inclusionAI/Ring-lite-rl-data huggingface dataset:
If the curve $y=\frac{a}{x}$ ($x>0$) has a common tangent line with the curve $y=2\ln x$, then the range of values for $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-\frac{2}{e}, 0)}
Let  $T$  be the answer to question  $18$ . Rectangle  $ZOMR$  has  $ZO = 2T$  and  $ZR = T$ . Point  $B$  lies on segment  $ZO$ ,  $O'$  lies on segment  $OM$ , and  $E$  lies on segment  $RM$  such that  $BR = BE = EO'$ , and  $\angle BEO' = 90^o$ . Compute  $2(ZO + O'M + ER)$ .

PS. You had better calculate it in terms of  $T$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7T}
Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(9,3)$, respectively. What is its area?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45 \sqrt{3}}
Find the radius of convergence for the power series: $$\sum_{i=0}^{\infty}\left({(-1)^{i}\over 2^i(i!)}\left(\prod_{j=1}^{i}\left(2j-1\right)\right)\left(-18x^4\right)^i\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\left(\frac{1}{18}\right)^{1/4}\)}
Given a sample set: $1, 3, 4, a, 7$. The mean of this sample set is $4$. Find the standard deviation of the sample set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the sequence {a_n} satisfying a_1 = 1/2, a_{n+1} = 3a_n + 1, find the sum S_{2016} of the first 2016 terms of the sequence. Express your answer as a fraction or integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3^{2016}-2017}{2}}
Calculate the volume of the solid formed by revolving the region bounded by the curve $x = y^2$ and the line $x = 9$ about the x-axis. The cross-sections perpendicular to the x-axis are isosceles right triangles with the hypotenuse as the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{81}{2}}
Given that the Mathematics College Entrance Examination scores distribution ξ closely follows the normal distribution N(100, 5^2), and P(ξ < 110) = 0.96, find the value of P(90 < ξ < 100). Express your answer as a decimal value between 0 and 1, rounded to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.46}
Calculate the residue of the function \( f(z) = \sin\left(\frac{z^2+1}{z}\right) \) at \( z = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{J_1(2)}
Determine the value of \(a > 2\) such that \(\int_{0}^{1}\frac{x^8-x^2}{x^{12}+ax^6+1}dx = -\frac{\pi}{24}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
How many positive integer solutions less than 71 are there for the equation $x^7 \equiv a \pmod{71}$, given that $a$ is an integer not divisible by 71?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
长为正整数 $l$ 的木棒每次都可锯成长均为正整数的 2 段，可连续进行这种操作，但要求任何时刻所有木棒中的最长者长小于最短者长的 2 倍。例如，一根长为 6 的木棒只能截成长均为 3 的两段；一根长为 7 的木棒只能截成长分别为 $2,2,3$ 的三段。试问:长为 30 的木棒最多可锯成多少段?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Quadrilateral $EFGH$ has right angles at $F$ and $H$, and $EG = 5$. If $EFGH$ has two sides with distinct integer lengths, and each side length is greater than 1, what is the area of $EFGH$? Express your answer in simplest radical form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let \( n = 2^{40}5^{15} \). How many positive integer divisors of \( n^2 \) are less than \( n \) but do not divide \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{599}
Consider the region defined by \((x,y,z)\) such that \(x^2+y^2<1, x>0, 0 \le z \le 5\). This describes one half of a cylinder. A wedge is cut from this half-cylinder along the plane \(z=2x\). Find the volume of the wedge.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Determine the values of $x$ for which the series \( \sum_{n=1}^\infty \frac{1}{n^2} \ln \left(1 + \frac{x}{n}\right) \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, \infty)}
Two curves are given by the equations \( y=2 \sin \frac{5 \pi}{6}x \) and \( y=\alpha x^2-3 \alpha x+2\alpha +1 \). If these curves touch each other at some point within the interval \( 0 \leq x \leq \frac{18}{5} \), find the value of \( \frac{2 \sqrt{3} \alpha}{5 \pi} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given an open set $M \subset \mathbb{R}^3$ and a continuous function $\rho: M \to \mathbb{R}$ representing a charge density, consider the vector field $v: M \to \mathbb{R}^3$ defined by

