inclusionAI/Ring-lite-rl-data huggingface dataset:
Evaluate the limit without using L'Hôpital's rule:
\[ \lim_{t\to\frac\pi4}\frac{\int_{\pi/4}^t (\sin t-\sin\left(\frac\pi4\right))dt-\frac{\left(\sin t-\sin\left(\frac\pi4\right)\right)\left(t-\frac\pi4\right)}2}{\frac{\left(\sin t-\sin\left(\frac\pi4\right)\right)\left(t-\frac\pi4\right)}2\left(t-\frac\pi4\right)} \]

Use the identity:
\[ \lim_{t\to0}\frac{\tan t - t}{t^3}=\frac13 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Two trains of different lengths run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. The time they take to cross each other is 26.99784017278618 seconds. If the shorter train is 250 m long, how long is the longer train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500}
Find the partial derivative \( \frac{\partial X}{\partial x_0}(t,0) \) for the function \( X: \mathcal{R} \times \mathcal{R} \rightarrow \mathcal{R} \) defined by the differential equation:
\[ \frac{dX}{dt}(t,x_0)=X(t,x_0)+\sin(X^2(t,x_0)), \]
with the initial condition \( X(0,x_0)=x_0. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^t}
Given the function f(x) = log2(x) + x - 2, which is an increasing function on the interval (0, +∞), find the solution interval where f(x) = 0. Express your answer as an interval in standard notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 2)}
In a box, there are 2 one-cent coins, 4 five-cent coins, and 6 ten-cent coins. Six coins are drawn in sequence without replacement, and each coin has an equal probability of being selected. Calculate the probability that the total value of the drawn coins is at least 50 cents. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{127}{924}}
Find the number of critical points of the function $h(x) = e^{3f(x) - 2x + 1} + e^{2f(x) + 3x - 2}$, given that $f(x)$ has a minimum of $-3$ at $x= 0$ and a maximum of $3$ at $x = -2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The real number \(x\) satisfies the equation \(x + \frac{1}{x} = \sqrt{3}\). Evaluate the expression \(x^{7} - 5x^{5} + x^{2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
On January 1, a bicycle dealer had an inventory of 200 bicycles. Starting February 1, the number of bicycles in stock began to decrease at unpredictable rates each month. The stock decreased by 4 for February, 6 for March, and 8 for April. In each following month, the decrease rate will be the sum of the previous two months' rates. 

In addition, the dealer experiences higher sales during holidays: 5 extra bicycles will be sold during an event in May, 10 during another event in July, and 15 during one in September. Furthermore, the dealer will receive new shipments of bicycles in June and August, which increases the stock by 50 and 60 bicycles, respectively. 

Considering the unpredictable rate of decrease in stock, the additional bicycles sold during holidays, and the new shipments, how many fewer bicycles will be in stock on October 1 than were in stock on January 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{162}
We will call two numbers almost equal if they are either equal to each other or differ from each other by no more than one unit. A grid rectangle with sides equal to the natural numbers \(a\) and \(b\) is such that it is impossible to cut out a rectangle along the grid lines that has an area almost equal to half of the original rectangle. What is the smallest value that the number \( |a-b| \) can take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $n$ be an odd number. In a circular arrangement of $n$ buttons, each colored either black or white, pressing a button changes the colors of its two adjacent buttons but not itself. Initially, each button is black or white with probability $\frac{1}{2}$, independently. What is the probability that all buttons can be turned to the same color through a sequence of moves?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given a matchstick of length 1 and an infinite grid of 1x1 squares, what is the probability that the matchstick lands overlapping at least one of the matchsticks in the grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{\pi}}
A standard deck of 52 cards is shuffled and dealt to 4 players (N, S, E, and W), each receiving 13 cards. If players N and S together hold exactly 10 cards of a specified suit, what is the probability that the remaining 3 cards of that suit are all in the hand of either player E or player W?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11}{50}}
Consider a Markov chain with state space $\mathbb{R}$. Its transition function is given by $P(x,A) = \mu([x − 1/4,x + 1/4] \cap A)$, where $\mu$ is the Lebesgue measure. If the initial distribution is concentrated at the origin, calculate the probability $P(|\omega_{3}| \leq 1/8)$. Provide your answer as a single probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{35}{96}}
Let $a, b, c$ be positive integers such that $0 < a, b, c < 11$. They satisfy the following congruences: \[ 3a + b + c \equiv abc \pmod{11} \] \[ a + 3b + c \equiv 2abc \pmod{11} \] \[ a + b + 3c \equiv 4abc \pmod{11} \] Find the sum of all possible values of $abc$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{198}
What is the name of the equivalence relation defined by $A \simeq B$ if $A = P^t BP$ for some invertible matrix $P$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{congruence}
$21$ bandits live in the city of Warmridge, each of them having some enemies among the others. Initially each bandit has $240$ bullets, and duels with all of his enemies. Every bandit distributes his bullets evenly between his enemies, this means that he takes the same number of bullets to each of his duels, and uses each of his bullets in only one duel. In case the number of his bullets is not divisible by the number of his enemies, he takes as many bullets to each duel as possible, but takes the same number of bullets to every duel, so it is possible that in the end the bandit will have some remaining bullets.
Shooting is banned in the city, therefore a duel consists only of comparing the number of bullets in the guns of the opponents, and the winner is whoever has more bullets. After the duel the sheriff takes the bullets of the winner and as an act of protest the loser shoots all of his bullets into the air. What is the largest possible number of bullets the sheriff can have after all of the duels have ended?

