inclusionAI/Ring-lite-rl-data huggingface dataset:
Given the third-order differential equation \(y''' + 7y'' + 16y' + 10y = 0\) and knowing that \(y = e^{-x}\) is a solution, find another solution to the equation without using a calculator.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-3x} \cos x}
A person walks at 14 km/hr instead of a slower speed, and as a result, he would have walked 20 km more. The actual distance travelled by him is 50 km. What is the slower speed he usually walks at?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.4}
Evaluate the limit as \( k \to \infty \):
\[ \lim_{k\rightarrow \infty} 2k(m+2k)\frac{e^{\frac{-m}{2k}}(m+2k)^{\frac{m}{2k}}}{(k!)^{\frac{1}{k}}\Gamma\left(k+\frac{p}{2}\right)^{\frac{1}{k}}c^{\frac{m}{2k}}} \]
where \(m, p \in \mathbb{N}\) and \(c > 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4e^2}
There are $20$ people in a certain community. $10$ of them speak English, $10$ of them speak German, and $10$ of them speak French. We call a committee to a $3$-subset of this community if there is at least one who speaks English, at least one who speaks German, and at least one who speaks French in this subset. At most how many commitees are there in this community?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1020
}
Let $a$, $b$, $c$ be three distinct real numbers such that the equation $x^2+ax+1=0$ and $x^2+bx+c=0$ have one common real root, and the equation $x^2+x+a=0$ and $x^2+cx+b=0$ also have one common real root. Find the value of $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Let \( S \) be the set of all points \((x_1, x_2, x_3, \dots, x_{2017})\) in \(\mathbb{R}^{2017}\) satisfying \(|x_i|+|x_j|\leq 1\) for any \(1\leq i< j\leq 2017\). The volume of \( S \) can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(100m+n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{201}
A pyramid has a square base with sides of length $1$ and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$5\sqrt{2} - 7$}
Find the inverse Fourier Transform of the function $2\pi \delta(x-\omega)$, where $\omega$ is a real parameter and $x \in \mathbb{R}$. Express your answer in terms of $\omega$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{i\omega t}}
Find the sum of the infinite series: \[ \sin(x) - \frac{1}{2}\sin(2x) + \frac{1}{3}\sin(3x) - \frac{1}{4}\sin(4x) + \cdots \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{x}{2}}
Let $X_1, \cdots, X_n$ be $n$ random variables (not necessarily independent) such that $E[X_i] > E[X_j]$ whenever $i < j$. What is a lower bound on the following probability: $P[ (X_1 > X_2) \wedge (X_1 > X_3) \wedge \cdots \wedge (X_1 > X_n) ]$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given a circle with 800 points labeled in sequence clockwise as \(1, 2, \ldots, 800\), dividing the circle into 800 arcs, initially, one point is painted red. Subsequently, additional points are painted red according to the following rule: if the \(k\)-th point is already red, the next point to be painted red is found by moving clockwise \(k\) arcs from \(k\). What is the maximum number of red points that can be obtained on the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
What is the universal covering space of a closed surface of genus 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{H}^2}
Tosha travels from point A to point B via point C. From A to C, Tosha travels at an average speed of 75 km/h, and from C to B, Tosha travels at an average speed of 145 km/h. The total time taken for the journey from A to B is 4 hours 48 minutes. The next day, Tosha travels back at an average speed of 100 km/h. The journey from B to C takes Tosha 2 hours, and from C to A, Tosha travels at an average speed of 70 km/h. Find the distance between points B and C.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{290 \, \text{km}}
There are $2010$ people sitting around a round table. First, we give one person $x$ a candy. Next, we give candies to the $1^{st}$ person, $1+2^{nd}$ person, $1+2+3^{rd}$ person, $\cdots$, and $1+2+\cdots+2009^{th}$ person clockwise from $x$. Find the number of people who get at least one candy.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{408}
Find the set of subsequential limits of the sequence \( \{x_n\} = \left\{1, \frac{1}{10}, \frac{2}{10},\cdots,\frac{9}{10}, \frac{1}{10^2}, \frac{2}{10^2},\cdots,\frac{99}{10^2}, \cdots, \frac{1}{10^n}, \frac{2}{10^n}\cdots \frac{10^n-1}{10^n}, \cdots\right\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 1]}
Given a uniformly distributed random variable $X$ with support $[0,1]$, consider the function:

