inclusionAI/Ring-lite-rl-data huggingface dataset:
Given that the three-digit numbers A88, 6B8, and 86C are divisible by 72, calculate the determinant of the matrix \( \begin{vmatrix} A & 6 & 8 \\ 8 & B & 6 \\ 8 & 8 & C \end{vmatrix} \) modulo 72.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A car that always travels in a straight line starts at the origin and goes towards the point $(8, 12)$. The car stops halfway on its path, turns around, and returns back towards the origin. The car again stops halfway on its return. What are the car’s final coordinates?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 3)}
Let $A$ be a full rank $m \times n$ matrix with $m < n$. Find the vector $x \in \mathbb{R}^n$ with the smallest 2-norm that satisfies the equation $Ax = b$. Express your answer in terms of the Moore-Penrose pseudoinverse of $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A^\dagger b}
One of the roots of
\[z^4 + az^3 + 5z^2 - iz - 6 = 0\]is $2i,$ where $a$ is a complex number.  Enter the other three roots, separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1,-1,-3i}
Determine the isomorphism class of the quotient group \( \frac{\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}}{\langle (2,0,1),(0,1,1) \rangle} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Find $x$ if
\[3 \arctan \frac{1}{4} + \arctan \frac{1}{20} + \arctan \frac{1}{x} = \frac{\pi}{4}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1985}
A  $23 \times 23$  square is tiled with  $1 \times 1, 2 \times 2$  and  $3 \times 3$  squares. What is the smallest possible number of  $1 \times 1$  squares?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of non-trivial ring homomorphisms from the ring \( \mathbb{Z}_{20} \) to the ring \( \mathbb{Z}_{44} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Triathlete Dan starts with a run along a 3-mile stretch of trail, of which 1 mile is a flat terrain followed by 1 mile uphill and another 1 mile downhill. He then swims back along a 3-mile stretch of river and finally completes a 3-mile bicycle ride with constant elevation. If Dan runs at a rate of 10 miles per hour on flat terrain, 6 miles per hour uphill, and 14 miles per hour downhill, swims at a rate of 4 miles per hour, and bikes at a rate of 12 miles per hour, what is his average rate for the entire trip in miles per minute?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.112}
Find the set of values of $b$ for which $y(x) = e^{bx}$ is a solution to the differential equation: $$x^2\frac{d^2y}{dx^2} + 2x\frac{dy}{dx} - 6y = 0, \quad x > 0.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Let proposition $p$: "The equation $x^{2}+mx+1=0$ has two real roots", and proposition $q$: "The equation $4x^{2}+4(m-2)x+1=0$ has no real roots". If $p \land q$ is false, and $\lnot q$ is false, find the range of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,2)}
In a $C^*$-algebra $A$, does the condition \((\forall b \in A)(b^*a^*ab = 0)\) for an element $a$ imply that $a = 0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the universal set $U=\mathbb{R}$, $A=\{x|2<x\leq 4\}$, and $B=\{x\in\mathbb{Z}|x^2-7x+10<0\}$, calculate the intersection $A\cap (\mathbb{C}_UB)$. Express your answer in interval notation, including all necessary details.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2,3)\cup(3,4)}
A dealer offers a cash discount of 10% and still makes a certain profit when he further allows 45 articles to be sold at the cost price of 40 articles to a particular sticky bargainer. His articles were listed 50% above the cost price. What is the profit percentage made by the dealer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35\%}
A month can have at most 5 Sundays. In a year of 12 months, the maximum number of months that can have 5 Sundays is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the line integral \( \oint_C (xy \, dx + xy^2 \, dy) \) using Stokes' Theorem, where \( C \) is the square in the xy-plane with vertices \((1,0), (-1,0), (0,1), (0,-1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Buses from town A to town B leave every hour on the hour (for example: 6:00, 7:00, …). Buses from town B to town A leave every hour on the half hour (for example: 6:30, 7:30, …). The trip between town A and town B takes 5 hours. Assume the buses travel on the same road. If you get on a bus from town A, how many buses from town B do you pass on the road (not including those at the stations)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10
}
Hyeonjoo stood on the 2nd prize podium with a height of 53 centimeters (cm) and 7 millimeters (mm) and measured his height from the floor, and it was 190 centimeters (cm). She stood on the 1st prize podium and measured his height from the floor, and it was 232 centimeters (cm) and 5 millimeters (mm). Find the height in centimeters (cm) of the first-place podium.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96.2}
Find the largest real number $c$ such that $$\sum_{i=1}^{101} x_{i}^{2} \geq c M^{2}$$ whenever $x_{1}, \ldots, x_{101}$ are real numbers such that $x_{1}+\cdots+x_{101}=0$ and $M$ is the median of $x_{1}, \ldots, x_{101}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5151}{50}}
The range of the inclination angle of the line $x\cos\alpha+ \sqrt{3}y+2=0$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, \frac{\pi}{6}]\cup[\frac{5\pi}{6}, \pi)}
In triangle $ABC,$ $AB=13,$ $BC=14,$ $AC=15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\circ$ rotation about $G.$ What is the area if the union of the two regions enclosed by the triangles $ABC$ and $A'B'C'?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{140}
Given that for every non-negative integer $k$ satisfying $0 \leq k \leq n$, $a_k$ is non-negative and \( \sum_{k=0}^n \frac{a_k}{2^k} < \delta \), find an upper bound for \( \sum_{k=0}^n a_k \) that is less than \( n2^n\delta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n \delta}
Determine the smallest positive integer $n$ such that in every two-coloring of the edges of the complete graph $K_n$, there always exist two monochromatic triangles that share exactly one common edge.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given that for a fixed $m$, the inequality $a_{n,m} < b_{n,m}$ holds for $n \geq 1-m$, and that $\limsup_{n\to\infty}\frac{1}{n}\ln b_{n,m} = c$, determine whether $\limsup_{n\to\infty}\frac{1}{n}\ln a_{n,m} < c$ or $\limsup_{n\to\infty}\frac{1}{n}\ln a_{n,m} \leq c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\limsup_{n\to\infty}\frac{1}{n}\ln a_{n,m} \leq c}
Find the remainder when$$\left \lfloor \frac{149^{151} + 151^{149}}{22499}\right \rfloor$$is divided by $10^4$.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
7800
}
The real numbers  $a_{1},a_{2},\ldots ,a_{n}$  where  $n\ge 3$  are such that  $\sum_{i=1}^{n}a_{i}=0$  and  $2a_{k}\le\ a_{k-1}+a_{k+1}$  for all  $k=2,3,\ldots ,n-1$ . Find the least  $f(n)$  such that, for all  $k\in\left\{1,2,\ldots ,n\right\}$ , we have  $|a_{k}|\le f(n)\max\left\{|a_{1}|,|a_{n}|\right\}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n+1}{n-1}}
A sequence of distinct circles $\omega_1, \omega_2, \cdots$ is inscribed in the parabola $y=x^2$ so that $\omega_n$ and $\omega_{n+1}$ are tangent for all $n$. If $\omega_1$ has diameter $1$ and touches the parabola at $(0,0)$, find the diameter of $\omega_{1998}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$3995$.

