inclusionAI/Ring-lite-rl-data huggingface dataset:
The captain's assistant, who had been overseeing the loading of the ship, was smoking one pipe after another from the very start of the process. When $2 / 3$ of the number of loaded containers became equal to $4 / 9$ of the number of unloaded containers, and noon struck, the old sea wolf started smoking his next pipe. When he finished this pipe, the ratio of the number of loaded containers to the number of unloaded containers reversed the ratio that existed before he began smoking this pipe. How many pipes did the second assistant smoke during the loading period (assuming the loading rate and the smoking rate remained constant throughout)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Example 2 There are $r$ people participating in a chess tournament, every 2 people play one game, the winner of each game gets 2 points, the loser gets 0 points, and in the case of a draw, each gets 1 point. After the tournament, there is exactly one person who has the fewest wins, and only he has the highest score. Find the minimum value of $r$. (16th All-Russian Mathematical Olympiad problem)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $\triangle ABC$ be a triangle with $\angle ABC = 76^\circ$ and $\angle ACB = 72^\circ$. Points $P$ and $Q$ lie on the sides $AB$ and $AC$, respectively, such that $\angle ABQ = 22^\circ$ and $\angle ACP = 44^\circ$. Find the measure of $\angle APQ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
在锐角三角形 $A B C$ 中, $A B$ 上的高 $C E$ 与 $A C$ 上的高 $B D$ 相交于点 $H$, 以 $D E$ 为直径的圆分别交 $A B 、 A C$ 于 $F 、 G$ 两点, $F G$ 与 $A H$ 相交于点 $K$, 已知 $B C=25, B D=20, B E=7$, 求 $25A K$ 的长。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{216}
Determine the number of homomorphisms from the symmetric group $S_4$ to the dihedral group $D_4$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half dollar. What is the probability that at least 25 cents worth of coins come up heads?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
In the tetrahedron P-ABC, PB=6, AC=3, and G is the centroid of triangle PAC. A section is made through point G such that the section is parallel to line PB and AC. The perimeter of the section is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
$P$ is a point inside triangle $A B C$, and lines $A P, B P, C P$ intersect the opposite sides $B C, C A, A B$ in points $D, E, F$, respectively. It is given that $\angle A P B=90^{\circ}$, and that $A C=B C$ and $A B=B D$. We also know that $B F=1$, and that $B C=999$. Find $A F$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{499 / 500}
5. In several packages, there are 20 candies, and there are no two packages with the same number of candies and no empty packages. Some packages may be inside other packages (then it is considered that a candy lying in the inner package also lies in the outer package). However, it is forbidden to place a package inside another package that already contains a package. What is the maximum possible number of packages?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
If $X$ is a random variable with a chi-squared distribution with $k$ degrees of freedom, and $Y=\sqrt{X}$, determine the distribution of $Y$. \( \text{Express your answer in terms of known distributions.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\chi(k)}
Determine the Galois group of the polynomial \(x^4 - 10\) over \(\mathbb{F}_7\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/2\mathbb{Z}}
Compute the degree of the zero cycle in the Fano variety of lines on a smooth quadric $Q \subset \mathbb{P}_k^3$, where $F_{osc} = \{[l] \in F(Q), \exists P\text{ plane such that } 2l \subseteq P \cap Q\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Compute the limit: \[ \lim_{n\to\infty}\int_{0}^{\infty}\frac{\sin\left(\frac{x}{n}\right)}{\left(1+\frac{x}{n}\right)^{n}}\,dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the expression \( \sum_{n=1}^{1009} \gcd(2n, 2020) - \sum_{n=0}^{1008} \gcd(2n+1, 2019) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6481}