$$v(x) = \int_{r\in M} \frac{\rho(r)(x-r)}{\|x-r\|^3} \, dr$$

where the integral is taken component-wise. Compute the divergence of $v$, $\nabla \cdot v$. Express your answer in terms of $\rho$. \(n = 3\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi \rho(x)}
Determine the greatest real number \( C \) ,such that for every positive integer \( n \geq  2 \) ,there exists \( {x}_{1},{x}_{2},\ldots ,{x}_{n} \in  \left\lbrack  {-1,1}\right\rbrack \) ,so that
 \( \mathop{\prod }\limits_{{1 \leq  i < j \leq  n}}\left( {{x}_{i} - {x}_{j}}\right)  \geq  {C}^{\frac{n\left( {n - 1}\right) }{2}} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1/2}
A toy consists of 9 differently sized rings and a tapering pole. The rings can be stacked on the pole in any order until the top of the pole is reached, at which point they must be in decreasing order of size. How many different configurations of the rings can be created without reaching the top of the pole?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{623530}
A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$. How many edges does the remaining solid have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Evaluate the following definite integral.
\[ 2^{2009}\frac {\int_0^1 x^{1004}(1 - x)^{1004}\ dx}{\int_0^1 x^{1004}(1 - x^{2010})^{1004}\ dx}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2010
}
The digits $1, 2, \cdots, 9$ are arranged in random order to form a nine-digit number. What is the probability that this number is divisible by $11$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11}{126}}
Compute the line integral \( \int (x^2-y^2) \, dx +(x^2+y^2) \, dy \) over the closed curve formed by the curves \( y=x^2+1 \) from \((1,2)\) to \((2,5)\), and \( y=2x+2 \) from \((2,5)\) to \((1,2)\), using Green's theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{49}{30}}
A set of three elements is called arithmetic if one of its elements is the arithmetic mean of the other two. Likewise, a set of three elements is called harmonic if one of its elements is the harmonic mean of the other two.

How many three-element subsets of the set of integers $\{z \in \mathbb{Z} \mid -2011 < z < 2011\}$ are arithmetic and harmonic?

**Remark:** The arithmetic mean $A(a,b)$ and the harmonic mean $H(a,b)$ are defined as:

\[A(a,b) = \frac{a+b}{2} \quad \text{and} \quad H(a,b) = \frac{2ab}{a+b} = \frac{2}{\frac{1}{a} + \frac{1}{b}},\]

