inclusionAI/Ring-lite-rl-data huggingface dataset:
Evaluate the surface integral \( \iint_S F \cdot n \,ds \) for the vector field \( F(x,y,z)=(z^2-x,-xy,3z) \) over the surface \( S \) of the solid bounded by the equations \( z=4-y^2, x=0, x=3 \), and \( z=0 \), with the normal vector pointing outward. Do not use the Divergence Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Let $U$ be the vector subspace of $\mathbb{R}^5$ generated by the set $S_U = \{(1,3,-3,-1,-4),(1,4,-1,-2,-2)\}$, and let $V$ be the vector subspace of $\mathbb{R}^5$ generated by the set $S_V = \{(1,6,2,-2,3),(2,8,-1,-6,-5),(1,3,-1,-5,-6)\}$. Determine the dimension of the intersection $U \cap V$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For any positive integer \( a,\sigma \left( a\right) \) denotes the sum of the
 positive integer divisors of \( a \) . Let \( n \) be the least positive int
 such that \( \sigma \left( {a}^{n}\right)  - 1 \) is divisible by 2021 for all positive
 integers \( a \) . Find the sum of the prime factors in the prime
 factorization of \( n \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
125}
Calculate the number of permutations \(\alpha\) such that \(\alpha\sigma\alpha^{-1} = \tau\), where \(\sigma = (14735)(698)(2)\) and \(\tau=(16973)(485)(2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the real values of $p$ for which the series \( \sum_{n=2}^{\infty}\frac{1}{(n^2\log n)\cdot(n^{\frac{1}{n}}-1)^p} \) is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1]}
Consider the polynomial \(f(x) = x^5 - 4x + 2\). Let \(L\) be the complex splitting field of \(f(x)\) over \(\mathbb{Q}\). Determine the number of subfields \(K\) of \(L\) such that \((K : \mathbb{Q}) = 2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
At a class reunion with 45 attendees, it is known that any two people with the same number of acquaintances among the attendees do not know each other. What is the maximum number of pairs of acquaintances that could exist among the attendees?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{870}
How many people do you need in a room so that you expect there to be a shared birthday on a Monday, assuming there are 52 Mondays in a year and 365 days in a year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{73}
Identify the location and direction of the tangent vector $t_y$ to the surface $S$ defined by $z = \frac{1}{1+x^2+y^2}$, restricted to the plane $y=0$. The surface is given by $\mathbf{r}(x,y) = x\mathbf{i} + y\mathbf{j} + \frac{1}{1+x^2+y^2}\mathbf{k}$. Determine $t_y$ in this plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbf{j}}
In a country, 3% of the population has a certain disorder. A test for this disorder has a 97% true positive rate and a 2% false positive rate. If a person takes the test twice independently and receives one positive result and one negative result, what is the probability that they actually have the disorder?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{205}}
Determine the number of ordered pairs \((x, y)\) that satisfy the following system of equations:
\[\begin{align*}
\sin(x+y) &= \cos(x+y) \\
x^2 + y^2 &= \left(\frac{1995\pi}{4}\right)^2
\end{align*}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2822}
Let \( f \) be a function defined on \(\{0, 1, \cdots, 2005\}\) that takes non-negative integer values. The function satisfies the following conditions for any variable \( x \) within its domain:

\[ f(2x+1) = f(2x), \]
\[ f(3x+1) = f(3x), \]
\[ f(5x+1) = f(5x). \]

Determine the maximum number of distinct values this function can take.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{536}
Calculate the flux of the vector field \( F = (x^3, 4y^3, z^3) \) through the surface defined by \( x^2 + 4y^2 + z^2 = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{6\pi}{5}}
Find the norm of the differential operator \( \frac{d}{dx} \) on the space of polynomials of degree \( \leq n \) on the interval \([-a,a]\), where \( a > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n^2}{a}}