In a universe $U$ shaped like $\mathbb{R}^3$, points are randomly distributed in a Poisson fashion with density parameter $\lambda$, such that in any measurable set $S$ of volume $V$, the expected number of points in $S$ is $V$. Consider a random line $L$ that intersects the origin. What is the probability that $L$ intersects at least one point in this distribution?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the supremum of the set \(\{(\omega+2)^k \mid k \in \omega\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\omega^{\omega}}
In a square room, there is a bulb on each wall that can shine one of the seven colors of the rainbow. There are no bulbs in the room that shine the same color. With each move, a person can change the color of one bulb to a color that no bulb is currently shining, while also changing the colors of the two other bulbs to the remaining two unused colors. (After the move, there still won't be two bulbs with the same color in the room). What is the minimum number of moves required so that eventually, each bulb has shone each of the seven colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Evaluate the double integral \( \int_{0}^{2}\int_{0}^{\sqrt{2x-x^2}} \sqrt{x^2 + y^2} \,dy\,dx \) using the change of variables \( x = \cos\theta \) and \( y=\sin\theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{16}{9}}
For each positive integer $n$, let $r_n$ be the smallest positive root of the equation $x^n = 7x - 4$. There are positive real numbers $a$, $b$, and $c$ such that \[\lim_{n \to \infty} a^n (r_n - b) = c.\] If $100a + 10b + c = \frac{p}{7}$ for some integer $p$, find $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1266}
Find the number of positive integers \( j \leq 3^{2013} \) such that
\[ j = \sum_{k=0}^{m} \left((-1)^k \cdot 3^{a_k}\right) \]
for some strictly increasing sequence of nonnegative integers \( \{a_k\} \). For example, we may write \( 3 = 3^1 \) and \( 55 = 3^0 - 3^3 + 3^4 \), but 4 cannot be written in this form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2013}}
Consider two distinct quadratic polynomials, $f(x)$ and $g(x)$. If $|f(x)| \geq |g(x)|$ for all $x \in \mathbb{R}$ and $f(x)$ has real roots, determine the number of distinct roots of the equation $h(x)h''(x) + (h'(x))^2 = 0$, where $h(x) = f(x)g(x)$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the radius of convergence of the function \( f(x) = \int_{0}^{x} \ln{(t+\sqrt{t^2+1})} \, dt \) using its Maclaurin series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the factorial of \(-\frac{1}{2}\) using the Gamma function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\pi}}
Determine the quotient group \( \mathbb{C}/P \), where \( \mathbb{C} \) is the multiplicative group of non-zero complex numbers and \( P \) is the subgroup of positive real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S^1}
Determine the value of \( k \) such that the number of monotonically increasing functions from the set \( \{1,2,3,4,5,6\} \) to itself, with the property that \( f(x) \ge x \) for all \( x \), is equal to \( \frac{2}{k} \cdot \binom{11}{5} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
The least positive angle \( \alpha \) for which
 \( \left( {\frac{3}{4} - {\sin }^{2}\left( \alpha \right) }\right) \left( {\frac{3}{4} - {\sin }^{2}\left( {3\alpha }\right) }\right) \left( {\frac{3}{4} - {\sin }^{2}\left( {{3}^{2}\alpha }\right) }\right) \left( {\frac{3}{4} - {\sin }^{2}\left( {{3}^{3}\alpha }\right) }\right)  = \frac{1}{256} \)
 has a degree measure of \( \frac{m}{n} \) ,where \( m \) and \( n \) are relatively prime positive integers. Find
 \( m + n \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{131}
Siva has the following expression, which is missing operations:\n$$\\frac12 \\,\\, \\_  \\,\\,\\frac14 \\,\\, \\_ \\,\\, \\frac18 \\,\\, \\_  \\,\\,\\frac{1}{16} \\,\\, \\_  \\,\\,\\frac{1}{32}.$$ For each blank, he flips a fair coin: if it comes up heads, he fills it with a plus, and if it comes up tails, he fills it with a minus. Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coinflips and operations. If the probability that the positive difference between his computed values is greater than $\\frac12$ can be expressed as $\\frac{a}{b}$ for relatively prime positive integers $a$, $b$, then find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
