inclusionAI/Ring-lite-rl-data huggingface dataset:
Find all pairs of natural numbers  $ (a, b)$  such that  $ 7^a \minus{} 3^b$  divides  $ a^4 \plus{} b^2$ .

*Author: Stephan Wagner, Austria*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(a, b) = (2, 4)}
满足方程 $\frac{1}{k_{1}}+\frac{1}{k_{2}}+\cdots+\frac{1}{k_{n}}=1, k_{1} k_{2} \cdots k_{n} \leq 200$ 的所有有序正整数组 $\left(n, k_{1}, \cdots, k_{n}\right)$的组数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the least number that must be subtracted from 899830 so that the remaining number is divisible by a certain number. The least number is 6. What is the divisor?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the number of Hahn-Banach extensions of a non-zero continuous linear functional \( f:(c_{00},\|\cdot\|_1)\to \mathbb{C} \) to the entire space \( (\ell^1,\|\cdot\|_1) \). Here, \( c_{00} \) is the set of all sequences with finitely many non-zero terms, and \( \|x\|_1 = \sum_{n=1}^\infty |x_n| \) for \( x = (x_1, x_2, \ldots) \in c_{00} \) or \( \ell^1 \), where \( \|x\|_1 < \infty \) for \( x \in \ell^1 \). Use the Hahn-Banach theorem on normed linear spaces, which states that a linear functional defined on a subspace can be extended to the whole vector space while preserving the norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
某足球赛有16个城市参加，每市派出2个队，根据比赛规则，每两队之间至多赛一场，同城两队之间不进行比赛.赛过一段时间后，发现除A城甲队外，其他各队已赛过的场数各不相同.问A城乙队已赛过几场？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Transform the equation \(x_{uu} + 2x_{uv} + x_{vv} = 0\) using the substitutions \(p = u + v\) and \(q = u - v\) to express it in terms of \(p\) and \(q\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x_{pp} = 0}
A regular dodecahedron is projected orthogonally onto a plane, and its image is an $n$-sided polygon. What is the smallest possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In a government institute, 20 staff members are employed. In a festival season, their boss gives a festival allowance to all. He instructs his accountant to calculate for 30 days at a certain rate. He also says the balance payment will be given to the driver and cleaning staff and to be adjusted from petty cash. He had given Rs. 65000 to the accountant. The amount taken from petty cash was Rs. 1000. What was the daily rate for the festival allowance?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110}
Find the smallest integer $k$ for which the conditions
 (1) $a_1,a_2,a_3\cdots$ is a nondecreasing sequence of positive integers
 (2) $a_n=a_{n-1}+a_{n-2}$ for all $n>2$
 (3) $a_9=k$
 are satisfied by more than one sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{748}
If $$A=\int_0^1(1-x^{2023})^{\frac{1}{2022}}dx$$ and $$B=\int_0^1(1-x^{2022})^{\frac{1}{2023}}dx$$ then find the value of $S$ if $$S=\frac{A^3-B^3}{B^3}+\frac{A^4+B^4}{B^4}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a set $A$ which contains $n$ elements. For any two distinct subsets $A_{1}$, $A_{2}$ of the given set $A$, we fix the number of elements of $A_1 \cap A_2$. Find the sum of all the numbers obtained in the described way.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \left( 2^{2n-3} - 2^{n-2} \right)}
Given a point  $P$  on a triangular piece of paper  $ABC,$  consider the creases that are formed in the paper when  $A, B,$  and  $C$  are folded onto  $P.$   Let us call  $P$  a fold point of  $\triangle ABC$  if these creases, which number three unless  $P$  is one of the vertices, do not intersect.  Suppose that  $AB=36, AC=72,$  and  $\angle B=90^\circ.$   Then the area of the set of all fold points of  $\triangle ABC$  can be written in the form  $q\pi-r\sqrt{s},$  where  $q, r,$  and  $s$  are positive integers and  $s$  is not divisible by the square of any prime.  What is  $q+r+s$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{597}
Ann made a 3-step staircase using 18 toothpicks. To complete a 5-step staircase, how many additional toothpicks does she need? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Let $A_{1} A_{2} \ldots A_{6}$ be a regular hexagon with side length $11 \sqrt{3}$, and let $B_{1} B_{2} \ldots B_{6}$ be another regular hexagon completely inside $A_{1} A_{2} \ldots A_{6}$ such that for all $i \in\{1,2, \ldots, 5\}, A_{i} A_{i+1}$ is parallel to $B_{i} B_{i+1}$. Suppose that the distance between lines $A_{1} A_{2}$ and $B_{1} B_{2}$ is 7 , the distance between lines $A_{2} A_{3}$ and $B_{2} B_{3}$ is 3 , and the distance between lines $A_{3} A_{4}$ and $B_{3} B_{4}$ is 8 . Compute the side length of $B_{1} B_{2} \ldots B_{6}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \sqrt{3}}