For a bounded set \(\Omega \subset \mathbb{R}^d\), determine the maximum value of \(\epsilon\) such that there exists an infinite subset \(\mathcal{A} \subset \mathcal{B}(\Omega)\) with the property that \(|A \triangle B| \geq \epsilon\) for all distinct sets \(A, B \in \mathcal{A}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{|\Omega|}{2}}
Let $M$ be a subset of the set of $2021$ integers $\{1, 2, 3, \ldots, 2021\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$, we have $|a + b - c| > 10$. Determine the largest possible number of elements of $M$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1006}
Asif has cubes of three colors: red, green, and blue. He builds a tower by stacking the cubes, with the restriction that no more than 4 cubes of each color can be used. Asif stops building the tower as soon as it contains 4 cubes of any two colors. How many different towers can Asif build?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34650}
Evaluate the series \( \sum_{k=1}^{\infty} e^{-\pi k^2} \left( \pi k^2 - \frac{1}{4} \right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{8}}
In the Monty Hall problem with an infinite number of doors, if the contestant chooses a door with an infinitesimally small probability, what is the probability of winning by switching?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find a bound for the probability that the sum of independent and identically distributed random variables is negative, given that for each variable $a_i$, $Pr(a_i < 0) < e^{-d}$, where $d$ is a positive integer. Specifically, determine a bound for $Pr(\sum_{i=1}^{n}{a_i} < 0)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n e^{-d}}
Evaluate $\int_e^{e^2} \frac{4(\ln x)^2+1}{(\ln x)^{\frac 32}}\ dx.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$3e^2\sqrt{2} - 2e$}
Determine the image of the set \( \mathcal{F} \) under the map that takes a function \( f(x) \) in \( \mathcal{F} \) to \( \int_{-1}^{1} f(x) \, dx \). Here, \( \mathcal{F} \) is the subset of real-valued continuous functions on \([-1, 1]\) such that \( f(x) \geq 0 \) for all \( x \) in \([-1, 1]\) and \( \limsup_{n\rightarrow \infty}\int_{-1}^{1}f(x)^{n}dx <\infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 2]}
If $i^2 = -1$, then the sum
\[ \cos{45^\circ} + i\cos{135^\circ} + \cdots + i^n\cos{(45 + 90n)^\circ} \]
\[ + \cdots + i^{40}\cos{3645^\circ} \]
equals
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \frac{\sqrt{2}}{2}(21 - 20i) \]}
Find a sufficient statistic \(T(X)\) that serves as an unbiased estimator for \(1/p\) in a geometric distribution with probability mass function \(f(x;p) = p(1-p)^x\), where \(x = 0, 1, 2, \ldots\) and \(0 < p < 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X + 1}
In a land of knights (who always tell the truth) and liars (who always lie), 10 people are seated at the vertices of a regular decagon at a round table. Among them, there are liars. A traveler can stand at any point outside the table and ask the seated individuals: "What is the distance from me to the nearest liar among you?" Each person will answer the question. What is the minimum number of questions the traveler must ask to guarantee identifying all the liars?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the lower bound for the limit below, given that the functions $u_\epsilon$ converge uniformly to $x \mapsto \xi \cdot x$:
$$\lim_{\epsilon \to 0} \epsilon \sum_{(x,y) \in (\epsilon \mathbb{Z}^3 \cap Q_1)^2, \ |x-y|=\epsilon} |u_\epsilon(x)-u_\epsilon(y)|^2$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|\xi|^2}
Determine the kernel of the homomorphism \( f: K[X,Y] \rightarrow K[T] \) defined by \( f(X) = T^2 \) and \( f(Y) = T^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(Y^2 - X^3)}
Given two matrices \( A \) and \( C \) of order \( n \times n \) and \( m \times n \) respectively, consider the block matrix:

\[ D=\left( \begin{matrix} C\\ CA\\ :\\ :\\ CA^{q-1} \end{matrix}\right) \]

where \( q \in \mathbb{N}^* \). Determine the smallest value of \( q \) such that \( \text{rank}(D) = n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Determine the order of growth of the function $f(z) = \sin(z^2)$ using the definition of order of growth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate

$$
\operatorname{tg} \frac{\pi}{47} \cdot \operatorname{tg} \frac{2 \pi}{47} + \operatorname{tg} \frac{2 \pi}{47} \cdot \operatorname{tg} \frac{3 \pi}{47} + \ldots + \operatorname{tg} \frac{k \pi}{47} \cdot \operatorname{tg} \frac{(k+1) \pi}{47} + \ldots + \operatorname{tg} \frac{2021 \pi}{47} \cdot \operatorname{tg} \frac{2022 \pi}{47}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2021}
Find an initial condition \( y(0) \neq (0,0,0) \) for the system of differential equations given by \( y_1'=12y_3 \), \( y_2'=y_1+13y_3 \), and \( y_3'=y_2 \) such that the solution \( y(x) \) approaches \( (0,0,0) \) as \( x \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-4, -3, 1)}
Determine the number of ways the sum \(1 + 2 + \cdots + 2007\) can be expressed as a sum of consecutive positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let  $ABC$  be an equilateral triangle, and let  $D$  and  $F$  be points on sides  $BC$  and  $AB$ , respectively, with  $FA=5$  and  $CD=2$ .  Point  $E$  lies on side  $CA$  such that  $\angle DEF = 60^\circ$ .  The area of triangle  $DEF$  is  $14\sqrt{3}$ .  The two possible values of the length of side  $AB$  are  $p \pm q\sqrt{r}$ , where  $p$  and  $q$  are rational, and  $r$  is an integer not divisible by the square of a prime.  Find  $r$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{989}
Compute the double integral \( I = \int_{-\infty}^{\infty}\int_{0}^{\infty} \cos k\xi \cdot u(\xi ) \, dkd\xi \), where \( u(\xi) = a_{0} \operatorname{sech}^{2} \left( \sqrt{\frac{a_{0}}{6+4a_{0}}} \, \xi \right) \) and \( a_0 \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi a_0}
Compute the area between the curves \( y = \cos(\sin(x)) \) and \( y = \sin(\cos(x)) \) on the interval \( 0 \leq x \leq 2\pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi J_0(1)}
Evaluate the limit:
$$ \lim_{x\to0}\frac{\sqrt[n]{a+x} - \sqrt[n]{a-x}}{x},\ a>0,\ n\in\Bbb N $$
without using L'Hospital's rule or generalized binomial expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{2\sqrt[n]{a}}{na}\)}
Evaluate the integral: $$\int_{0}^{\infty} \frac{\sin^4x}{x^4}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
How many different triangular pyramids can be formed using the vertices of a triangular prism?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
What is the probability that a subset $X$ of $\mathbb{R}$, formed by independently including each real number with a coin toss, is a Lebesgue measurable subset of $\mathbb{R}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the limit \( \lim_{n \rightarrow \infty} \frac{\text{ex}(n; P)}{\binom{n}{2}} \), where \( \text{ex}(n; P) \) is the extremal number for the Petersen graph \( P \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
存在 x_0<0，使得 x^2+|x-a|-2<0 恒成立，其中 a\in\bZ，则所有满足条件的 a 的和为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the function \( f(x) \) such that for all \( a > 1 \) and \( b > 1 \), the following equation holds:
\[ \int_{3}^{ab+2} f(x) \, dx = \int_{3}^{a+2} f(x) \, dx + \int_{3}^{b+2} f(x) \, dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{x - 2}}
给定以正方形内的 2022 个点，且这 2022 个点以及正方形的顶点中任三点不共线。在这 2022 以及正方形的顶点之间连一些线段，将正方形划分为一些三角形，并要求这些线段之间只可以在端点处有公共点，则所连线段（不包含原正方形的边）的总数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6067}
Determine the interval in which the constant \( k \) must lie if the curve \( f(x) = k^3 x + k^3 - 2 \) intersects the curve \( g(x) = \frac{1}{2} \ln x^2 \) at exactly one point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, \infty)}
Which positive integers satisfy that the sum of the number's last three digits added to the number itself yields $2029$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019}
Let $ p_{1} <p_{2} <\ldots <p_{24}$ be the prime numbers on the interval $ \left[3,100\right]$. Find the smallest value of $ a\ge 0$ such that $ \sum \limits_{i=1}^{24}p_{i}^{99!}  \equiv a\, \, \left(mod\, 100\right)$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
48
}
Determine the largest integer $n$ such that $99^{(n+1)} > 100^n$ without using logarithms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{457}
