inclusionAI/Ring-lite-rl-data huggingface dataset:
Express \((2207 - \frac{1}{2207 - \frac{1}{2207 - \frac{1}{2207 - \cdots}}})^{\frac{1}{8}}\) in the form \(\frac{a + b\sqrt{c}}{d}\), where \(a\), \(b\), \(c\), and \(d\) are integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3 + \sqrt{5}}{2}}
How many $1 \times 10 \sqrt{2}$ rectangles can be placed in a $50 \times 90$ rectangle without overlapping, with the sides of the smaller rectangles being parallel to the sides of the larger one?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
给定正整数r和n，求最小正整数m，满足：若将集合S={1,2,⋯,m}任意分成r个两两不相交的子集$A _ { 1 }$,$A _ { 2 } ,$,⋯,$, A _ { r } ,$，则必存在两个正整数a，b属于同一个子集$A _ { i } ( 1 \leq i \leq r )$且$b < a \leq \frac { n + 1 } { n } - b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{( n + 1 ) r}
(What percent of 1265) ÷ 6 = 438.53333333333336?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{208\%}
16. Given a tetrahedron $ABCD$, the areas of faces $ABC$ and $ABD$ are 4 and 7, respectively, and the dihedral angle between them is $60^{\circ}$. Then the area of the section through edge $AB$ and the center of the inscribed sphere of the tetrahedron is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{28\sqrt{3}}{11}}
Consider a $2021$-by-$2021$ board of unit squares. For some integer $k$, we say the board is tiled by $k$-by-$k$ squares if it is completely covered by (possibly overlapping) $k$-by-$k$ squares with their corners on the corners of the unit squares. What is the largest integer k such that the minimum number of $k$-by-$k$ squares needed to tile the $2021$-by-$2021$ board is exactly equal to $100$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{224}
6. Through the midpoints of sides $A B$ and $A D$ of the base of a regular quadrilateral pyramid $S A B C D$, a plane is drawn parallel to the median of the lateral face $S D C$, drawn from vertex $D$. Find the area of the section of the pyramid by this plane, if the side of the base of the pyramid is $\sqrt{2}$, and the lateral edge is 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{2}}
In how many different ways can five students stand in a straight line if three of the students refuse to stand next to each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Given that point $B$ is the intersection of the left asymptote of the hyperbola $\frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ and the $x$-axis, and point $A$ has coordinates $(0, b)$. If point $P$ on the hyperbola satisfies $\overrightarrow{AP}=3 \overrightarrow{AB}$, then the eccentricity of the hyperbola is $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{5}}{5}}
Given the functions $f(x)$ and $g(x)$ ($x \in \mathbb{R}$), let the solution set of the inequality $|f(x)| + |g(x)| < a$ ($a > 0$) be $M$, and the solution set of the inequality $|f(x) + g(x)| < a$ ($a > 0$) be $N$. Then, compare the sets $M$ and $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{M \subseteq N}
A region $S$ in the complex plane is defined by
<cmath>
S = \{x + iy: - 1\le x\le1, - 1\le y\le1\}.
</cmath>
A complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\left(\frac34 + \frac34i\right)z$ is also in $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac79}
Given the joint probability density function (pdf) of random variables $X$ and $Y$:

$$f(x, y) = \frac{1}{2\pi} e^{-\sqrt{x^2 + y^2}} \quad \text{for} \quad x,y \in \mathbb{R}$$