Being someones enemy is mutual. If two opponents have the same number of bullets in their guns during a duel, then the sheriff takes the bullets of the bandit who has the wider hat among them.

Example: If a bandit has $13$ enemies then he takes $18$ bullets with himself to each duel, and they will have $6$ leftover bullets after finishing all their duels.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4800
}
Evaluate the definite integral \( \int_{0}^{2} (x^2 + 1) \, d\lfloor x \rfloor \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
How many $3 \times 3$ matrices $A$ with entries either 0 or 1 exist such that the system $A\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ has exactly two distinct solutions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{126}
Let $A$ be an $n \times n$ diagonal matrix with the characteristic polynomial $(x-a)^p(x-b)^q$, where $a$ and $b$ are distinct real numbers. Determine the dimension of the real vector space $V$ consisting of all $n \times n$ matrices $B$ such that $AB = BA$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2 + q^2}
Determine the smallest integer $m$ such that the wheel graph $W_4$, consisting of a triangle and a central hub, can be embedded in $\mathbb{R}^m$ with every edge having length 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the number of triples(a,b,c)of positive integers such that
 (a) \( {ab} \) is a prime;
 (b) \( {bc} \) is a product of two primes;
 (c) \( {abc} \) is not divisible by square of any prime and
 (d) \( {abc} \leq  {30} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
17
}
Compute the length of the $\mathbb{R}[x]$-module $M=\mathbb{C}[x]/\mathbb{R}[x]$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
计算下列实变函数定积分．