}
Determine the number of ways to arrange $K$ elements, numbered from $1$ to $K$, in $N$ places such that no two identical numbers are adjacent, except for the number specified as $X$. For example, if $N = 3$, $K = 2$, and $X = 1$, the possible arrangements are $\{1,1,1\}$, $\{1,1,2\}$, $\{1,2,1\}$, $\{2,1,1\}$, and $\{2,1,2\}$. Find a general formula for this arrangement for any given $N$, $K$, and $X$. Provide the number of possible arrangements for $N = 5$, $K = 4$, and $X = 3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{469}
Find the square root of 49. Express your answer using the ± symbol to indicate both positive and negative values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±7}
What is the maximum number of points that can be placed on a segment of length 1 such that on any subsegment of length \( d \) contained in this segment, there are no more than \( 1 + 1000 d^2 \) points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
A train 130 meters long, traveling at 54 km/hr crosses a bridge in 30 seconds. At the same time, another train, which is 90 meters long and traveling at 72 km/hr, starts to cross the bridge from the opposite direction. Calculate the length of the bridge such that both trains can cross the bridge safely without collision.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{510}
Let $G$ be a graph on $n \geq 6$ vertices where every vertex has a degree of at least 3. If $C_{1}, C_{2}, \dots, C_{k}$ are all the cycles in $G$, determine all possible values of $\gcd(|C_{1}|, |C_{2}|, \dots, |C_{k}|)$, where $|C|$ denotes the number of vertices in the cycle $C$. Provide your answer as a set of possible values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{1, 2\}}
Consider a terminal with fifteen gates arranged in a straight line with exactly $90$ feet between adjacent gates. A passenger's departure gate is assigned at random. Later, the gate is changed to another randomly chosen gate. Calculate the probability that the passenger walks $360$ feet or less to the new gate. Express the probability as a simplified fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, and find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Let $X_1$ and $X_2$ be independent random variables, each following a normal distribution with mean 0 and variance 4, i.e., $X_1, X_2 \sim \mathcal{N}(0,4)$. Calculate the probability $P(X_1^2 < 36.84 - X_2^2)$. Provide your answer as a decimal rounded to four significant figures.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.9900}
Evaluate the limit:
\[ \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{D(\epsilon)} \frac{1 + \sin(x) + \sin(y)}{\sqrt{x^2 + y^2}} \, d(x,y) \]
where \( D(\epsilon) = \{(x,y) \in \mathbb{R}^2 \mid 0 < x^2 + y^2 \le \epsilon^2\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Given the curve \( y = 5^{(x-2)} - \frac{1}{25} \), bounded by \( y = 5 \) and \( x = 0 \), find the height \( x \) at which the volume of the solid formed by rotating the curve about the y-axis is half of the total volume. The total volume is given by \( \int_0^5 \pi \left( \log_5(y + \frac{1}{25}) + 2 \right)^2 \, dy \approx 95.976 \). Determine the height \( x \) such that \( \int_0^x \pi \left( \log_5(y + \frac{1}{25}) + 2 \right)^2 \, dy = \frac{95.976}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.72}
Determine the maximum possible radius of a ball centered at 0 that is contained in the image of a univalent function $f$ with a continuous extension to the boundary, given that $f(0)=0$, $f'(0)=1$, and $f(-1)=-1$. Provide your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a fair die game, you roll a die for the first time as a preparatory roll, and the result is P. You continue rolling the die, and if you roll a number greater than or equal to P, the game ends. If you roll a number less than P, you add it to your total gain and continue rolling. What is the expected total gain from this game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{153}{40}}
Compute the residue of the differential form \( \frac{dz}{w^3} \) at the point \( p \) on the Riemann surface defined by \( w^3 = z(z-1)(z-2) \), where \( w = 0 \) and \( z = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Determine the values of $b$ for which any solution $y(x)$ of the differential equation $y'' + 2y' + by = 0$ satisfies the condition \( \lim_{x \to \infty} \frac{y(x)}{e^{2x}} = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-8, \infty)}
Find the limit of the expression \( \sqrt[q]{\sum_j {b_j}^q} \) as \( q \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\max_j b_j}
The school plans to organize a movie viewing for the students either on January 4th or January 10th. After finalizing the date, the teacher informs the class leader. However, due to the similarity in pronunciation between "four" and "ten," there is a 10% chance that the class leader hears it incorrectly (mistaking 4 for 10 or 10 for 4). The class leader then informs Xiaoming, who also has a 10% chance of hearing it incorrectly. What is the probability that Xiaoming correctly believes the movie date?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.82}
The picture shows several circles connected by segments. Tanya chooses a natural number \( n \) and places different natural numbers in the circles so that the following property holds for all these numbers:

