrt{2x+23}}(-x-4)\). For which values of \(x\) are two of these numbers equal and the third one greater by 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-9}
For each positive integer \( n \), define \( A_{n} = \frac{20^{n} + 11^{n}}{n!} \), where \( n! = 1 \times 2 \times \cdots \times n \). Find the value of \( n \) that maximizes \( A_{n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Find the number of subsets of $\{1,3,5,7,9,11,13,15,17,19\}$ where the elements in the subset add to $49$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
22
}
Let \( s(n) \) denote the sum of the digits (in base ten) of a positive integer \( n \). Compute the number of positive integers \( n \) of at most \( 10^4 \) that satisfy
\[ s(11n) = 2s(n). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2530}
Determine the sum of absolute values for the complex roots of \( 20x^8 + 7ix^7 - 7ix + 20 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In a finite sequence of real numbers, the sum of any 7 consecutive terms is negative, while the sum of any 11 consecutive terms is positive. What is the maximum number of terms this sequence can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Consider the rectangular hyperbola \(xy = c^2\). Normals at points \(P, Q, R,\) and \(S\) on the curve are concurrent and meet at point \(O(h, k)\). Find the value of \(OP^2 + OQ^2 + OR^2 + OS^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3(h^2 + k^2)}
Determine the number of points of discontinuity for the function \( f(\theta) = \int_{-1}^{1} \frac{\sin\theta \, dx}{1-2x\cos\theta +x^2} \) where \( \theta \in [0,2\pi] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the extremal function $y(x)$ that satisfies the integral \( \int^{3}_{0}(x(y')^3 -3y(y')^2)dx \) with the boundary conditions $y(3) = 1$ and $y(0) = 7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = -2x + 7}
Calculate the cardinality of the stabilizer \( \operatorname{Stab}_{G}(\langle e_1, e_3 \rangle) \) for the action of \( G=GL(V) \) on \( X=\{W\subset V:\text{sub-vector space,}\dim =2\} \), where \( V=\Bbb F_2^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
In a ten-story building, you live on the ground floor, and each of your nine friends lives on a different floor. You randomly draw the numbers 1 through 9 from a hat, determining the order in which you visit your friends' floors. If the distance between each floor is 4 meters, calculate the average number of meters you travel by elevator, moving from one floor to the next in the order drawn.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{380}{3}}
已知关于$x$的不等式$$(2ax-\ln x)\left(x^{2}-(a+1)x+1\right)\geqslant0$$ 对于任意$x\in(0,+\infty)$恒成立。则实数$a$的取值范围是_______。标准答案格式为$\left[\frac{a}{be},c\right]$，计算a+b+c的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the limit of the sequence \((z_n)\) defined by \[ z_n = \frac{(1+2)^{p+1}+(1+2+3)^{p+1}+\cdots+(1+2+\cdots+n)^{p+1}}{n^{p+2}} \] using the Stolz-Cesaro theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the inverse of the element \( \beta \) in the finite field \( GF(81) \), where \( \beta \) satisfies the equation \( \beta^4 + \beta^2 + 3 = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\beta}
There is exactly one real number \( x \) with the property that both \( x^{2} - t x + 36 = 0 \) and \( x^{2} - 8 x + t = 0 \).

What is the value of \( x \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Rhombus $ABCD$ is similar to rhombus $BFDE$. The area of rhombus $ABCD$ is $24$ and $\angle BAD = 60^\circ$. What is the area of rhombus $BFDE$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
As shown in the diagram, in square $ABCD$, extend $BA$ to $G$ such that $AG = BD$. What is the measure of $\angle BCG$ in degrees?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67.5^\circ}
On a sheet of paper, a grid of \( n \) horizontal and \( n \) vertical lines is drawn. How many distinct closed \( 2n \)-segment broken lines can be drawn along the grid lines such that each broken line passes through all horizontal and all vertical lines at least once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n!(n-1)!}{2}}
Given propositions $p$ and $q$:
- Proposition $p$: The function $f(x)=x^{2}+ax-2$ has exactly one root in the interval $[-1,1]$.
- Proposition $q$: The inequality $x^{2}+3(a+1)x+2 \leqslant 0$ holds true for all $x$ in the interval $[\frac{1}{2}, \frac{3}{2}]$.
If the compound proposition "$p$ and $q$" is false, find the range of values for the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > -\frac{5}{2}}
Find the monotonic increasing interval of the function $f(x)=( \frac {1}{2})^{ \sqrt {x-x^{2}}}$. Express your answer as an interval using standard interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\frac{1}{2}, 1]}
A cube with a volume of 1 cubic centimeter (cm3) was cut by the number of vertexes from a cube with one edge 3 centimeters (cm) long. Find the volume of the remaining part.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$(m, n)=(0, 0)$}
Determine the appropriate limits for the parameter $t$ in the parametrization $(a\cos t, a\sin t, b-b\cos t)$ to describe the curve of intersection between the cylinder $x^2+y^2=a^2$ and the plane $\frac{x}{a}+\frac{z}{b}=1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 2\pi]}
Find a set of numbers \( S = \{s_1, s_2, \ldots, s_n\} \) such that the sum of the greatest common divisors of all subsets of \( S \) equals the sum of the elements of \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{1\}}
Given the infinite series \( \alpha = \frac{5}{2!\cdot 3} + \frac{5\cdot 7}{3!\cdot 3^2} + \frac{5\cdot 7 \cdot 9}{4!\cdot 3^3} + \cdots \), find the value of \( \alpha^2 + 4\alpha \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Let $P_k(x) = (x-k)(x-(k+1))$. Kara picks four distinct polynomials from the set $\{P_1(x), P_2(x), P_3(x), \ldots ,$ $P_{12}(x)\}$ and discovers that when she computes the six sums of pairs of chosen polynomials, exactly two of the sums have two (not necessarily distinct) integer roots! How many possible combinations of four polynomials could Kara have picked?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108
}
Evaluate the limit: $$\lim_{n \to \infty} \int_{0}^{1}\frac{n\cos(x)}{1+n^2x^{3/2}}\, dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}