Find the probability $P(X < 2Y)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5}
Fill integers $1, 2, 3, \cdots, 2016^{2}$ into a $2016 \times 2016$ grid, with each cell containing one number, and all numbers being different. Connect the centers of any two cells with a vector, with the direction from the smaller number to the bigger number. If the sum of numbers in each row and each column of the grid is equal, find the sum of all such vectors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Lucas, Emma, and Noah collected shells at the beach. Lucas found four times as many shells as Emma, and Emma found twice as many shells as Noah. Lucas decides to share some of his shells with Emma and Noah so that all three will have the same number of shells. What fraction of his shells should Lucas give to Emma?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{24}}
There are 8 white balls and 2 red balls in a bag. Each time a ball is randomly drawn and then a white ball is put back. What is the probability that all red balls are drawn exactly at the 4th draw?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["0.0434"]}
Given $f(x) = x^3 - 3x + 2 + m$ with $m > 0$, in the interval $[0, 2]$ there exist three distinct real numbers $a$, $b$, and $c$, such that the triangle with sides $f(a)$, $f(b)$, and $f(c)$ is a right-angled triangle. The range of $m$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 < m < 4 + 4\sqrt{2}}
Find the smallest natural number $n$ with the following property: in any $n$-element subset of $\{1, 2, \cdots, 60\}$, there must be three numbers that are pairwise coprime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
\section*{Task 2 - 011012}

In the USSR, in 37 minutes today, as much gas is produced as was produced in the entire year 1913 in Tsarist Russia.

Calculate the increase in percent!
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1420441}
Given that three balls are randomly and independently tossed into bins numbered with the positive integers such that for each ball, the probability that it is tossed into bin i is $3^{-i}$ for i = 1,2,3,..., find the probability that all balls end up in consecutive bins.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/702}
How many ways are there to write $657$ as a sum of powers of two, where each power of two is used at most twice in the sum? For example, $256 + 256 + 128 + 16 + 1$ is a valid sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
what is the sum between the place values of three 7 ' s in the numeral 87953.0727
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7,000.0707}
求最小的整数 n\geq3，使得在平面内任意无三点共线的 n 个点中，必有三个点是一个非等腰三角形的顶点.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
sides of a rectangular park are in the ratio 3 : 2 and its area is 3750 sq m. Inside the park, there is a circular flower bed with a diameter equal to half the length of the shorter side of the park. The cost of fencing the park at 70 ps per meter and the flower bed at 90 ps per meter is ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{245.65}
A high school second-year class has a total of 1000 students. To investigate the vision condition of the students in this grade, if 50 samples are selected using the systematic sampling method, and all students are randomly numbered from 000, 001, 002, …, to 999, if it is determined that the second number is drawn in each group, then the student number drawn in the 6th group is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
Calculate the integral