$$f(X) = \frac{1}{n^2}\sum_{l=1}^n \left[\frac{1}{2}\left(|x_l - \frac{1}{2}| - |1 - x_l|\right)\right]$$

where $n$ is a positive integer, and $x_l$ denotes the $l$th draw from $X$. Determine an upper bound for $f(X)$ using mathematical inequalities, such as Markov's or Chebyshev's inequality. Note that $x_l$ is nonnegative, but $f(X)$ may not be.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{4n}\)}
There are 4 sections A, B, C, and D in a class, consisting of 36, 44, 50, and 30 students respectively. The average weight of students in section A is 40 kg, in section B is 35 kg, in section C is 42 kg, and in section D is 38 kg. Furthermore, 5 students in section A have an additional 10 kg of average weight, and 7 students in section B have 8 kg less average weight compared to their sections' averages. Find the average weight of the entire class considering these additional differences in student weights.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38.84}
Calculate the sum of the series \( T = 1+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}+\frac{1}{17}+\frac{1}{19}+\frac{1}{23}+\cdots \), where each term is the reciprocal of a prime number that is one more than a multiple of 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
A pupil's two marks were wrongly entered: the first as 83 instead of 63 and the second as 75 instead of 85. Due to these errors, the average marks for the class got increased by half (1/2) when considering a weighted average of 3:2 for the first and the second marks, respectively. Find the number of pupils in the class.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Evaluate the limit: \[ \lim_{x\to 0} \frac{\tan( [-\pi^2]x^2)- \tan^2( [-\pi^2]x^2)}{\sin^2x} \] where \([\cdot]\) denotes the greatest integer function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-10}
Let  $M$  be a set of six distinct positive integers whose sum is  $60$ . These numbers are written on the faces of a cube, one number to each face. A *move* consists of choosing three faces of the cube that share a common vertex and adding  $1$  to the numbers on those faces. Determine the number of sets  $M$  for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Mara has 90 crayons and 15 percent of her crayons are pink. Luna has 120 crayons and 25 percent of them are pink. Sara has 45 crayons, with 5 percent of them being pink. In total, how many pink crayons do Mara, Luna, and Sara have together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
There are 6 different books: one math book, two English books, and three music books. They are to be arranged in a row such that the English books are not adjacent, and the music books are also not adjacent. Find the number of different arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
In the expansion of $(3+x)^5+(2-x)^6$, the binomial coefficient of the term containing $x^4$ is the term of the sequence $\{a_n\}$ with the general term $a_n=2n+2$. Which term of the sequence is it? Express your answer as the term number (e.g., 6th, 7th, etc.).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9th}
Let $Z_n\sim \chi^2(n)$. What is the limiting distribution of $V_n= \dfrac{\sqrt{Z_n+2}-\sqrt{n+2}}{\sqrt{2}}?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N(0,1/4)}
Find the limit of the sum \( \displaystyle \lim_{n \to \infty} \sum_{x=0}^{n-1} \frac{n-x}{n+x} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Suppose that \( n \) is a positive integer and that the set \( S \) contains exactly \( n \) distinct positive integers. If the mean of the elements of \( S \) is equal to \( \frac{2}{5} \) of the largest element of \( S \) and is also equal to \( \frac{7}{4} \) of the smallest element of \( S \), determine the minimum possible value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
From the numbers \(1, 2, 3, \cdots, 7\), select several numbers such that the sum of the even numbers is equal to the sum of the odd numbers. How many ways are there to satisfy this condition?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Count the number of non-negative integer solutions to the equation $$x_1 + x_2 + x_3 + x_4 + x_5 = 23$$ such that $0 \leq x_i \leq 9$ for all $i$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6000}
通项为$a _ { n } = b [ \sqrt { n + c } ] + d$的数列，逐次算得各项是

