inclusionAI/Ring-lite-rl-data huggingface dataset:
Choose positive integers \( b_{1}, b_{2}, \ldots \) satisfying
\[
1=\frac{b_{1}}{1^{2}}>\frac{b_{2}}{2^{2}}>\frac{b_{3}}{3^{2}}>\frac{b_{4}}{4^{2}}>\cdots
\]
and let \( r \) denote the largest real number satisfying \( \frac{b_{n}}{n^{2}} \geq r \) for all positive integers \( n \). What are the possible values of \( r \) across all possible choices of the sequence \( \left( b_{n} \right) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 \leq r \leq \frac{1}{2}}
Ajay spends 40% of his monthly income on household items, 20% on buying clothes, 10% on medicines, 5% on entertainment, and 10% on transportation. He invests 5% of his monthly income. Over a period of 6 months, his savings percentages are 10%, 15%, 8%, 12%, 7%, and 18% of his monthly income. Given the total savings for the entire 6-month period as Rs. 58,500, calculate his average monthly income during this time.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13,928.57}
Determine the smallest positive integer \( n \) such that for any \( n \) points \( A_1, A_2, \ldots, A_n \) on the circumference of a circle, there are at least 2007 angles \( \angle A_i O A_j \) (where \( 1 \leq i < j \leq n \)) that do not exceed \( 120^\circ \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
Problem 6. Each of the two sides of a triangle is divided into six equal parts; the corresponding division points are connected by segments. Find the sum of the lengths of these segments if the third side of the triangle is 12.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
There are 4 different books on the bookshelf. Three people, A, B, and C, go to choose books, with each person choosing at least one book. How many different ways are there to choose?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
The triangular prism \( S-ABC \) has three lateral edges that are pairwise perpendicular. \( O \) is a point inside the base triangle \( \triangle ABC \). What is the minimum value of \( W = \tan \angle OSA \cdot \tan \angle OSB \cdot \tan \angle OSC \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Calculate the product of the non-zero eigenvalues of the matrix \(\begin{pmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{pmatrix}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Consider an $n$-dimensional sphere of radius $\sqrt{n}$. A point $(x_1, x_2, \ldots, x_n)$ is chosen randomly on this sphere. Determine the probability that the first coordinate $x_1$ is less than or equal to a given value $a$ as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Phi(a)}
Calculate the sum \( \sum_{i=1}^{200} f(i) \) where the function \( f(x) \) is defined as follows:
\[
f(x) = \begin{cases} 
\left[\frac{1}{\{\sqrt{x}\}}\right] & \text{if } x \neq k^2 \\
0 & \text{if } x = k^2
\end{cases}
\]
Here, \( x, k \in \mathbb{N} \), \([.]\) denotes the greatest integer function, and \(\{.\}\) denotes the fractional part. Determine the value of the sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{629}
How many $k$-simplices in a triangulation of the simplex $\sigma_k:=\{(x_1,...,x_{k+1}):x_1+\cdots+x_{k+1}=1, x_i\ge 0\}$ does a non-boundary $(k-1)$-simplex lie in?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$.
Clarifications
The problem should read $|a+b+c| = 21$. An earlier version of the test read $|a+b+c| = 7$; that value is incorrect.
$|b|^2+|c|^2$ should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers $m$ and $n$. Find $m+n$.''
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
132

}
A teacher received a number of letters from Monday to Friday, which were 10, 6, 8, 5, 6, respectively. The variance $s^2$ of this set of data is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["3.2"]}
Determine the tangent space of the manifold \( M = \{(x,y,z):z=\sqrt{y-x}\} \) at the point \((x,y,z) = (2,6,*)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x - y + 4z = 4}
Determine the radius of convergence for the Maclaurin series:
\[
(-1)^{n+1} \frac{\left(\left|\frac{1-x}{1+x}\right|-1\right)^n}{n}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Find the infimum of the integral \[ \iint_{x^2+y^2\leqslant1}\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2\,\mathrm{d}x\,\mathrm{d}y \] for \( C^\infty \)-functions \( u \) that vanish at the origin and are equal to 1 on the boundary \( x^2+y^2=1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the expected time until a bank balance, modeled by a Brownian motion with positive drift, first becomes negative.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
What fraction of regular hexagon \( {ZUMING} \) is covered by both quadrilateral \( {ZUMI} \) and quadrilateral \( {MING} \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{6}$}
Find a nontrivial example of a function \( f \) defined on a neighborhood of \( z \in \mathbb{C} \) and a path from \( z \) to \( z \) such that the analytic continuation of \( f \) along the path is \( f' \). Ensure that \( f \neq f' \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(z) = \sqrt{z}}
In triangle $ABC$, $AX = XY = YB = BC$ and the measure of angle $ABC$ is 120 degrees. What is the number of degrees in the measure of angle $BAC$?