39
}
Determine the number of different 6-digit codes that can be formed using the digits 4, 6, and 9, with the following conditions: the code must use all the digits, it ends in 4, and 4 and 9 are never consecutive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38}
Find the value of $c$ that minimizes the sum of the areas of two regions bounded by the curve $f(x) = x^3 - x^2$ and the line $y = c + x$. Assume $c$ is such that exactly two of these regions have finite area.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{11}{27}}
Given two fixed points A(-2, 0) and B(1, 0), if a moving point P satisfies $|PA| = \sqrt{3}|PB|$, then the area of the figure enclosed by the trajectory of point P is equal to \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{27\pi}{4}}
Compute the factor group: $\mathbb{Z} \times \mathbb{Z} \big / \langle (1,2) \rangle$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\mathbb{Z}$}
Find all positive integers \( n \) such that \( n \) is equal to 100 times the number of positive divisors of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
If the eight digits \(1, 2, 3, 4, 5, 6, 7, 8\) are randomly permuted, what is the probability that the resulting eight-digit number is divisible by 8?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{8}}
用 3 个 1,2 个 2,5 个 3 这十个数字能构成多少个四位数?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{能构成 71 个四位数.}
Given a convex quadrilateral \(ABCD\) where \(AB=AD=1\), \(\angle A=80^\circ\), and \(\angle C=140^\circ\), find the length of the diagonal \(AC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
$x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } , x _ { 5 } , x _ { 6 } \in \{ 1 , 2 , 3 , 4 , 5 , 6 \}$，且$x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } , x _ { 5 } , x _ { 6 }$各不相同，

则满足$x _ { 1 } - 5 x _ { 2 } + 1 0 x _ { 3 } - 1 0 x _ { 4 } + 5 x _ { 5 } - x _ { 6 } = 0$的组数为多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
(JBMO 2000 - Shortlist, Problem G6), treated in class

Let $A B C D$ be a quadrilateral such that $\widehat{D A B}=60^{\circ}, \widehat{A B C}=90^{\circ}$ and $\widehat{B C D}=120^{\circ}$. Let $M$ be the intersection point of the diagonals $[A C]$ and $[B D]$. Given that $B M=1$ and $D M=2$, calculate the area of the quadrilateral $A B C D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{2}}
Given \( S = \frac{2}{1 \times 3} + \frac{2^2}{3 \times 5} + \frac{2^3}{5 \times 7} + \cdots + \frac{2^{49}}{97 \times 99} \) and \( T = \frac{1}{3} + \frac{2}{5} + \frac{2^2}{7} + \cdots + \frac{2^{48}}{99} \), find \( S - T \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \frac{2^{49}}{99}}
A permutation of \{1,2, \ldots, 7\} is chosen uniformly at random. A partition of the permutation into contiguous blocks is correct if, when each block is sorted independently, the entire permutation becomes sorted. For example, the permutation $(3,4,2,1,6,5,7)$ can be partitioned correctly into the blocks $[3,4,2,1]$ and $[6,5,7]$, since when these blocks are sorted, the permutation becomes $(1,2,3,4,5,6,7)$. Find the expected value of the maximum number of blocks into which the permutation can be partitioned correctly.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{151}{105}}
If $x < 0$ and $0 < y < 1$, then the order from smallest to largest of $\frac{y^2}{x}$, $\frac{y}{x}$, $\frac{1}{x}$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{x} < \frac{y}{x} < \frac{y^2}{x}}
A plane is drawn through the midpoint of the diagonal of a cube and is perpendicular to this diagonal. Find the area of the resulting cross-section if the edge length of the cube is \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3a^2\sqrt{3}}{4}}
Evaluate \(\lim_{n \to \infty} \frac{1}{n^5} \sum (5r^4 - 18r^2s^2 + 5s^4)\), where the sum is over all \(r, s\) satisfying \(0 < r, s \leq n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