1,3,3,3,5,5,5,5,5,···

其中每一个正奇数m恰好连续出现m次.上述b，c，d是待定的整数.求b+c+d 的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
在透明的球袋中有 1 个黑球和 2 个白球.接下来从袋中摸球，每次摸出一个球，然后放回袋中，并继续往袋中再放入 2 个与摸出球同色的球.记 $S_{n}$ 为前 $n$ 次中摸出黑球的次数. 已知第 2 次摸出的球是黑球，记 $S_{100}$ 的期望为 $E$ ，则 $[E]=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Given that \( \frac{dx}{dy} \cdot \frac{d^2y}{dx^2} = k \cdot \frac{dy}{dx} \) for a constant \( k \), and the gradient \( \frac{dy}{dx} = 1 \) at the point \( (0,1) \), find the value of the gradient \( \frac{dy}{dx} \) at the point where \( y = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^k}
Find the sum

$$
\frac{1}{2 \cdot 5}+\frac{1}{5 \cdot 8}+\frac{1}{8 \cdot 11}+\ldots+\frac{1}{(3n-1)(3n+2)}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{N}{2(3N+2)}}
A train of a certain length is running at a speed of 55 km/hr. It crossed a platform of length 300 m in 35.99712023038157 seconds. What is the length of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250}
Given a continuous non-constant function $f: \mathbb R \to \mathbb R$ that satisfies $(e^x-1)f(2x) = (e^{2x}-1)f(x)$ and $f'(0) = 1$, determine $\lim_{x\to 0}\left(\frac{f(x)}{x}\right)^{\frac 1x}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(e^{\frac{1}{2}}\)}
Find all pairs of positive integers $ (x,y)$ such that
\[ x^y = y^{x - y}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,1), (8,2), (9,3)}
Let $a, b, c, x$ be reals with $(a+b)(b+c)(c+a) \neq 0$ that satisfy $$\frac{a^{2}}{a+b}=\frac{a^{2}}{a+c}+20, \quad \frac{b^{2}}{b+c}=\frac{b^{2}}{b+a}+14, \quad \text { and } \quad \frac{c^{2}}{c+a}=\frac{c^{2}}{c+b}+x$$ Compute $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-34}
The café has enough chairs to seat $312_8$ people. If $3$ people are supposed to sit at one table, how many tables does the café have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Given the function $y=\sin 3x$, determine the horizontal shift required to obtain the graph of the function $y=\sin \left(3x+\frac{\pi }{4}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{12}}
Each row of a seating arrangement seats either 9 or 10 people. A total of 100 people are to be seated. How many rows seat exactly 10 people if every seat is occupied?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
On each square of a $9 \times 9$ board, there is a chip. At a certain signal, each chip jumps to one of the squares diagonally adjacent to the one it is currently on. Find the minimum number of empty squares after this movement.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
The price of a certain model of mobile phone in January of this year was $a$ yuan per unit. After that, the price decreased by 3% every month. Using the geometric sequence formula, calculate the price of this mobile phone in October of this year. Express your answer in yuan, including the variable $a$, and use the boxed notation for your final answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \cdot (0.97)^9 \text{ yuan}}
Let $B_{k}(n)$ be the largest possible number of elements in a 2-separable $k$-configuration of a set with $2n$ elements $(2 \leq k \leq n)$. Find a closed-form expression (i.e. an expression not involving any sums or products with a variable number of terms) for $B_{k}(n)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{2n}{k} - 2\binom{n}{k}}
Twenty people, including \( A, B, \) and \( C \), sit randomly at a round table. What is the probability that at least two of \( A, B, \) and \( C \) sit next to each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{57}}
Determine if the curve given by the equation \(x^3 + y^3 = 3xy\) has an asymptote of the form \(x + y = k\). If it does, find the value of \(k\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Consider a sequence $\{a_n\}$ of integers, satisfying $a_1=1, a_2=2$ and $a_{n+1}$ is the largest prime divisor of $a_1+a_2+\ldots+a_n$. Find $a_{100}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the largest real number $k$ such that \[x_1^2 + x_2^2 + \dots + x_{11}^2 \geq kx_6^2\] whenever $x_1, x_2, \ldots, x_{11}$ are real numbers such that $x_1 + x_2 + \cdots + x_{11} = 0$ and $x_6$ is the median of $x_1, x_2, \ldots, x_{11}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{66}{5}}
C6