In triangle \(ABC\), the angle bisectors \(AK\) and \(CL\) and the median \(BM\) are drawn. It turns out that \(ML\) is the angle bisector of \(\angle AMB\) and \(MK\) is the angle bisector of \(\angle CMB\). Find the angles of triangle \(ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^
\circ, 30^
\circ, 120^
\circ}
A random variable $X_1$ has a uniform distribution on the interval $(0, 1)$. Another random variable $X_2$ has a uniform distribution on the interval $(0, X_1)$. A third random variable $X_3$ has a uniform distribution on the interval $(0, X_2)$. Calculate the variance of $12(X_1 + X_2 + X_3)$. \( \text{Provide your answer as a single numerical value.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{565}{12}}
On the board, the numbers 25 and 36 are written. Two students play alternately: on each turn, they choose any two numbers already written on the board and add a new number equal to the absolute value of their difference (it is not allowed to rewrite a number that is already on the board). The player who cannot make a move loses. Who has a winning strategy: the first or the second player?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Second player}}
Determine the number of elements of order 15 in the symmetric group $S_{13}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79567488}
Determine the number of ways to arrange the five vowels—A, E, I, O, U—along with 20 X's in a row such that no X is at an extreme position and there are at least 4 X's between any two vowels.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4200}
How many of the first 1000 positive integers can be written as the sum of finitely many distinct numbers from the sequence $3^{0}, 3^{1}, 3^{2}, \ldots$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
A deer passed a certain tree at a constant speed of 50 miles per hour while being chased by a cheetah. Then, 2 minutes later, the cheetah passed the same tree at a certain speed. If both animals maintained their speeds, the cheetah caught up with the deer 1 minute after the cheetah passed the tree. What was the speed of the cheetah?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Square $EFGH$ is inside the square $ABCD$ so that each side of $EFGH$ can be extended to pass through a vertex of $ABCD$. Square $ABCD$ has side length $\sqrt {50}$ and $BE = 1$. What is the area of the inner square $EFGH$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
There are 96 matches divided into three piles. First, take from the first pile the same number of matches as in the second pile and add them to the second pile. Then, take from the second pile the same number of matches as in the third pile and add them to the third pile. Finally, take from the third pile the same number of matches as in the first pile and add them to the first pile. At this point, all three piles have the same number of matches. Determine how many matches were originally in the first pile, second pile, and third pile.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44, 28, 24}
若$2$是${{4}^{a}}$与${{2b}}$的等比中项，则$a\left( a+b \right)$的最大值为______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1$}
Let $N=30^{2015}$.  Find the number of ordered 4-tuples of integers $(A,B,C,D)\in\{1,2,\ldots,N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3+Bn^2+2Cn+D$ is divisible by $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A pyramid $SABC$ has a triangular base $ABC$ with a height of $9\sqrt{3}/2$. Triangle $SAB$ is equilateral with side length $6\sqrt{3}$, and the other sides of the pyramid are equal ($SC=AC=BC$). Calculate the volume of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27\sqrt{69}}{2}}
Let \( K = \mathbb{R}(X) \) be the rational function field, and \( F = \mathbb{R}(X^4 - \frac{1}{X^4}) \). Let \( L \) be the Galois closure of the extension \( K/F \). Determine the extension degree \([L:F]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
The octagon $P_1P_2P_3P_4P_5P_6P_7P_8$ is inscribed in a circle, with the vertices around the circumference in the given order.  Given that the polygon $P_1P_3P_5P_7$ is a square of area 5, and the polygon $P_2P_4P_6P_8$ is a rectangle of area 4, find the maximum possible area of the octagon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{5}}
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form $y=ax+b$ with a and b integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{810
}
Determine the number of countable non-isomorphic models of the theory $\text{ACF}_p$. Note that $\text{ACF}_p$ is $\kappa$-categorical for all uncountable $\kappa$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
On the side \( BC \) of triangle \( ABC \), point \( K \) is marked.