In a regular truncated quadrilateral pyramid with the height of the lateral face \( a \), a sphere can be inscribed. Find the area of its lateral surface.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4a^2}
Find the largest positive integer value of $n < 1000$ such that $\phi(36n) = \phi(25n) + \phi(16n)$, where $\phi(n)$ denotes the number of positive integers less than $n$ that are relatively prime to $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{996}
a, b, and c started a business with different capitals. At the end of the year, the profit share of b is Rs. 1800, and the difference between the profit shares of a and c is Rs. 720. If b invested Rs. 10000 and c invested Rs. 12000, how much did a invest?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16000}
Let \( a < b < c < d < e \) be real numbers. We calculate all the possible sums of two distinct numbers among these five numbers. The three smallest sums are 32, 36, and 37, and the two largest sums are 48 and 51. Find all possible values of \( e \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{55}{2}}
3. In the equation $A \times B \times \overline{B B C}=\overline{A D A A}$, the same letters represent the same digits, different letters represent different digits, and $D$ is the smallest natural number, $A$ is the smallest prime number. Using the digits $A$, $B$, and $C$ to form natural numbers without repeated digits, the number of prime numbers formed is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given circle $O$: $x^{2}+y^{2}=4$ and curve $C$: $y=3|x-t|$, there are two points $A(m,n)$ and $B(s,p)$ on curve $C$ ($m$, $n$, $s$, $p$ are all positive integers). The ratio of the distance from any point on circle $O$ to point $A$ and point $B$ is a fixed value $k(k > 1)$. Find $m^{s}-n^{p}=$ _____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
One hundred friends, including Petya and Vasya, live in several cities. Petya found out the distance from his city to the city of each of the remaining 99 friends and summed these 99 numbers. Vasya did the same. Petya obtained 1000 km. What is the largest number Vasya might have obtained? (Consider the cities as points on a plane; if two people live in the same city, the distance between their cities is considered to be zero.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99000 \text{ km}}
A turtle crawled out of its home and crawled in a straight line at a constant speed of 5 m/hour. After an hour, it turned $90^{\circ}$ (either right or left) and continued crawling, then crawled for another hour, then turned again $90^{\circ}$ (either right or left), and so on. It crawled for a total of 11 hours, turning $90^{\circ}$ at the end of each hour. What is the minimum distance it could be from its home?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 \text{ m}}
Given three sequences $\{F_n\}$, $\{k_n\}$, $\{r_n\}$ satisfying: $F_1=F_2=1$, $F_{n+2}=F_{n+1}+F_n$ ($n\in\mathbb{N}^*$), $r_n=F_n-3k_n$, $k_n\in\mathbb{N}$, $0\leq r_n<3$, calculate the sum $r_1+r_3+r_5+\ldots+r_{2011}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1509}
A regular pentagon with a unit side, by drawing two diagonals from one vertex, results in the net of a pyramid with a regular triangle base. What is the volume of the pyramid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1+\sqrt{5}}{24}}
已知p为质数，$\sqrt{p}$的小数部分为x$, \frac { 1 } { x }$的小数部分为$\frac { \sqrt { p } - 3 1 } { 7 5 }$.求所有满足条件的质数p的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2011}
Calculate the double integral of the function \( f(x,y) = (x+y)^9(x-y)^9 \) over the region \( R = \{(x,y) \mid 0 \leq x \leq 1/2, x \leq y \leq 1-x\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{400}}
In a particular game, each of $4$ players rolls a standard $6{ }$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo's first roll was a $5,$ given that he won the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{41}{144}}