respectively, where $H(a,b)$ is not defined for some $a$, $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1004}
Evaluate the limit: \[ \lim_{(x, y, z) \to (0, 0, 0)} \frac{x^2 y^2 z^2}{(x^4 + y^4 + z^4)^{a/4}}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral \( \int_0^1 \ln(1-x^{1/n}) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-H_n}
For positive integer \( n \), consider the ordinary differential equation
\[ y'' + \frac{1}{x}y' + \left(1 - \frac{n^2}{x^2}\right)y = 0 \]
which has a solution in the form of the Bessel function of order \( n \), \( J_n(x) = x^n \sum_{k=0}^{\infty} a_kx^k \). Determine the coefficient \( a_1 \) by substituting the series representation of the Bessel function into the differential equation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \( g: \mathbb{R}^3 \to \mathbb{R}^3 \) be defined by \( g(x,y,z) = (3x+4z, 2x-3z, x+3y) \) and let \( S = \{ (x,y,z) \in \mathbb{R}^3 : 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1 \} \). It is given that
\[ \iiint_{g(S)} (2x + y - 2z) \, dx \, dy \, dz = \alpha \iiint_S z \, dx \, dy \, dz. \]
Find the value of \( \alpha \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{255}
Determine the interval on which the polynomial \( y = -\frac{3\pi}{2} + x - \frac{(2x-3\pi)^3}{48} \) best approximates the function \( y = \cos(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\pi, 2\pi]}
Determine the box-counting dimension (Minkowski Dimension) of the set \( S = \left\{ \frac{1}{n^2} \mid n \in \mathbb{Z}^+ \right\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Given the quartic polynomial \(x^4 - 3x^3 + ... - 10 = 0\) with real coefficients, and two complex roots \(a + 2i\) and \(1 + bi\), where \(a\) and \(b\) are real numbers, determine the sum of the real roots of the polynomial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Five bags, each of a distinct color (Blue, Green, Pink, Red, and Yellow), contain 5 balls of the same color as the bag. A ball is drawn from each bag and transferred to one of the other bags, ensuring that each bag still contains 5 balls at the end. Calculate the probability that, after all transfers, each bag contains 5 balls of its original color.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Convert the following limit to a Riemann sum and evaluate the resulting integral:
\[ \lim_{N \to \infty} \sum_{n=-N}^{N} \left( \frac{1}{N+in} + \frac{1}{N-in} \right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
A fair $6$-sided die is repeatedly rolled until a $1, 4, 5$, or $6$ is rolled. What is the expected value of the product of all the rolls?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
16
}
Consider a unit segment [0, 1] that is colored randomly using two colors, white and black. What is the probability that a randomly chosen point on the segment is black after n steps?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the dimension of the set \( A = \left\{ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} \in M(2, \mathbb{R}) \mid a, c \neq 0 \right\} \), given that \( A \) is an open subset of the differential manifold \( GL(2, \mathbb{R}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Compute \(2^{2475} \bmod 9901\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
Suppose $X$ and $Y$ are independent random variables, each following a normal distribution with mean $8$ and variance $10$, denoted as $X,Y\sim\mathcal{N}(8,10)$. Calculate the probability $\mathbb{P}(X\le 2Y)$. \(\boxed{?}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.871}
Consider a smooth projective variety $M$ of dimension $n$ over the complex numbers, where the cotangent bundle $T^*M$ forms a completely integrable system with $n$ independent regular functions $f_1, \ldots, f_n$ in involution. The moment map is given by $\mu: T^*M \to \mathbb{C}^n$, defined by $f_1, \ldots, f_n$. Let $Y \subset \mathbb{C}^n$ be the set of points $p \in \mathbb{C}^n$ such that the fiber $\mu^{-1}(p)$ is reducible. Determine the co-dimension of $Y$ and verify if it is always $\ge 2$. \( \text{Provide your answer in the form of a co-dimension value.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider a line of $n$ bugs, each with a unique size, moving in the same direction. If a larger bug is behind a smaller bug, it will eat the smaller bug. What is the expected number of bugs remaining after a long enough time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H_n}
Find the angle of rotation that eliminates the cross term in the conic section defined by the equation \(5x^2 + 5y^2 - 6xy - 8 = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Evaluate the surface integral \( \iint_S \mathbf{F} \cdot \mathbf{n} \, dS \) where \( \mathbf{F}(x,y,z) = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \), \( S \) is the plane surface whose boundary is the triangle with vertices at \((1,0,0), (0,1,0),\) and \((0,0,1)\), and \( \mathbf{n} \) is the unit normal to \( S \) having a nonnegative \( z \)-component. Use the vector representation \( \mathbf{r}(u,v) = (u+v)\mathbf{i} + (u-v)\mathbf{j} + (1-2u)\mathbf{k} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the number of integer solutions \((a, b)\) for the equation \( \frac{ab}{a+b} = 3600 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{449}
Evaluate the integral \( \frac{1}{2 \pi i} \int_{\partial B_1(0)} \frac{6z^{98}}{23z^{99}-2z^{81}+z^4-7}dz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{6}{23}}
A server processes events from clients, where each client sends an event every second. The server processes an event in t seconds, where t < 1 second (e.g., 0.01 seconds). We want to serve 70,000 clients. The server can process up to n simultaneous requests, and a timeout occurs if this limit is exceeded. What should be the value of n so that a timeout occurs on less than 1% of requests?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{762}