If numbers \( a \) and \( b \) are not connected by a segment, then the sum \( a^2 + b^2 \) must be coprime with \( n \). If they are connected, then the numbers \( a^2 + b^2 \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
In the calculation 
$$ HE \times EH = WHEW, $$
where different letters stand for different nonzero digits. Find the values of all the letters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{E=8, H=7, W=6}
A capacitor is made with two square plates, each with side length $L$, of negligible thickness, and capacitance $C$. The two-plate capacitor is put in a microwave which increases the side length of each square plate by $ 1 \% $. By what percent does the voltage between the two plates in the capacitor change? 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
decreases by \( 2\% \)}
Evaluate
$\frac{\log_{10}20^2 \cdot \log_{20}30^2 \cdot \log_{30}40^2 \cdot \cdot \cdot \log_{990}1000^2}{\log_{10}11^2 \cdot \log_{11}12^2 \cdot \log_{12}13^2 \cdot \cdot \cdot \log_{99}100^2}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\dfrac{3}{2}$}
Let $A$ and $B$ be orthogonal projection matrices. Determine if the norm of the product $AB$ is always less than or equal to 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\leq 1}
Let $X=\ell_1$ and $Y=\ell_2$. Define an operator $B$ as follows:

$$B(x_1,x_2,\ldots)=(x_2,x_3-x_2,x_4-x_3,\ldots)$$

Is $B$ a bounded linear operator from $X$ to $Y$? If it is, find the operator norm $\|B\|$. Provide your answer as a decision and, if applicable, a numerical value for the norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
What is the value of $$\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1?$$