It is known that \( \angle B + \angle C = \angle AKB \), \( AK = 4 \), \( BK = 9 \), and \( KC = 3 \).

Find the area of the circle inscribed in triangle \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{35}{13} \pi}
A permutation $a_1, a_2, ..., a_{13}$ of the numbers from $1$ to $13$ is given such that $a_i > 5$ for $i=1,2,3,4,5$. Determine the maximum possible value of\[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \]Evan Chen
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
45
}
Toss a die 6 times, and let the number obtained on the $i$-th toss be $a_{i}$. If there exists a positive integer $k$ such that $\sum_{i=1}^{k} a_{i}=6$, what is the probability $p=\frac{n}{m}$, where $m$ and $n$ are coprime positive integers. Find the value of $\log _{6} m - \log _{7} n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider an infinite sequence of tosses of a fair coin. Using a Markov Chain approach with a modified state-change matrix, calculate the expected number of tosses until the pattern HHTT appears.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
对于一个实数 $x$, 令 $\{x\}=x-[x]$. 记 $S=\int_{0}^{2024} \min \left(\left\{\frac{x}{8}\right\},\left\{\frac{x}{4}\right\}\right) \mathrm{d} x$, 则 $[S]=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{759}
Given the sequence \(a_k\) where \(a_k a_{k-1} + a_{k-1} a_{k-2} = 2a_k a_{k-2}\) for \(k \geq 3\) and \(a_1 = 1\), let \(S_q = \sum_{k=1}^{q} \frac{1}{a_k}\). If \(\frac{S_{2q}}{S_q}\) is independent of \(q\), find \(\frac{1}{a_{2016}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4031}
Let $X$ and $Y$ be independent random variables, each following a standard normal distribution, $N(0,1)$. Calculate the probability $P(|Y| \ge \frac{1}{\sqrt{3}}|X|)$. Provide your answer as a single probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Find the asymptotic equivalent of the sequence \((u_n)_{n \geq 1}\) defined by \(u_1 > 0\) and \(u_{n+1} = \frac{u_n}{n} + \frac{1}{n^2}\) as \(n \to +\infty\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{n^2}}
Determine the interval of convergence for the power series representation of the function \( f(x) = x^{0.6} \) centered at \( a = 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 8]}
已知下列结论成立：在复平面上的多项式 $f(z), g(z)$ 和实数 $r>0$ ，若对 $|z|=r$ ，都有 $|g(z)|<|f(z)|$ ，则在 $\{z||z|<r\}$ 中， $f(z)$ 与 $f(z)+g(z)$ 的零点数相等 (计算重数)。现已知多项式 $z^{9}+2 z^{5}-8 z^{3}+3 z+1$, 求其在 $\{z|1<|z|<2\}$ 上的零点个数（计算重数）.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the number of different possible rational roots of the polynomial:
\[6x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 10 = 0.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Calculate the curvature of a surface with isometric coordinates given by \(\frac{1}{x^2}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) using the Laplace–Beltrami operator.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
20. [10] Find the volume of the set of points $(x, y, z)$ satisfying
$$
\begin{array}{l}
x, y, z \geq 0 \\
x+y \leq 1 \\
y+z \leq 1 \\
z+x \leq 1
\end{array}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{1}{4}"]}
A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$30\pi - 9\sqrt3$}
Determine the dimension of the vector space $V_r$ over the complex numbers $\mathbb{C}$, where $V_r$ consists of all holomorphic functions $f: \mathbb{C} \rightarrow \mathbb{C}$ such that $|f(z)| < \infty$ for all $z \in \mathbb{C}$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
$\underline{\text { Gordrin P.K. }}$.