Taylor Swift made the song $15$ in $2008$, and the song $22$ in $2012$. If she had continued at the same rate, in what year would her song $1989$ have come out?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3136}
Find the minimum value of the sum of distances $PX + PY$, where $P$ is a variable point on the line $2x + y + z = 1$. The fixed points are $X(1,1,1)$ and $Y(2,2,4)$. What is the minimum value of $PX + PY$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{29}}
If Player 1 and Player 2 each independently sample from a Uniform(a, b) distribution, where a < b, what is the expected number of samples Player 2 must make to obtain a number higher than Player 1's sample?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In a company with 100 shareholders, any group of 66 shareholders collectively owns at least 50% of the company's shares. What is the maximum percentage of shares that a single shareholder can own?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25\%}
Evaluate the integral \( \iint_{D}\left(x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}\right)\mathrm{d}x\mathrm{d}y \) where \( f(x,y) \) is defined on \( D=\{(x,y)\mid x^2+y^2\le1\} \) and has continuous second-order partial derivatives in \( D \). Given that \( \frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}=\mathrm{e}^{-(x^2+y^2)} \), find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2e}}
Consider three disjoint line segments of equal length labeled AB, CD, and EF. Determine the size of the group of symmetries of this configuration, where symmetries include exchanging line segments and flipping segments end for end.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Sidney Crosby is standing on the goal line, three feet outside one of the goal posts of a hockey goal that is six feet wide. How far should he move out perpendicular to the goal line to maximize the angle at which he can shoot at the goal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{3}}
Siva has the expression \( \frac{1}{2} \_ \frac{1}{4} \_ \frac{1}{8} \_ \frac{1}{16} \_ \frac{1}{32} \), where each blank is filled with either a plus or a minus based on a fair coin flip. He computes the value of the expression and repeats the process with new coin flips. What is the probability that the positive difference between his computed values is greater than \( \frac{1}{2} \)? If this probability can be expressed as \( \frac{a}{b} \) for relatively prime positive integers \( a \) and \( b \), find \( a + b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
In a group of 50 people, what is the minimum number of rotations required to seat them at tables of 5 so that each person meets every other person at least once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Calculate the limit: \[ \lim_{n\to\infty}\left(\frac{\sin(2\sqrt{1})}{n\sqrt{1}\cos\sqrt{1}} + \cdots + \frac{\sin(2\sqrt{n})}{n\sqrt{n}\cos\sqrt{n}}\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
设$x, y , z \in R ^ { + }$，求$\frac { 2 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } } { x y + \sqrt { 3 } y z }$的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt 2$}
In the isosceles trapezoid \(ABCD\), the bases \(AD\) and \(BC\) are related by the equation \(AD = (1 + \sqrt{15}) BC\). A circle is constructed with its center at point \(C\) and radius \(\frac{2}{3} BC\), which cuts a chord \(EF\) on the base \(AD\) of length \(\frac{\sqrt{7}}{3} BC\). Determine the ratio in which the circle divides side \(CD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:2}
Determine the number of transitive $G$-sets of the symmetric group $S_3$ up to isomorphism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Approximate the value of \( a = \frac{2.04^2}{\sqrt[3]{0.97 \sqrt[5]{1.02^2}}} \) using differential approximation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.19}
In the rhombus $ABCD,$ the measure of $\angle{B}=40^{\circ}, E$ is the midpoint of $BC,$ and $F$ is the base of the perpendicular dropped from $A$ on $DE.$ Find the measure of $\angle{DFC}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$110^\circ$
}
Evaluate the integral \( \int_A \frac{dz}{4z^2+1} \), where \( A = \{ z \in \mathbb{C} \mid |z-i|=1 \} \), in the negative direction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{2}}
Determine the value of the Ramsey number \( R(K_3, C_5) \), which is the smallest number of vertices \( n \) such that any graph on \( n \) vertices contains either a triangle \( K_3 \) or a cycle of length 5 \( C_5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the dimension of the kernel of the linear transformation on the cyclic module \( M \), where \( M \cong \frac{D}{(p^k)} \) for a principal ideal domain \( D \), an irreducible element \( p \), and an integer \( k \geq 1 \). The transformation is defined as multiplication by \( p \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the value of the improper integral $ \int_0 ^ {\pi} \log (\sin (x)) dx$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$-\pi \log 2$}
February 2, 2020, is an interesting "palindrome day" because the date written as 20200202 reads the same from left to right and right to left. What is the next "palindrome day" after this date?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20211202}
In an equilateral triangle with an area of 1, color parts of the triangle red, blue, and green such that each color forms exactly one connected region strictly inside the triangle, and no line parallel to any side contains points of multiple colors. Determine the maximum possible value of the minimum area among the red, blue, and green regions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Determine the smallest number of data points, \( n \), such that the kernel matrix \( K \) is guaranteed to be singular for any matrix \( X \in \{1, 2, \ldots, m\}^{d \times n} \) and any kernel function \( k : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m^d + 1}
Define a sequence $\{a_{n}\}$ by
$$a_{0}=\frac{1}{2}, \quad a_{n+1}=a_{n}+\frac{a_{n}^{2}}{2012},(n=0, \quad 1, \quad 2, \quad \cdots),$$