Evaluate the sum \(\sum_{k=2}^{n}{(k-1)(n-k)!{n \choose k}}\) using properties of binomial coefficients.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n! - 1}
Evaluate the integral: $$\int_{-\infty}^{\infty} \frac{1}{x^6+1}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\pi}{3}}
Given that the probability of \(\gcd(a,b) \in P\) is \(0.2749\ldots\), the probability that \(\gcd(a,b) = 1\) is \(0.6079\ldots\), and the probability that \(\gcd(a,b) \in \{1\} \cup P\) is \(0.8822 \ldots\), find the probability that \(\gcd(a,b) \in 2P\), where \(2P = \{2p : p \in P\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0687}
Evaluate the limit:
\[ \lim_{n \to \infty} \left( \frac{\sqrt[n+1]{(n+1)! (2n+1)!!}}{n+1} - \frac{\sqrt[n]{n!(2n-1)!!}}{n} \right) \]
where \( n! \) is the factorial and \((2n-1)!!\) is the double factorial. Use Stirling's formula, \( n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n \), to simplify the expression and determine the limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{e^2}}
If the graph of the function $f(x)=a+b\cos x+c\sin x$ passes through the points $(0,1)$ and $(-\frac{π}{4},a)$, and when $x\in[0,\frac{π}{2}]$, $|f(x)|≤\sqrt{2}$ always holds, then the range of real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0,4+2\sqrt{2}]}
A point light source is positioned at a distance from a sphere equal to three times the radius of the sphere. Determine the ratio of the illuminated area of the sphere to the lateral surface area of the cone of light formed by the light source and the sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{5}}
A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a right triangle $ABC$ with a right angle at $C$, the leg $BC$ has a length of 26. A circle is constructed with $BC$ as its diameter. From point $A$, a tangent $AP$ is drawn to this circle, distinct from $AC$. A perpendicular $PH$ is dropped to segment $BC$, intersecting segment $AB$ at point $Q$. If $BH : CH = 4 : 9$, find the area of triangle $BPQ$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Suppose there are 3 bank tellers, all currently occupied with a customer. You are waiting in line (and there are no other customers that will come in after you). All service times are iid exponential random variables with parameter $\lambda$. What is the probability that you are the last to leave?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
Determine the cardinality of the set \( \mathcal{R}[0,1] \) of all Riemann integrable real functions on the interval [0,1].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\mathfrak{c}}}
Determine the number of triples of natural numbers \((m, n, k)\) that satisfy the equation \( m + \sqrt{n+\sqrt{k}} = 2023 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2757680773}
Given the pressure function \( P(x, y, z) = \frac{x^2 + 2y^2}{1 + z^2} \) and the temperature function \( T(x, y, z) = 5 + xy - z^2 \), and the airplane's position described by \( (x(t), y(t), z(t)) = (2t, t^2 - 1, \cos t) \), find the time derivative of \((PT)^2\) at \(t = 0\) as observed from the airplane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-16}
Compute \( \lim_{|h| \to \infty} \int_{\mathbb{R}^n} |f(y+h) + f(y)|^p \, dy \) for a function \( f \in L^p(\mathbb{R}^n) \) where \( 0 < p < \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \|f\|_p^p}
Find a constant \( C > 0 \) such that for Hardy's integral operator \( T \) defined on \( L^1(0, +\infty) \) by \( Tf(x) = \frac{1}{x}\int_0^x f(t) \, dt \), the inequality \( \lambda m(|Tf| > \lambda) \leq C{\Vert f \Vert}_1 \) holds for all \( \lambda > 0 \) and \( f \in L^1(0, +\infty) \), where \( m \) denotes the Lebesgue measure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In triangle \(ABC\), a bisector \(BD\) is drawn. In triangles \(ABD\) and \(CBD\), bisectors \(DE\) and \(DF\) are drawn, respectively. Given that \(EF \parallel AC\), determine the measure of angle \(DEF\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