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{11}{7}$

}
4. How many 6-tuples $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}\right)$ are there such that each of $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ is from the set $\{1,2,3,4\}$ and the six expressions

$$
a_{j}^{2}-a_{j} a_{j+1}+a_{j+1}^{2}
$$

for $j=1,2,3,4,5,6$ (where $a_{7}$ is to be taken as $a_{1}$ ) are all equal to one another?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Let $ABC$ be a triangle with $\angle BAC = 90^o$ and $D$ be the point on the side $BC$ such that $AD \perp BC$. Let$ r, r_1$, and $r_2$ be the inradii of triangles $ABC, ABD$, and $ACD$, respectively. If $r, r_1$, and $r_2$ are positive integers and one of them is $5$, find the largest possible value of   $r+r_1+ r_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
A $4\times 4\times 4$ open cubical box contains 64 identical small cubes that exactly fill the box. How many of these small cubes touch the bottom or one of the four lateral sides of the box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
Circles  $C_1$  and  $C_2$  intersect at points  $X$  and  $Y$  . Point  $A$  is a point on  $C_1$  such that the tangent line with respect to  $C_1$  passing through  $A$  intersects  $C_2$  at  $B$  and  $C$ , with  $A$  closer to  $B$  than  $C$ , such that  $2016 \cdot AB = BC$ . Line  $XY$  intersects line  $AC$  at  $D$ . If circles  $C_1$  and  $C_2$  have radii of  $20$  and  $16$ , respectively, find  $\sqrt{1+BC/BD}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
A train 90 m long running at a certain speed crosses a platform in 18 sec. The length of the platform is 190.0224 m. What is the speed of the train in kmph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56.00448}
Let \(a, b > 0\). Define the stopping time \(T_{a,b}\) for a Brownian motion with drift as \[ T_{a,b} := \inf\{t > 0 : B(t) = at - b\} \] where \(B(t)\) is a standard Brownian motion. Compute \(E[T_{a,b}]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{b}{a}}
A number is chosen at random from among the first $100$ positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by $11$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{9}{200}$}
Determine for how many positive integers $n\in\{1, 2, \ldots, 2022\}$ it holds that $402$ divides at least one of
\[n^2-1, n^3-1, n^4-1\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Given two similar triangles $\triangle ABC\sim\triangle FGH$, where $BC = 24 \text{ cm}$ and $FG = 15 \text{ cm}$. If the length of $AC$ is $18 \text{ cm}$, find the length of $GH$. Express your answer as a decimal to the nearest tenth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11.3}
Given: $\sqrt{23.6}=4.858$, $\sqrt{2.36}=1.536$, then calculate the value of $\sqrt{0.00236}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.04858}
Calculate the volume bounded by the surfaces $z = 2-x^2-y^2$ and $z^2=x^2+y^2$ in the region where $z \ge 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi }{2}\)}
Given the dihedral angle α-l-β is 60°, and m, n are skew lines with m ⊥ α and n ⊥ β, find the angle formed by m and n. Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Determine the degree of the field extension \([\mathbb{Q}(a) : \mathbb{Q}(a+1/a)]\), where \(a\) is an algebraic number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
What is the maximum number of points that can be specified in a projective transformation while preserving collinearity and incidence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed—a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
If $f(c)=\frac{3}{2c-3}$, find $\frac{kn^2}{lm}$ when $f^{-1}(c)\times c \times f(c)$ equals the simplified fraction$\frac{kc+l}{mc+n}$, where $k,l,m,\text{ and }n$ are integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Two numbers $a$ and $b$ with $0 \leq a \leq 1$ and $0 \leq b \leq 1$ are chosen at random. The number $c$ is defined by $c=2a+2b$. The numbers $a, b$ and $c$ are each rounded to the nearest integer to give $A, B$ and $C$, respectively. What is the probability that $2A+2B=C$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{16}}
In triangle \(ABC\), the bisector \(AL\) is drawn. Points \(E\) and \(D\) are marked on segments \(AB\) and \(BL\) respectively such that \(DL = LC\) and \(ED \parallel AC\). Find the length of segment \(ED\) if it is known that \(AE = 15\) and \(AC = 12\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a laboratory, the speed at which sound propagates along rods made of different materials is measured. In the first experiment, it turns out that the entire path, consisting of three rods connected sequentially, is traversed by sound in $a$ seconds. The path consisting of the second and third rods is traversed by sound twice as fast as the first rod alone.

In another experiment, the second rod is replaced with a new one, and then the sequential connection of the three rods is traversed by sound in $b$ seconds. The connection of the first and second rods is twice as slow as the third rod alone.

Find the speed of sound in the new rod if its length is $l$ meters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3l}{2(b-a)} \text{ m/s}}
In a regular quadrilateral pyramid \(P-ABCD\), the lateral faces are equilateral triangles with side length 1. \(M\) and \(N\) are the midpoints of edges \(AB\) and \(BC\) respectively. Find the distance between skew lines \(MN\) and \(PC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\cfrac{\sqrt{2}}{4}}
8.60 What is the maximum number of "rooks" that can be placed on a $3 n \times 3 n$ chessboard such that each rook is attacked by at most one other rook?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4n}
The lines $l_1: x+3y-7=0$ and $l_2: kx-y-2=0$ and the positive halves of the x-axis and y-axis enclose a quadrilateral that has a circumscribed circle. Find the value of $k$ that satisfies this condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}