[asy]
pair A,X,Y,B,C;
X = A + dir(30); Y = X + dir(0); B = Y + dir(60); C = B + dir(-30);
draw(B--Y--X--B--C--A--X);
label("$A$",A,W); label("$X$",X,NW); label("$Y$",Y,S); label("$B$",B,N); label("$C$",C,E);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
On the island of Truth and Lies, there are knights who always tell the truth, and liars who always lie. One day, 20 inhabitants of the island lined up in order of their height (from tallest to shortest, the tallest being the first) for a game. Each of them had to say one of the following phrases: "There is a liar below me" or "There is a knight above me". As a result, those standing in positions from the third to the seventh said the first phrase, while the others said the second phrase. How many knights were there among these 20 people, if all the inhabitants have different heights?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Evaluate the iterated integral: $$\int_0^8\int_{y^{1/3}}^2 4e^{x^4} \,dx\,dy$$ for the region defined by $0 \leq y \leq 8$ and $y^{1/3} \leq x \leq 2$. Determine the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{16} - 1}
In triangle ABC, points A', B', and C' are located on sides BC, AC, and AB, respectively. The cevians AA', BB', and CC' are concurrent at point O. Given that the sum of the ratios AO:OA', BO:OB', and CO:OC' is 92, determine the value of the product AO:OA' × BO:OB' × CO:OC'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{94}
Determine the values of \(\alpha\) for which the integral \(\int\int_{D}\frac{1}{(x+y)^{\alpha}}dxdy\) converges, where the region \(D\) is defined by \(0 \le y \le 1 - x\) and \(0 \le x \le 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha < 2}
Given that $0 < a \leqslant b \leqslant c$ and $abc = 1$, find the minimum value of $\frac{1}{a^{2}(b+c)} + \frac{1}{b^{3}(a+c)} + \frac{1}{c^{3}(a+b)}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Zhendarov R.G.

In the cells of a $4 \times 4$ table, numbers are written such that the sum of the neighbors of each number is 1 (cells are considered neighbors if they share a side).

Find the sum of all the numbers in the table.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Kelvin the Frog lives in the 2-D plane. Each day, he picks a uniformly random direction (i.e., a uniformly random bearing $\theta \in [0, 2\pi]$) and jumps a mile in that direction. Let $D$ be the number of miles Kelvin is away from his starting point after ten days. Determine the expected value of $D^4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{190}
Consider a  $6 \times 6$  grid of squares. Edmond chooses four of these squares uniformly at random. What is the probability that the centers of these four squares form a square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{561}}
In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{63}{146}}
Given the sequence $\{a_n\}$ where $a_n = (\sqrt{2}+1)^n - (\sqrt{2}-1)^n \quad (n \in \mathbf{N})$, and let $[x]$ denote the greatest integer not exceeding the real number $x$, compute the units digit of $[a_{2017}]$. Express your answer as a single digit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that the function $f(x)$ is an even function on $\mathbb{R}$, and $f(x)$ is monotonically increasing on $[0, +\infty)$, let $m=f(-1)$, $n=f(a^2+2a+3)$, then the relationship between $m$ and $n$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m < n}
Let \( A = \{1, -1, \mathrm{i}, -\mathrm{i} \} \) (where \( \mathrm{i} \) is the imaginary unit), and \( f(x) \) be a function whose domain and range are both \( A \). Given that for any \( x, y \in A \), \( f(x y) = f(x) f(y) \), how many such functions \( f(x) \) satisfy this condition?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of conjugacy classes in a non-abelian group of order $p^3$, where $p$ is a prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2 + p - 1}
We write integers into an $n \times k$ table in such a way that every integer from 1 to $k$ appears in each of the $n$ rows. Let $S$ denote the largest of the $k$ column sums. For every $n$ and $k$, determine the smallest possible value of $S$!
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lceil\frac{1}{2}n(k+1)\rceil}
Glue two small cuboids, each with a length of 3cm, a width of 2cm, and a height of 1cm, into one large cuboid. Then, cut it into two small cuboids of the same size. The surface area of the final small cuboid could be at most how many square centimeters larger than the surface area of the original small cuboid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The positions of cyclists in the race are determined by the total time across all stages: the first place goes to the cyclist with the shortest total time, and the last place goes to the cyclist with the longest total time. There were 500 cyclists, the race consisted of 15 stages, and no cyclists had the same times either on individual stages or in total across all stages. Vasya finished in seventh place every time. What is the lowest position (i.e., position with the highest number) he could have taken?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
现有 7 把钥匙，用这些钥匙随机开锁，则 $D_{1}, D_{2}, D_{3}$ 这三把钥匙不能打开对应的锁的概率是 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{172}
Compute the double integral \( \iint\limits_{G} \sqrt{x^{2} - y^{2}} \, dx\, dy \), where the region \( G \) is defined as \( G=\{(x,y) \in \mathbb{R_{+}^{2}}: 0 \leq x + y \leq 2, 0 \leq x - y \leq 2\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{9}}
Evaluate the double integral: \[ J = \int\limits_0^1 \int\limits_0^1 \dfrac{x-y}{(x+y)^3} \, dy \, dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
$4 \cdot 35$ Find the smallest positive integer $n$, such that in any two-coloring of the complete graph $K$ with $n$ vertices, there always exist $m$ edges of the same color, which are pairwise non-adjacent.