What is the least positive integer $k$ such that, in every convex 101 -gon, the sum of any $k$ diagonals is greater than or equal to the sum of the remaining diagonals?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4900}
Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$, where $S_n = (-1)^n a_n - \frac{1}{2^n}$, $n \in \mathbb{N}^{*}$. Calculate the value of $S_1 + S_2 + \ldots + S_{100}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3} \left( \frac{1}{2^{100}} - 1 \right)}
Let $N$ be a positive integer. Suppose given any real $x\in (0,1)$ with decimal representation $0.a_1a_2a_3a_4\cdots$, one can color the digits $a_1,a_2,\cdots$ with $N$ colors so that the following hold:
    1. each color is used at least once;
    2. for any color, if we delete all the digits in $x$ except those of this color, the resulting decimal number is rational.
Find the least possible value of $N$.

[i]~Sutanay Bhattacharya[/i]

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A rectangular piece of land is to be fenced in using two kinds of fencing. Two opposite sides will be fenced using standard fencing that costs $\$6/\text{m}$, while the other two sides will require heavy-duty fencing that costs $\$9/\text{m}$. What are the dimensions of the rectangular lot of greatest area that can be fenced in for a cost of $\$9000$?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250 \text{ meters by } 375 \text{ meters}}
Determine whether the limit \( \lim_{n\rightarrow\infty} \frac{P(n)}{A(n)} \) exists, where \( P(n) \) is the count of perfect numbers less than or equal to \( n \), and \( A(n) \) is the count of amicable pairs with both numbers less than \( n \). If the limit exists, find its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $(B_t)$ be a Brownian motion on $\mathbb{R}$. For a fixed $x >0$, let $\tau$ be a stopping time defined by $$ \tau = \inf \{t \geq 0 : |B_t| = x \}.$$ Compute, for $\lambda \in \mathbb{R}$, $$ \mathbb{E} \bigg(\exp \bigg\{ \lambda B_\tau -\frac{\lambda ^2}{2} \tau \bigg\} \bigg).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine whether or not there are two digits other than \( a,b \) such that the number \( \overline{ab} \) is a multiple of the number \( \overline{ba} \) (both
 written in decimal notation).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
A rectangle, whose one sidelength is twice the other side, is inscribed inside a triangles with sides $3$ cm, $4$ cm and $5$ cm, such that the long sides lies entirely on the long side of the triangle. The other two remaining vertices of the rectangle lie respectively on the other two sides of the triangle. Find the lengths of the sides of this rectangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{60}{49}, \frac{120}{49}$

}
A certain bacteria colony grows at a specific rate every day for 25 days, at which point it reaches the limit of its habitat and can no longer grow. If two bacteria colonies start growing simultaneously, they will take 24 days to reach the habitat's limit. What is the growth rate of the bacteria colony?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2,}
In America, temperature is measured in degrees Fahrenheit. This is a linear scale where the freezing point of water is $32^{\circ} \mathrm{F}$ and the boiling point is $212^{\circ} \mathrm{F}$.