$$
I=\int_{0}^{\infty} \frac{x \sin a x}{x^{2}+k^{2}} \, dx \quad(a>0, k>0)
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{2} e^{-a k}}
The monotonically increasing interval of the function $y=-x^{2}+4x$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(-\infty ,2\right]}
2. Twenty numbers are arranged in a circle. It is known that the sum of any six consecutive numbers is 24. What is the number in the 12th position if the number in the 1st position is 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
设 5\times5 矩阵 A 的元素均取自 \{0,1,\cdots,14\}，满足 (\det A,15)=1 的矩阵 A 的个数记为 N，则 \nu_2(N)=__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
A pulley system with two blocks is released from rest. The block on the table, with mass $m = 1.0 \, \text{kg}$, slides and hits the pulley, coming to a stop. The initial distance between the block and the pulley is $1.0 \, \text{m}$. During the slide, the rope makes a constant angle of $10.0^\circ$ with the horizontal. The hanging block has a mass of $M = 2.0 \, \text{kg}$. The table has a coefficient of friction of $0.50$ with the block. The pulley is frictionless, and the block remains on the ground throughout. Let $t$ be the time in seconds for the $1.0\text{-m}$ slide. Find $100t$, rounded to two significant figures.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
Given that $x = 1$ is a solution to the inequality $k^2x^2 - 6kx + 8 \geq 0$ ($k \neq 0$), the range of values for $k$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 0) \cup (0, 2] \cup [4, +\infty)}
Let $\sigma(n)$ be the number of positive divisors of $n$, and let $\operatorname{rad} n$ be the product of the distinct prime divisors of $n$. By convention, $\operatorname{rad} 1 = 1$. Find the greatest integer not exceeding 
\[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{164}
Given a twentieth-degree polynomial with integer coefficients. On the plane, all points with integer coordinates whose ordinates are no less than 0 and no more than 10 are marked. What is the maximum number of marked points that can lie on the graph of this polynomial?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20 points}
Compute the sum of all positive integers $n$ for which $9 \sqrt{n}+4 \sqrt{n+2}-3 \sqrt{n+16}$ is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
In the cartesian coordinate plane $(xOy)$, the asymptote of the hyperbola $\frac{x^2}{3} - y^2 = 1$ coincides with the asymptote of the parabola $y^2 = 2px (p > 0)$. Find the value of the real number $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given a tetrahedron $ABCD$ with each vertex on the surface of sphere $O$, $AB=AC=5$, $BC=8$, $AD\perp$ the base plane $ABC$, $G$ is the centroid of $\triangle ABC$, and the tangent of the angle formed by line $DG$ and the base plane $ABC$ is $\dfrac{1}{2}$, then the surface area of sphere $O$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{634\pi}{9}}
In the three words "尽心尽力", "力可拔山", and "山穷水尽", each Chinese character represents a digit from 1 to 8. Identical characters represent the same digit, and different characters represent different digits. If the sum of the digits represented by the characters in each word is 19, and "尽" > "山" > "力", what is the maximum possible value of "水"?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
What is the area of the portion of the unit sphere that is enclosed within the cylinder with equation $x^2 + y^2 = \frac{1}{4}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi (2 - \sqrt{3})}
If the solution set of the inequality regarding $x$, $a \leq \frac{3}{4}x^2 - 3x + 4 \leq b$, is exactly $[a, b]$, then $a+b=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the symmetry center of the function $y=3\tan\left(\frac{1}{2}x+\frac{\pi}{3}\right)$. Express your answer as an ordered pair (x, y).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(-\frac{2\pi}{3},0\right)}
For any 2016 complex numbers \( z_1, z_2, \ldots, z_{2016} \), it holds that
\[ 
\sum_{k=1}^{2016} |z_k|^2 \geq \lambda \min_{1 \leq k \leq 2016} \{ |z_{k+1} - z_k|^2 \}, 
\]
where \( z_{2017} = z_1 \). Find the maximum value of \( \lambda \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{504}
已知四面体 $A B C D$ 的外接球 $\Gamma$ 的半径为 3, 球心为 $O$, 其中 $A B$ 为球 $\Gamma$ 的直径, $C D=3, \angle A O C=\theta \in\left[\frac{\pi}{3}, \frac{5}{6} \pi\right]$, 四面体 $A B C D$的体积的最大值记为 $V(\theta)$. 则 $4\sqrt{3}V(\theta)$ 的最小值是 $\qquad$。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Roberto is raising chickens for eggs. He buys 4 chickens of different breeds, each with a variable initial cost based on their breed and egg-laying consistency:

- Chicken 1: $25 initially, 4 eggs per week
- Chicken 2: $30 initially, 3 eggs per week
- Chicken 3: $22 initially, 5 eggs per week
- Chicken 4: $35 initially, 2 eggs per week

The weekly cost to feed each chicken is as follows:

- Chicken 1: $1.50
- Chicken 2: $1.30
- Chicken 3: $1.10
- Chicken 4: $0.90

Each chicken produces a different number of eggs per week based on their breed, and these egg production rates are expected to last forever. Roberto used to buy 1 dozen eggs a week and spent $2 per dozen. After how many weeks will the chickens be cheaper than buying his eggs at the store?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Given vectors $\vec{a}=(3,2)$ and $\vec{b}=(x,1-y)$, where $\vec{a}$ is parallel to $\vec{b}$ and both $x$ and $y$ are positive numbers, find the minimum value of $\frac{3}{x}+\frac{2}{y}$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Consider the number obtained by writing the numbers $1, 2, \ldots, 1990$ one after another. In this number, every digit at an even position is omitted. In the resulting number, every digit at an odd position is omitted. Then, in the new number, every digit at an even position is omitted, and so on. What will be the last remaining digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Calculate the volume of the solid enclosed by the surface $(x^2+y^2+z^2)^2=y$ using spherical coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{3}\)}
Given a function $f(x)$ on the interval $[a,b]$, if there exists an $x\_0∈[a,b]$ such that $f(x\_0)=∫_a^bf(x)dx$, then $x\_0$ is called an "integral point" of the function $f(x)$ on the interval $[a,b]$. Find the "integral point" of the function $f(x)=cos(2x+π/6)$ on the interval $[0,π/2]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{π/4}
Marie is painting a $4 \times 4$ grid of identical square windows. Initially, they are all orange, but she wants to paint $4$ of them black. How many ways can she do this up to rotation and reflection?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}
A student rolls two dice simultaneously, with the resulting numbers being $a$ and $b$. Determine the probability that the eccentricity $e$ of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ satisfies $e > \frac{\sqrt{3}}{2}$. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{6}}
Determine the maximum number of algebraically independent homogeneous polynomials of degree $d \geq 2$ in the polynomial ring $\mathbb{Z}[x_1, \dots, x_n]$. Can this number be as large as $n^2$ or $n^3$, or is it limited to $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
William writes the number $1$ on a blackboard. Every turn, he erases the number $N$ currently on the blackboard and replaces it with either $4N + 1$ or $8N + 1$ until it exceeds $1000$, after which no more moves are made. If the minimum possible value of the final number on the blackboard is $M$, find the remainder when $M$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{173}
Let $X$ be a projective Noetherian scheme over $\mathbb{C}$. Determine whether $H^0(\mathcal{O}_X(-t))=0$ for any $t>0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Paul bakes a pizza in $15$ minutes if he places it $2$ feet from the fire. The time the pizza takes to bake is directly proportional to the distance it is from the fire and the rate at which the pizza bakes is constant whenever the distance isn’t changed. Paul puts a pizza $2$ feet from the fire at $10:30$. Later, he makes another pizza, puts it $2$ feet away from the fire, and moves the first pizza to a distance of $3$ feet away from the fire instantly. If both pizzas finish baking at the same time, at what time are they both done?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10:50}
Find the minimum natural number $n$ with the following property: between any collection of $n$ distinct natural numbers in the set $\{1,2, \dots,999\}$ it is possible to choose four different $a,\ b,\ c,\ d$ such that: $a + 2b + 3c = d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{835}
Count the number of functions \( f: \{1, 2, \ldots, 12\} \rightarrow \{1, 2, \ldots, 12\} \) such that if \((a, b, c)\) is a multiplication pair, then \((f(a), f(b), f(c))\) is also a multiplication pair.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7776}
Let $P$ be an orthogonal projection onto a subspace $E$ of an inner product space $V$. If the dimensions of $V$ and $E$ are $n$ and $m$ respectively, determine the dimensions of the eigenspaces of $P$ corresponding to the eigenvalues $0$ and $1$. Provide your answer as a pair of dimensions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(m, n - m)}
If $a > 1$, then the sum of the real solutions of 
$\sqrt{a - \sqrt{a + x}} = x$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{4a- 3} - 1}{2}}