Between two parallel lines, a circle of radius 1 is placed, touching both lines, and an isosceles triangle, the base of which lies on one of the lines, and the vertex on the other. It is known that the triangle and the circle have exactly one common point and that this point lies on the inscribed circle of the triangle. Find the radius of the inscribed circle of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
In the following figure, the square \(ABCD\) is divided into three rectangles of equal area.

If the length of the segment \(BM\) is equal to 4, calculate the area of the square \(ABCD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
A jar was filled with jelly beans so that  $54\%$  of the beans were red,  $30\%$  of the beans were green, and  $16\%$  of the beans were blue. Alan then removed the same number of red jelly beans and green jelly beans from the jar so that now  $20\%$  of the jelly beans in the jar are blue. What percent of the jelly beans in the jar are now red?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55\%}
Compute the sum
 \( \mathop{\sum }\limits_{{k = 0}}^{{1273}}\frac{1}{1 + {\tan }^{2548}\left( \frac{k\pi }{2548}\right) } \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{637.5
}
When Cheenu was a young man, he could run 20 miles in 4 hours. In his middle age, he could jog 15 miles in 3 hours and 45 minutes. Now, as an older man, he walks 12 miles in 5 hours. What is the time difference, in minutes, between his current walking speed and his running speed as a young man?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
A unit square is divided into rectangles, each colored either yellow or blue. For a blue rectangle, its associated number is the width divided by the height. For a yellow rectangle, the number is the height divided by the width. Let $x$ be the sum of all these numbers. If the total area of the blue rectangles equals the total area of the yellow rectangles, find the smallest possible value of $x$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{2}}
Find the maximal value of the sum \( \sum_{n=0}^{\infty} a_n \), where \( a_n \) is the distance from a real number \( a \) to the nearest number of the form \( \frac{m}{2^n} \) with \( m \in \mathbb{Z} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Calculate the expectation \( \mathbb{E}\left[|\langle a, Cx \rangle|^2\right] \) where \( a \sim \mathcal{N}(0,I_m) \) is independent and \( C \in \mathbb{R}^{m \times n} \) has iid \( \mathcal{N}(0,1) \) entries. Fix \( x \in \mathbb{R}^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \|x\|^2}
在一个圆周上给定十二个红点. 求 $n$的最小值，使得存在以红点为顶点的 $n$ 个三角形，满足以红点为端点的每条弦，都是其中某个三角形的一条边。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Find the value of $x$ in terms of $a$ that maximizes the function $$y=\left(1-e^{-\frac{x}{a}}\right)^{\frac{x}{a}}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \ln 2}
Find the minimum value of \( M_{a,b} \) for the function \( f_{a,b}(x) = \{x+a\} + 2\{x+b\} \), where \( \{x\} \) denotes the fractional part of \( x \), and the range of \( f_{a,b}(x) \) is \([m_{a,b}, M_{a,b})\). Consider all real numbers \( a \) and \( b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{3}}
Let \( U, V, W, Y, Z \) be independent random variables with the following distributions:

- \( U \sim \text{Exp}(1/5) \)
- \( V \sim \text{Exp}(2/3) \)
- \( W \sim \mathcal{U}(1, 5) \) (uniform distribution on \([1, 5]\))
- \( Y \sim \mathcal{N}(-13, 1) \)
- \( Z \sim \mathcal{N}(0, 5) \)

Determine the expected value:
\[ \mathbb{E}[(Z + \min(U+W, V+W)) \cdot (Y+Z)] \]

Hint: Use the properties of independence and linearity of expectation to simplify the computation. You do not need to derive any density or distribution functions by hand; all necessary means can be computed directly via the parameters of the distributions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-49}