$\int _ { 0 } ^ { \infty } \frac { \sin m x } { x ( x ^ { 2 } + a ^ { 2 } ) } d x ( m > 0 , a > 0 )$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$( 1 - e ^ { - m a } ) \pi / 2 a ^ { 2 }$}
Determine the radius of convergence for the power series \( \sum_{n=1}^\infty \left[\frac {z(z+n)}{n}\right]^{n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that f(x) is an even function defined on R, and for x >= 0, f(x) = (x^2 - 2x)e^x, find the number of extreme points of the function f(x). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a $5$-regular graph $G$ with $m$ edges and girth at least $6$, how many paths of length $5$ are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256m}
Given that $|\vec{a}|=|\vec{b}|=1$ and $\vec{a} \perp \vec{b}$, find the projection of $2\vec{a}+\vec{b}$ in the direction of $\vec{a}+\vec{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\sqrt{2}}{2}}
Find the inverse Z-transform of the function \( \frac{1}{z-a} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^{n-1} u[n-1]}
Let \( a \) and \( b \) be nonnegative real numbers such that
\[
\sin (ax + b) = \sin 17x
\]
for all integers \( x \). Find the smallest possible value of \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Equilateral triangles $ABC$ and $A_{1}B_{1}C_{1}$ with a side length of 12 are inscribed in a circle $S$ such that point $A$ lies on the arc $B_{1}C_{1}$, and point $B$ lies on the arc $A_{1}B_{1}$. Find $AA_{1}^{2} + BB_{1}^{2} + CC_{1}^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
Find the infimum of the expression \( \inf\limits_{k \ge n} k^{2}\chi_{[0,\frac{1}{k}]} \), where \( \chi \) is the indicator function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
What is the difference between the place value of 2 in the numeral 7669?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{540}
For a real number $x$, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of $x$. The sum of all real numbers $\alpha$ that satisfy the equation $$\alpha^2 + \{\alpha\} = 21$$ can be expressed in the form $$\frac{\sqrt{a} - \sqrt{b}}{c} - d$$ where $a$, $b$, $c$, and $d$ are positive integers, and $a$ and $b$ are not divisible by the square of any prime. Compute $a + b + c + d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{169}
Find the characteristic polynomial of the $n \times n$ matrix $C$, where $C$ has 1s on the superdiagonal and the last entry of the first row, and 0s elsewhere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t^n}
Suppose that \( T_1 \) is a \( \alpha \times 100\% \) lower confidence limit for \( \theta \) and \( T_2 \) is a \( \alpha \times 100\% \) upper confidence limit for \( \theta \). Further assume that \( P(T_1<T_2)=1 \). Determine a \( (2\alpha-1) \times 100\% \) confidence interval for \( \theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[T_1, T_2]}
1. The sequence is defined by the following conditions: $x_{1}=1, x_{n+1}=\cos \left(\operatorname{arcctg}\left(x_{n}\right)\right)$. Find $x_{4} 000000$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0005}
Given two matrices $A$ and $B$ of size $n \times m$ with integer entries, determine the minimum number of moves required to make them identical. In each move, you can choose one of the matrices and either increment all elements of a selected row or column by 1. If it is impossible to make the matrices identical, state that it is impossible. For example, for $A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 2 \\ 0 & 0 \end{bmatrix}$, the matrices can be made identical in 2 moves.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the sets $M=\{a|a=(1,2)+\lambda(3,4), \lambda\in\mathbb{R}\}$ and $N=\{a|a=(-2,-2)+\lambda(4,5), \lambda\in\mathbb{R}\}$, calculate the intersection $M\cap N$. Express your answer as a set of points, including coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{(-2,-2)\}}
The value of log 2 ( log of a certain number ) is 2. What is that number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A reduction in the price of petrol by 20% enables three motorists A, B, and C to buy additional gallons for their budgets of $200, $300, and $400 respectively. Motorist A was able to buy 8 more gallons, motorist B was able to buy 15 more gallons, and motorist C was able to buy 22 more gallons. Find the original price of petrol per gallon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$6.25}
设数列 \{a_n\} 满足 a_0=\df{1}{2}, a_{n+1}=a_n+\df{a_n^2}{2012}, n\geq0. 求所有的整数 k，使得 a_k<1<a_{k+1}.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2012}
Evaluate the integral \( \int_0^{2\pi} \log |1-re^{i\theta}| \, d\theta \) where \( r \) is a real number such that \( |r| < 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
After a storm deposits some gallons of water into the city reservoir, the reservoir is 85% full. If the original contents of the reservoir totaled 220 billion gallons, and the reservoir was approximately 55.00000000000001% full before the storm, how many gallons of water did the storm deposit into the reservoir?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
there exists a $k_{0} \in \mathbb{N}$ such that for every $k \in \mathbb{N}, k > k_{0}$, there exists a finite number of lines in the plane not parallel to one of them, that divide the plane exactly in k regions. Find $k_{0}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. Determine the value of the sum:
\[
\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}
\]
This sum can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Given $\pi < \alpha + \beta < \frac{4}{3}\pi$ and $-\pi < \alpha - \beta < -\frac{\pi}{3}$, the range of values for $2\alpha$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \pi)}
Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to form such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
If \(a, b, c\) satisfy the equations \(a+b+c=1\), \(a^{2}+b^{2}+c^{2}=2\), and \(a^{3}+b^{3}+c^{3}=3\), what is \(a b c\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{6}}
Let $p_{}$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of $5$ heads before one encounters a run of $2$ tails. Given that $p_{}$ can be written in the form $m/n$ 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{037}
Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence of positive real numbers that satisfies $$\sum_{n=k}^{\infty}\binom{n}{k} a_{n}=\frac{1}{5^{k}}$$ for all positive integers $k$. The value of $a_{1}-a_{2}+a_{3}-a_{4}+\cdots$ can be expressed as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Compute $100a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{542}
Nine children, numbered 1 to 9, each have a character pinned on their chest. The characters are: 惠 (Hui), 州 (Zhou), 西 (Xi), 湖 (Hu), 丰 (Feng), 鳄 (E), 平 (Ping), 菱 (Ling), 南 (Nan). They are standing in five circles, with the sum of the numbers of children in each circle being 13. What is the highest possible number for the child with the character "丰" (Feng) on their chest?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given $a > 0$, $b < 0$, and $a + b > 0$, arrange $a$, $b$, $-a$, $-b$ in ascending order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-a < b < -b < a}
A dishonest dealer claims to sell a product at its cost price. He uses a counterfeit weight which is 20% less than the real weight. Further greed overtook him and he added 15% impurities to the product. What is the net profit percentage of the dealer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25\%}
The real number $x$ satisfies
\[3x + \frac{1}{2x} = 3.\]Find
\[64x^6 + \frac{1}{729x^6}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{416}{27}}
In the tetrahedron $ABCDE$ with congruent side edges and a volume of one unit, the base is the rectangle $ABCD$. A plane intersects the tetrahedron through edge $BC$, creating a cross-section that intersects side edges $EA$ and $ED$ at points $A_1$ and $D_1$, respectively. The lines $BA_1$ and $CD_1$ intersect at point $M$. If $EM = 2AB$, find the volume of the tetrahedron $EMA_1D_1$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{9}}
How many even integers are there between \( \frac{12}{3} \) and \( \frac{50}{2} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
8.1. For all triples $(x, y, z)$ satisfying the system