Given two independent exponential random variables $X_1$ and $X_2$ with mean 1, find the probability that the larger of the two, $X_{(2)}$, is greater than three times the smaller, $X_{(1)}$. That is, calculate $P(X_{(2)} > 3X_{(1)})$. Assume $X_1$ and $X_2$ are identically distributed with rate parameter $\lambda = 1$. Provide your answer as a probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the minimum number of steps required for a random walker, who moves with a 50% probability in either direction, to have at least a 99% probability of being at least 10 meters away from the starting point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{640000}
For the interval $\left[a,b\right]$, we define $b-a$ as the "length" of this interval. Given that $A$ and $B$ are both subsets of the set $\{x\left|\right.-1\leqslant x\leqslant 1\}$, $A=\{x\left|\right.a\leqslant x\leqslant a+1\}$, $B=\left\{{x|b-\frac{3}{2}≤x≤b}\right\}$, then the range of the "length" of the set $A\cap B$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\frac{1}{2}, 1]}
Given a Markov chain \( \{X_n, n \geq 0\} \) with states 1, 2, 3 and the transition probability matrix \( P = \begin{bmatrix}0 & 0.4 & 0.6 \\ 1 & 0 & 0 \\ 0.3 & 0.3 & 0.4\end{bmatrix} \), and an initial distribution \( (0.5, 0, 0.5) \), calculate \( P(X_1=3|X_2=1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.3}
The drag force on a boat varies jointly as the wetted surface area and the square of the velocity of a boat. If a boat going $6.5 \mathrm{mph}$ experiences a drag force of $86 \mathrm{~N}$ when the wetted surface area is $41.2 \mathrm{ft}^{2}$, how fast must a boat with $28.5 \mathrm{ft}^{2}$ of wetted surface area go in order to experience a drag force of $94 \mathrm{~N} ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$8.2 \mathrm{mph}$}
Let $O$ be a point with three other points $A,B,C$ and $\angle AOB=\angle BOC=\angle AOC=2\pi/3$.  Consider the average area of the set of triangles $ABC$ where $OA,OB,OC\in\{3,4,5\}$.  The average area can be written in the form $m\sqrt n$ where $m,n$ are integers and $n$ is not divisible by a perfect square greater than $1$.  Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Given \(a, b, c, d \in \mathbf{R}\) such that \(a^{2} + b^{2} + c^{2} + d^{2} = 1\), find the maximum value of \(ab + bc + cd + da + ac + 4bd\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{2}}
7.2. Find the volume of a regular triangular pyramid, the midpoint of the height of which is at a distance of 2 and $\sqrt{11}$ from the lateral face and the lateral edge, respectively. Round your answer to two decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{335.64}
On a natural number $n$, you are allowed two operations: 
1. Multiply $n$ by $2$, or 
2. Subtract $3$ from $n$. 

For example, starting with $8$, you can reach $13$ as follows: $8 \longrightarrow 16 \longrightarrow 13$. You need two steps, and you cannot do it in less than two steps. 

Starting from $11$, what is the least number of steps required to reach $121$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the number of positive integers less than or equal to $2017$ whose base-three representation contains no digit equal to $0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{222}
Let the set
\[ S=\{1, 2, \cdots, 12\}, \quad A=\{a_{1}, a_{2}, a_{3}\} \]
where \( a_{1} < a_{2} < a_{3}, \quad a_{3} - a_{2} \leq 5, \quad A \subseteq S \). Find the number of sets \( A \) that satisfy these conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{185}
Let $ABC$ be a triangle with $AB=7$, $BC=9$, and $CA=4$.  Let $D$ be the point such that $AB\parallel CD$ and $CA\parallel BD$.  Let $R$ be a point within triangle $BCD$.  Lines $\ell$ and $m$ going through $R$ are parallel to $CA$ and $AB$ respectively.  Line $\ell$ meets $AB$ and $BC$ at $P$ and $P^\prime$ respectively, and $m$ meets $CA$ and $BC$ at $Q$ and $Q^\prime$ respectively.  If $S$ denotes the largest possible sum of the areas of triangle $BPP^\prime$, $RP^\prime Q^\prime$, and $CQQ^\prime$, determine the value of $S^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Given the differential equation: $$y^{2}dy = x\left ( x dy - y dx \right ) e^{\frac{x}{y}}$$, determine the integrating factor that makes the equation exact.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{y^3}}