Player A has a utility function given by \( u(x) = \sqrt{x} \). Determine the maximum amount that player A would be willing to pay to play first in the game.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Let  $k>1$  be a given positive integer. A set  $S$  of positive integers is called *good* if we can colour the set of positive integers in  $k$  colours such that each integer of  $S$  cannot be represented as sum of two positive integers of the same colour. Find the greatest  $t$  such that the set  $S=\{a+1,a+2,\ldots ,a+t\}$  is *good* for all positive integers  $a$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2k-1}
The feet of the altitudes of an acute-angled triangle $ABC$ form the vertices of another triangle, whose perimeter is $2p$. Find the area of triangle $ABC$ if the radius of its circumscribed circle is $\boldsymbol{R}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Rp}
Find the number of subsets $\{a, b, c\}$ of $\{1,2,3,4, \ldots, 20\}$ such that $a<b-1<c-3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{680}
I have chosen five of the numbers $\{1,2,3,4,5,6,7\}$. If I told you what their product was, that would not be enough information for you to figure out whether their sum was even or odd. What is their product?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{420}
Pat Peano has plenty of 0's, 1's, 3's, 4's, 5's, 6's, 7's, 8's and 9's, but he has only twenty-two 2's. How far can he number the pages of his scrapbook with these digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$119$}
Find all positive integers $n > 1$ such that the polynomial $x^4 + 3x^3 + x^2 + 6x + 10$ belong to the ideal generated by the polynomial $x^2 + x + 1$ in $\Bbb Z_n[x]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2, 3, 6}
For which values of \( x \) does the series \( \sum_{n=1}^{\infty} \frac{\sin(nx)}{n^2} \) converge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Find the greatest value of \( n \) such that the equation \( 2\tan^2x - 5\sec x = 1 \) has exactly 7 distinct solutions for \( x \in [0, \frac{n\pi}{2}] \), where \( n \) is a natural number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Find the derivative with respect to the time-varying variable $x_t$ of the integral $\int_0^T y(x_s) \, ds$, where $T > t$ and $y(x_t)$ is a function of $x_t$, and $x_t$ is a function of $t$. Express your answer in terms of $x_t$ and $y(x_t)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y'(x_t)}
Shivani has a single square with vertices labeled $ABCD$. She is able to perform the following transformations:
$\bullet$ She does nothing to the square.
$\bullet$ She rotates the square by $90$, $180$, or $270$ degrees.
$\bullet$ She reflects the square over one of its four lines of symmetry.
For the first three timesteps, Shivani only performs reflections or does nothing. Then for the next three timesteps, she only performs rotations or does nothing. She ends up back in the square’s original configuration. Compute the number of distinct ways she could have achieved this.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
784
}
Determine the number of possible ordered pairs \((a,b)\) of integers such that the function \(f(x)=\sqrt{\frac{x^2+ax+4}{x^2+bx+16}}\) is defined for all real \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135}
In the quadrilateral  $ABCD$ ,  $AB = BC = CD$  and  $\angle BMC = 90^\circ$ , where  $M$  is the midpoint of  $AD$ . Determine the acute angle between the lines  $AC$  and  $BD$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find all ordered pairs  $(a,b)$  of positive integers that satisfy  $a>b$   and the equation  $(a-b)^{ab}=a^bb^a$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(4, 2)}
Find the equation of the hyperplane that contains the skew lines given by \( \mathbf{p}(s) = (s, 2s, 0, 1) \) and \( \mathbf{r}(t) = (1, 0, t, t+1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{w - z = 1}