Someone provides the temperature rounded to whole degrees Fahrenheit, which we then convert to Celsius and afterwards round to whole degrees. What is the maximum possible deviation of the obtained value from the original temperature in Celsius degrees?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13/18}
在 \triangle ABC 中，\angle A>90^\circ，\angle B=20^\circ，作 AD\bot AB 交 BC 于点 D. 已知 AB=1, CD=4，设 S 为 \triangle ABC 的面积，则 S^2 写成最简分数之后，分子分母之和为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given that the odd function $f(x)$ is defined on the domain $\mathbb{R}$ and satisfies $f(x) = f(2 - x)$ for any real number $x$. When $x \in (0, 1)$, $f(x) = 2^x + 1$. Find the value of $f(\log_{\frac{1}{2}} \frac{1}{15})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{31}{15}}
8. Let $a_{1}, a_{2}, a_{3}, a_{4}$ be 4 distinct numbers from $1,2, \cdots, 100$, satisfying
$$
\left(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}\right)\left(a_{2}^{2}+a_{3}^{2}+a_{4}^{2}\right)=\left(a_{1} a_{2}+a_{2} a_{3}+a_{3} a_{4}\right)^{2},
$$

then the number of such ordered quadruples $\left(a_{1}, a_{2}, a_{3}, a_{4}\right)$ is $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Given a tetrahedron \(ABCD\) with the following edge lengths:
\[
AB = 2, \quad AD = \frac{11}{2}, \quad BC = 8, \quad CD = \frac{19}{2}
\]
find the sine of the angle between the skew lines \(AC\) and \(BD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that a random variable $X$ follows a standard normal distribution ($X\sim N(0,1)$) and $Y = X^2$, and knowing that the density function of $Y$ matches that of a Chi-squared distribution with 1 degree of freedom, can you conclude that $Y$ is a Chi-squared distributed random variable with 1 degree of freedom ($Y\sim \chi^2_1$)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Y \sim \chi^2_1}
Determine whether the problem of finding the partition function of a 3D Ising model is in the complexity class #P or NP.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\#P}
Let $\omega = -\frac{1}{2} + \frac{i\sqrt{3}}{2}$ and $\omega^2 = -\frac{1}{2} - \frac{i\sqrt{3}}{2}$. Define $T$ as the set of all points in the complex plane of the form $a + b\omega + c\omega^2 + d$, where $0 \leq a, b, c \leq 1$ and $d \in \{0, 1\}$. Find the area of $T$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{3}}
Determine whether the integral \( \int_{0}^{1}\frac{\ln(1+n^2x^2)}{n^2}~dx \) converges as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the Laplace transform of the function \( g(t) = \int_{0}^t \frac{1}{e^s\sqrt{s}}ds \) and evaluate it at \( t = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The school club sells 200 tickets for a total of $2500. Some tickets are sold at full price, and the rest are sold for one-third the price of the full-price tickets. Determine the amount of money raised by the full-price tickets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1250}
Dave's Amazing Hotel has $3$ floors. If you press the up button on the elevator from the $3$rd floor, you are immediately transported to the $1$st floor. Similarly, if you press the down button from the $1$st floor, you are immediately transported to the $3$rd floor. Dave gets in the elevator at the $1$st floor and randomly presses up or down at each floor. After doing this $482$ times, the probability that Dave is on the first floor can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is the remainder when $m+n$ is divided by $1000$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{803}
由 1,2,\cdots,9 这九个正整数构成的所有圆排列中，任意相邻两数之积均不超过 60 的圆排列的个数为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21600}
Misha has accepted a job in the mines and will produce one ore each day. At the market, he is able to buy or sell one ore for \ $3, buy or sell bundles of three wheat for \$ 12 each, or  $\textit{sell}$  one wheat for one ore. His ultimate goal is to build a city, which requires three ore and two wheat. How many dollars must Misha begin with in order to build a city after three days of working?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
If, in the expression $ x^2 - 3$, $ x$ increases or decreases by a positive amount of $ a$, the expression changes by an amount:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$ \pm2ax + a^2$
}
在平面上，整点（纵、横坐标都是整数的点）到直线 $y=\frac{5}{3} x+\frac{4}{5}$ 的距离中的最小值是多少？原始答案的形式为\frac{\sqrt{n}}{m}，请给出n+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{119}