find integer $k$ such that $a_{k}<1<a_{k+1}$.

(September 29, 2012, Hohhot)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2012

}
Determine the number of singular matrices of the form \(A=\begin{bmatrix} 1 & a & b\\ w&1 &c \\ w^2 &w & 1 \end{bmatrix}\) where \(a, b, c\) are elements of the set \(S=\{w, w^2\}\), and \(w\) is a complex cube root of unity with \(w \neq 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the limiting value of \( \mathcal{P}_\infty \) as \( n \) approaches infinity, where \( \mathcal{P}_n \) is the ratio of twin primes \((p_1, p_2)\) with \( p_2 = p_1 + 2 \) and \( p_1 - 1 = a_1 \times b_1 \), \( p_2 + 1 = a_2 \times b_2 \), \( p_1 + 1 = p_2 - 1 = a \times b \) such that \( \min\{|b_1 - a_1|, |b - a|, |b_2 - a_2|\} \neq |b - a| \). Exclude the pair \((2, 3)\) from consideration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Identify the smallest non-abelian group that is not a complex reflection group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Q_8}
Evaluate the surface integral \(\iint_S (x, y, 3z) \, dS\) over the region \(S\), where \(S\) is the finite portion of the surface \(z = 4x^2 + y^2\) cut off by the plane \(z = 8x + 4y - 4\), and the normal to \(S\) points upwards.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\pi}
Given the expression \(\frac{1}{(1-x)^4(1+2x^3+x^6)^4}\), simplify it and find the sum of the coefficients of \(x^{20}\) and \(x^{21}\) in the resulting expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3829}
Determine the degree of the minimal polynomial of $\cos \frac{\pi}{48}$ over $\mathbb{Q}$, given that it is a root of the $96$-th cyclotomic polynomial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Consider 8 distinct boxes placed in a line, each containing between 1 and 8 red balls. The difference in the number of balls between any two adjacent boxes must be at most 1. Determine the number of distinct ways to distribute the balls among the boxes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11814}
Find the number of strongly monotonic increasing functions \(f:[n]\to \mathcal{P}([k])\) with respect to the subset relation, where \(n > k+1\). Use the pigeonhole principle to determine the value of \(a_{n,k}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
How many 13-card bridge hands, drawn from a standard 52-card deck, contain one or more quads (four cards of the same rank)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21717689136}