 untranslated parts:
求最小正整数 $n$, 使在任何二染色的 $n$ 个顶点的完全图 $K$,中,总存在同色的 $m$ 条线段,它们两两之间没有公共端点。

translated:
Find the smallest positive integer $n$, such that in any two-coloring of the complete graph $K$ with $n$ vertices, there always exist $m$ edges of the same color, which are pairwise non-adjacent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3m-1}
Compute the number of ordered quadruples of positive integers $(a, b, c, d)$ such that 
\[ a! \cdot b! \cdot c! \cdot d! = 24! \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise was $6:57\textsc{am}$, and the sunset as $8:15\textsc{pm}$. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$5:21\textsc{pm}$}
A city bus moving at a certain speed covers a distance of 600.048 meters in 30 seconds. What is the speed of the bus in kmph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72.006}
Find the derivative of the expression $b^tAc$ with respect to the matrix $A$ in matrix notation. What is the resulting derivative?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b c^T}
Let $S = \{ A = (a_1, \ldots, a_s) \mid a_i = 0 \text{ or } 1, \ i = 1, \ldots, 8 \}$. For any two elements of $S$, $A = \{ a_1, \ldots, a_8 \}$ and $B = \{ b_1, \ldots, b_8 \}$. Let $d(A,B) = \sum_{i=1}^{8} |a_i - b_i|$. Call $d(A,B)$ the distance between $A$ and $B$. At most, how many elements can $S$ have such that the distance between any two sets is at least 5?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Calculate the value of the Eisenstein series $G_2(\Lambda)$ for the lattice $\Lambda=\mathbb{Z}[i]$ in the complex plane when $k=1$. Provide your answer as a complex number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A line passing through the left focus $F_1$ of a hyperbola at an inclination of 30° intersects with the right branch of the hyperbola at point $P$. If the circle with the diameter $PF_1$ just passes through the right focus of the hyperbola, determine the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Find the minimum value of the expression \( \sqrt{x^2+1}+ \sqrt {4+(y-z)^2} + \sqrt{1+ (z-x)^2} + \sqrt{9+(10-y)^2} \) where \( x, y, z \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{149}}
Find the number of automorphisms of the group $\mathbb{Z}_4 \oplus \mathbb{Z}_2$ that map the element $(1, 0)$ to an element of order 4 and the element $(0, 1)$ to an involution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In a building with 1,800 light bulbs, the lifetime of each bulb (in days) is modeled as a Poisson process with an expected lifetime of 200 days. Calculate the probability that at most 10 bulbs fail on any given day.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.706}
Given a quadratic function \( f(x) \), define \( F(x) = \int_0^x f(t) \, dt \). Assume \( a \) is a positive number and \( F(x) \) has extreme values at \( x = -2a \) and \( x = 2a \). Find the local maximum value of the function \( \frac{F(x)}{F'(0)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4a}{3}}
397. What is the maximum number of queens that can be placed on an $8 \times 8$ chessboard so that each queen is attacked by at least one other queen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
There are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $\gcd(a, b, c, d) = 77$ and $\operatorname{lcm}(a, b, c, d) = n$. What is the smallest possible value for $n$? Please provide your answer as an integer without any commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27720}
The ant starts at point A and travels $1+\frac{1}{10}$ cm north, then $2+\frac{2}{10}$ cm west, then $3+\frac{3}{10}$ cm south, then $4+\frac{4}{10}$ cm east, then $5+\frac{5}{10}$ cm north, then $6+\frac{6}{10}$ cm west, and so on. After 1000 steps, the ant reaches point B. Find the straight-line distance between points A and B in centimeters. Record the square of the found distance as your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{605000}
已知正方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 的棱长为 1, 以顶点 $A$ 为球心， $\frac{2 \sqrt{3}}{3}$ 为半径作一个球，则球面与正方体的表面相交所得到的曲线的长等于 $\frac{m \sqrt{k}}{n} \pi$，请给出 m + n + k 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Alex starts with a rooted tree with one vertex (the root). For a vertex  $v$ , let the size of the subtree of  $v$  be  $S(v)$ . Alex plays a game that lasts nine turns. At each turn, he randomly selects a vertex in the tree, and adds a child vertex to that vertex. After nine turns, he has ten total vertices. Alex selects one of these vertices at random (call the vertex  $v_1$ ). The expected value of  $S(v_1)$  is of the form  $\tfrac{m}{n}$  for relatively prime positive integers  $m, n$ . Find  $m+n$ .**Note:** In a rooted tree, the subtree of  $v$  consists of its indirect or direct descendants (including  $v$  itself).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9901}
A shipment of 250 smartphones contains 3 different types: Type A, Type B, and Type C. There are 100 Type A smartphones with 30 defective ones, 80 Type B smartphones with 25 defective ones, and 70 Type C smartphones with 21 defective ones. If a customer buys two smartphones at random from the shipment, what is the approximate probability that both phones are defective?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0916}
Assume the Generalized Continuum Hypothesis (GCH). Calculate \( \aleph_{\omega_1}^{\aleph_0} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_{\omega_1}}
Determine the dimension of the ring \( \mathbb{K}[x,y,z]/(xy, xz, yz) \), where \( \mathbb{K} \) is a field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Bookman purchased some copies of a new book released recently, 10 of which are hardback and sold for $20 each, and the rest are paperback and sold for $10 each. If 14 copies were sold and the total value of the remaining books was $360, how many total copies did Bookman purchase initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
The perimeter of triangle $APM$ is $180$, and angle $PAM$ is a right angle. A circle of radius $20$ with center $O$ on line $\overline{AP}$ is drawn such that it is tangent to $\overline{AM}$ and $\overline{PM}$. Let $\overline{AM} = 2\overline{PM}$. Find the length of $OP$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
已知数列 $\left\{a_{n}\right\}$ 满足: $a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right]\left(n \in \mathbf{N}^{*}\right)$ ，其中 $[x]$ 表示不超过实数 $x$ 的最大整数. 设 $C$ 为实数, 且对任意的正整数 $n$, 都有 $\sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C$, 则 $C$ 的最小值是 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{289}
Problem 11.8. In a $28 \times 35$ table, some $k$ cells are painted red, another $r-$ in pink, and the remaining $s-$ in blue. It is known that