A set of data is arranged in ascending order as follows: $9$, $10$, $12$, $15$, $x$, $17$, $y$, $22$, $26$. The median of this data set is $16$, and the $75\%$ percentile is $20$. Find the sum of $x$ and $y$. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
8.4. In the tetrahedron $E F G H$, it is known that $E F=G H=7, E G=F H=10, E H=F G=11$. Points $K, L, M, N$ are the centers of the circles inscribed in triangles $E F G, E F H$, $E G H$ and $F G H$. Find the volume of the tetrahedron $K L M N$. Round the answer to the nearest hundredth if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.09}
The non-terminating periodic decimal $0.124124 \ldots = 0.\overline{124}$ has period three and is abbreviated by placing a bar over the shortest repeating block.
 If only digits $0$ and $1$ are allowed, how many distinct periodic decimals $0.\overline{d_1d_2\ldots d_{12}}$ have period exactly $12$?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$4,020$}
Evaluate
$$
S=\sum_{k=1}^{n} k(k+1) \cdots(k+p),
$$
where \( n \) and \( p \) are positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n(n+1) \cdots (n+p+1)}{p+2}}
对有限非空实数集 $X$, 记 $X$ 的元素个数为 $|X|$ ，均有  
$f(X)=\frac{1}{|X|} \sum\limits_{a \in X} a$.  
集合对 $(A, B)$ 满足  
$A \cup B=\{1,2, \cdots, 100\}, A \cap B=\varnothing$,  
且 $1 \leqslant|A| \leqslant 98$.  
任取 $p \in B$ ，令  
$A_{p}=A \cup\{p\}, B_{p}=\{x \mid x \in B, x \neq p\}$.  
对所有满足上述条件的集合对 $(A, B)$ 与 $p \in$ $B$ ，求 $\left(f\left(A_{p}\right)-f(A)\right)\left(f\left(B_{p}\right)-f(B)\right)$ 的最大值时的$\frac{a}{b}$中的a+b.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{821}
Let's call a natural number special if one of its digits can be changed to another such that all the digits in the resulting number are different. Numbers in which all the digits are already different are also considered special. How many special ten-digit numbers exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{414 \cdot 9!}
Given the spherical metric \(ds^2 = dr^2 + r^2(d\theta^2 + \sin^2(\theta) d\phi^2)\), determine if the covariant derivative \(\nabla_\theta \theta\) is equal to 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The alphabet in its natural order $\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$.  We apply a permutation to $T_0$ to get $T_1$ which is $\text{JQOWIPANTZRCVMYEGSHUFDKBLX}$.  If we apply the same permutation to $T_1$, we get $T_2$ which is $\text{ZGYKTEJMUXSODVLIAHNFPWRQCB}$.  We continually apply this permutation to each $T_m$ to get $T_{m+1}$.  Find the smallest positive integer $n$ so that $T_n=T_0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}
Determine whether the limit \( \lim_{x \to \pm \infty} \partial_x u(x,t) \) exists for the Burgers equation given by \( u_t + u u_x = \epsilon u_{xx} \), where \( u_t \) is the partial derivative of \( u \) with respect to \( t \), \( u_x \) is the partial derivative of \( u \) with respect to \( x \), and \( \epsilon \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
You plot weight $(y)$ against height $(x)$ for three of your friends and obtain the points $(x_{1},y_{1})$, $(x_{2},y_{2})$, $(x_{3},y_{3})$. If\[x_{1} < x_{2} < x_{3}\quad\text{ and }\quad x_{3} - x_{2} = x_{2} - x_{1},\]which of the following is necessarily the slope of the line which best fits the data? "Best fits" means that the sum of the squares of the vertical distances from the data points to the line is smaller than for any other line.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$ \frac{y_{3} - y_{1}}{x_{3} - x_{1}}$}
Suppose that $(a_1,\ldots,a_{20})$ and $(b_1,\ldots,b_{20})$ are two sequences of integers such that the sequence $(a_1,\ldots,a_{20},b_1,\ldots,b_{20})$ contains each of the numbers $1,\ldots,40$ exactly once.  What is the maximum possible value of the sum \[\sum_{i=1}^{20}\sum_{j=1}^{20}\min(a_i,b_j)?\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5530}
Evaluate whether the following limit is true:
\[ \lim_{x\to 0}(x!\,x!!\,x!!!\,x!!!!\cdots)^{-1/x} = e \]
where \( x!! := ((x!)!) \) and \( x!!! = (((x!)!)!) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
A conical glass is in the form of a right circular cone. The slant height is $21$ and the radius of the top rim of the glass is $14$. An ant at the mid point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If $d$ the shortest distance it should crawl to reach the honey drop, what is the integer part of $d$ ?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
18
}
将所有非完全平方的正奇数与所有正偶数的立方从小到大排成一列（前若干项依次为 $3,5,7,8,11,13, \cdots)$ ，则该数列的第 2023 项的值为 $\qquad$。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4095}