Evaluate the double integral: \[ J = \int\limits_0^1 \int\limits_0^1 \dfrac{x-y}{(x+y)^3} \, dy \, dx. \] Determine the correct value of \( J \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Given the ellipse $\Gamma_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ and $c=\sqrt{a^{2}-b^{2}}$, the parabola $\Gamma_{2}: y^{2}=2c\left(x-\frac{c}{2}\right)$ intersects with $\Gamma_{1}$ at point $A$. Let $O$ be the origin. The line $l: x=\frac{a^{2}}{c}$ intersects the x-axis and the line $OA$ at points $D$ and $E$, respectively. A point $B$ on the ellipse $\Gamma_{1}$ satisfies $\frac{S_{\triangle ABE}}{S_{\triangle ABD}}=\frac{a}{c}$, with points $A$ and $B$ on opposite sides of the y-axis. Find the slope of line $AB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A factory produces rattles shaped in the form of a ring with 3 red and 7 blue spherical beads on it. Two rattles are said to be of the same type if one can be obtained from the other one by moving a bead along the ring or by flipping the ring over in space. How many different types of rattles can be manufactured?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
已知正四棱雉 $\Gamma$ 的高 $h=3$, 侧面与底面所成角 $\theta=\frac{\pi}{3}$ 。先在正四棱雉 $\Gamma$ 内放人一个内切球 $O_{1}$, 然后依次放人球 $O_{2}, O_{3}, \cdots$, 使得后放入的各球均与前一个球及正四棱雉 $\Gamma$的四个侧面都相切。则放入所有球的体积之和为$\frac{m \pi}{n}$，求$m+n$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Given the hyperbola x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0), one of its asymptotes intersects the ellipse x^2/4 + y^2 = 1 and the length of the chord obtained is 4*sqrt(3)/3. Find the eccentricity of this hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Alex drives from point \( P \) to point \( Q \), then immediately returns to \( P \). Bob drives from point \( Q \) to point \( P \), then immediately returns to \( Q \). They travel on the same road, start at the same time, and each travels at a constant speed. Alex's speed is three times Bob's speed. They meet each other for the first time 15 minutes after the start. How long after the start will they meet each other for the second time? Express your answer in minutes as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
A man saves 25% of his income during a year and spends the remaining portion on his personal expenses. Next year his income increases by 25%, but his savings increase by some percentage. His total expenditure in 2 years is double his expenditure in the 1st year. By what percentage did his savings increase in the second year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100\%}
A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in 2007. A set of plates in which each possible sequence appears exactly once contains $N$ license plates. Find $\frac{N}{10}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{372}
Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number $1$, then Todd must say the next two numbers ($2$ and $3$), then Tucker must say the next three numbers ($4$, $5$, $6$), then Tadd must say the next four numbers ($7$, $8$, $9$, $10$), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number $10,000$ is reached. Find the $2019$th number said by Tadd.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5979}
Compute the tensor product \( \mathbb{Z} \otimes_{\mathbb{Z}[x]} \mathbb{Z} \), where the left \( \mathbb{Z} \) is endowed with a \( \mathbb{Z}[x] \)-module structure via the ring homomorphism \( \phi_n: \mathbb{Z}[x] \to \mathbb{Z} \) defined by \( \phi_n(f) = f(n) \), and the right \( \mathbb{Z} \) uses the structure from \( \phi_m \). The answer should be expressed in terms of the integers \( n \) and \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/(n - m)\mathbb{Z}}
There are 4 willow trees and 4 poplar trees planted in a row. How many ways can they be planted alternately?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1152}
In the symmetric group $S_5$, how many equivalence classes of subgroups isomorphic to the non-cyclic group of order 4 are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
$已知椭圆E：x^2/a^2 + y^2/b^2 = 1 (a > b > 0) 的一个顶点为 A(0，1)，离心率 e = \sqrt{6}/3.$
$过点P(-\sqrt{3}，1)作斜率为k的直线与椭圆E交于不同的两点B，C，直线AB，AC分别与x轴交于点M，N.设椭圆的左顶点为D，求\frac{|MD|}{|MN|}的值.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{2}$}