- $k \geqslant r \geqslant s$
- each boundary cell has at least 2 neighbors of the same color;
- each non-boundary cell has at least 3 neighbors of the same color.

What is the smallest value that the quantity $k-s$ can take?

(A cell is called boundary if it is adjacent to the edge of the table. Neighbors are cells that share a common side.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Jessy plans to finish a 140-page book in a week. She initially decides to read twice daily at a speed of 10 pages an hour, spending 30 mins each time, every day of the week. By mid-week, she realizes she is behind. She decides to increase her reading speed to 15 pages per hour and add more reading time to her schedule. How many extra minutes should she read per day for the rest of the week to finish the book on time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Evaluate the limit $$\lim_{x \to 0}\frac{\sin(\tan x)-\tan(\sin x)}{x-\sin x}$$ using Taylor series or L'Hôpital's rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the gradient with respect to the matrix $\pmb{A}$ of the function $\log(\det(\pmb{A}\pmb{X}))$, where both $\pmb{X}$ and $\pmb{A}$ are positive definite matrices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pmb{A}^{-1}}
An individual contracted influenza and after two rounds of transmission, a total of 100 people were infected. Calculate the average number of people who contracted the flu from one infected person per round of transmission.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Calculate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \), where \( S \) is the boundary of the solid \( E \) enclosed by the paraboloids \( z = x^2 + y^2 \) and \( z = 12 - 2x^2 - 2y^2 \), and \( \mathbf{F}(x, y, z) = \langle x^3 + y^2, 2yz + e^z, y^2 - z^2 \rangle \). Use the Divergence Theorem to find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48\pi}
In a circle, there are two mutually perpendicular chords $AB$ and $CD$. Determine the distance between the midpoint of segment $AD$ and the line $BC$, given that $AC=6$, $BC=5$, and $BD=3$. If necessary, round the answer to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.24}