$$
\left\{\begin{array}{l}
2 \sin x=\operatorname{tg} y \\
2 \cos y=\operatorname{ctg} z \\
\sin z=\operatorname{tg} x
\end{array}\right.
$$

find the smallest value of the expression $\cos x-\sin z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{5\sqrt{3}}{6}}
记 m=7997\times7999^3-7998\times7996^3，求 \sqrt[3]{m}.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15995}
John is flipping his favorite bottle, which currently contains $10$ ounces of water. However, his bottle is broken from excessive flipping, so after he performs a flip, one ounce of water leaks out of his bottle. When his bottle contains k ounces of water, he has a $\frac{1}{k+1}$ probability of landing it on its bottom. What is the expected number of number of flips it takes for John’s bottle to land on its bottom ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6
}
Suppose that \( f(x) \) is a function defined for every real number \( x \) with \( 0 \leq x \leq 1 \) with the properties that

- \( f(1-x)=1-f(x) \) for all real numbers \( x \) with \( 0 \leq x \leq 1 \),
- \( f\left(\frac{1}{3} x\right)=\frac{1}{2} f(x) \) for all real numbers \( x \) with \( 0 \leq x \leq 1 \), and
- \( f(a) \leq f(b) \) for all real numbers \( 0 \leq a \leq b \leq 1 \).

What is the value of \( f\left(\frac{6}{7}\right) \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
How many numbers should there be in a lottery for the probability of getting an ambo to be $\frac{5}{473}$, when drawing five numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
On the New Year's table, there are 4 glasses in a row: the first and third contain orange juice, while the second and fourth are empty. While waiting for guests, Valya absent-mindedly and randomly transfers juice from one glass to another. Each time, she can take a full glass and pour all its contents into one of the two empty glasses.

Find the expected number of transfers needed for the first and third glasses to become empty and the second and fourth glasses to become full.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In the card game War, assuming cards are put into the bottom of the pile when taken, calculate the probability that a game will end and not result in an infinite loop.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In rectangle $ABCD$, point $E$ is chosen on $AB$ and $F$ is the foot of $E$ onto side $CD$ such that the circumcircle of $\vartriangle ABF$ intersects line segments $AD$ and $BC$ at points $G$ and $H$ respectively. Let $S$ be the intersection of $EF$ and $GH$, and $T$ the intersection of lines $EC$ and $DS$. If $\angle SF T = 15^o$ , compute the measure of $\angle CSD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105^\circ}
In the two regular tetrahedra \(A-OBC\) and \(D-OBC\) with coinciding bases, \(M\) and \(N\) are the centroids of \(\triangle ADC\) and \(\triangle BDC\) respectively. Let \(\overrightarrow{OA}=\boldsymbol{a}, \overrightarrow{OB}=\boldsymbol{b}, \overrightarrow{OC}=\boldsymbol{c}\). If point \(P\) satisfies \(\overrightarrow{OP}=x\boldsymbol{a}+y\boldsymbol{b}+z\boldsymbol{c}\) and \(\overrightarrow{MP}=2\overrightarrow{PN}\), find the value of the expression \(9x+81y+729z\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{439}
Given an even function $f(x)$ with the domain $(-1,0)\cup(0,1)$, and $f(\frac{1}{e})=0$. When $0 < x < 1$, $(1-x^{2})\ln x(1-x^{2})f′(x) > 2xf(x)$. Find the range of $x$ for which $f(x) < 0$. Express your answer as an interval or union of intervals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1,-\frac{1}{e})\cup(\frac{1}{e},1)}
Let $m$ and $n$ be distinct even numbers. Determine the value of the following expression:

$$\sum_{i=0}^{\frac{(m-2)}{2}} [\frac{ni}{m}] + \sum_{i=0}^{\frac{(n-2)}{2}} [\frac{mi}{n}]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{(m-2)(n-2)}{4}\)}