inclusionAI/Ring-lite-rl-data huggingface dataset:
Given that \( n > 3 \), and \( a_1 = a_2 = \cdots = a_{n-1} = 1 \) and \( a_n = -2 \), find the value of the expression:
\[
f_n = \sum_{i=1}^{n}|a_{i}| - \sum_{1 \le i < j \le n}|a_{i} + a_{j}| + \sum_{1 \le i < j < k \le n}|a_{i} + a_{j} + a_{k}| - \cdots + (-1)^{n-1}|a_{1} + a_{2} + \cdots + a_{n}|
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 - 2n}
Evaluate the limit: $$\lim_{n\to\infty}\left(\frac{1}{n!} \int_0^e \log^n x \ dx\right)^n$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of idempotents in the ring \( \mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{Z}[i] \), where \( \mathbb{Z}[i] = \{a+bi : a,b \in \mathbb{Z} \} \) is the ring of Gaussian integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of polynomials of degree 3 contained in the ideal \( I = (x^5 + x^2 + x + 1, x^7 + x^5 + x^3 + x) \) in \( \mathbb{F}_2[x] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the exterior derivative $d\beta$ of the two-form $\beta = z(y^2 - x^2)\,dx\,dy - x^2y\,dx\,dz - xy^2\,dy\,dz$ in three-dimensional real space and simplify the result if possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In the figure, \(\triangle ABC\) is an equilateral triangle. Points \(D\) and \(E\) are on side \(BC\) such that \(BC = 3DE\). Construct an equilateral \(\triangle DEF\). Connect \(AF\), draw \(DG\) parallel to \(AF\) at point \(G\), and draw \(EH\) parallel to \(AF\) intersecting side \(AC\) at point \(H\). Let \(GI \perp AF\) and \(HJ \perp AF\). If the area of \(\triangle BDF\) is \(45\) and the area of \(\triangle DEF\) is \(30\), find the value of \(\frac{GI}{HJ}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Identify a group $G$ that contains subgroups isomorphic to the cyclic group $\mathbb{Z}_n$ for every positive integer $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Q}/\mathbb{Z}}
Determine the norm of the function \( f(x) = \sum_k x_k a_k \), where \( x = (x_1, x_2, \dots) \in l^p \) and \( a = (a_1, a_2, \dots) \in l^q \). Assume that \( p \) and \( q \) are appropriate indices for the \( l^p \) and \( l^q \) spaces, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\|a\|_q}
Evaluate the integral \( I = \int_{0}^{1}\int_{0}^{1} \left\{\frac{x}{y}\right\}\mathrm{d}x\mathrm{d}y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4} - \dfrac{\gamma}{2}}
Determine the winding number of the loop \( \alpha(z) = 3z^3 + 5z^2 + 2 \) around the origin \( 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $0<t<\frac{\pi}{2}$.

Evaluate

\[\lim_{t\rightarrow \frac{\pi}{2}} \int_0^t \tan \theta \sqrt{\cos \theta}\ln (\cos \theta)d\theta\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
-4
}
Evaluate the integral \( \int_0^\frac{\pi}{2}\cos ^2x\log(\tan x)dx. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{4}}
In triangle \(ABC\), where \(AB = 8\), \(AC = 7\), and \(BC = 10\), let \(I\) be the incenter. If \(AI\) intersects \(BC\) at \(K\) and the circumcircle of \(\triangle ABC\) at \(D\), find \(\frac{DK}{DI}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Let \( V = \left\lbrack  {\left( {x,y,z}\right) |0 \leq  x,y,z \leq  {2008}}\right\rbrack \) be a set of points in a 3-D space.
 If the distance between two points is either \( 1,\sqrt{2},2 \) , we color the two points differently.
 How many colors are needed to color all points in \( V \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
7
}
Let $a, b, c$ be three distinct positive integers. Define $S(a, b, c)$ as the set of all rational roots of $px^2 + qx + r = 0$ for every permutation $(p, q, r)$ of $(a, b, c)$. Determine the maximum number of elements in $S(a, b, c)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the value of \(\alpha\) such that the roots of the quintic equation \(x^5 + 5\lambda x^4 -x^3 + (\lambda \alpha -4)x^2 - (8\lambda +3)x + \lambda\alpha - 2 = 0\) are independent of the parameter \(\lambda\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Let $\triangle{ABC}$ be an isoceles triangle with $\angle A=90^{\circ}$. There exists a point $P$ inside $\triangle{ABC}$ such that $\angle PAB=\angle PBC=\angle PCA$ and $AP=10$. Find the area of $\triangle{ABC}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250}
Let \( a,b,c \) be real nonzero numbers such that \( a + b + c = {12} \) and
 \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{abc} = 1 \)
 Compute the largest possible value of \( {abc} - \left( {a + {2b} - {3c}}\right) \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Find a tight upper bound for the function \( f(n) \) defined by the recurrence relation \( f(n) = f(\sqrt{n}) + \frac{1}{n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(1)}
Find the asymptotic upper bound on the number of elementary matrices required to decompose a matrix in $GL(n,K)$ as $n$ approaches infinity, where $K$ is an infinite field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^2)}
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{1}{z^{4}-1}\,dz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{2}}
Evaluate the integral \( \int_{-1}^1 f(x) \,dx \) for the function \( f(x) \) defined as follows:

\[f(x) = \begin{cases} \frac{|\sin(x)|}{x}, & \text{if } x \neq 0 \\
1, & \text{if } x = 0 \end{cases} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the conic hull of the set \( T = \left\{(x_1,x_2) : (x_1 + 1)^2 + x_2^2 = 4 \right\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}^2}
Given that \( \int^{\infty}_{-\infty}e^{tx}f(x)dx = \sin^{-1}\left(t-\sqrt{\frac{1}{2}}\right) \), find the value of \( \left(\int^{\infty}_{-\infty}xf(x)dx\right)^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the value of \( x \) such that the matrix \( M_x = \begin{pmatrix} 1 & x+1 & 0 & 0 \\ 0 & x & x-1 & 0 \\ 0 & x-1 & x & 0 \\ 1 & 0 & x+1 & x \end{pmatrix} \) is diagonalizable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given a sample $X_1, X_2, \ldots, X_{15}$ from a distribution with probability density $\exp(-(x-\theta))$ for $x \ge \theta$, we test $H_0: \theta=0$ against $H_1: \theta>0$. The test statistic is $T = \min\{x_1, x_2, \ldots, x_{15}\}$. If the observed value of $T$ is $t=0.1$, calculate the p-value of this test.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-1.5}}
Calculate the line integral \(\oint_C \vec{F} \cdot d\vec{R}\) for the vector field \(\vec{F}=(ye^x, x^2+e^x, z^2e^z)\) and the curve \(C\) defined by \(\vec{r}(t)=(1+\cos t, 1+\sin t, 1-\cos t-\sin t)\), where \(0\leq t\leq 2\pi\). Given that the curl of \(\vec{F}\) is \((0, 0, 2x)\), apply Stokes' theorem to solve this problem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Determine the Lebesgue measure of the set of games that have a finite and odd number of Nash equilibria, given a fixed finite player set and finite action sets for these players.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the value of \(a\) in the interval \((2, \infty)\) 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}
Start by writing the integers $1, 2, 4, 6$ on the blackboard. At each step, write the smallest positive integer $n$ that satisfies both of the following properties on the board.
$n$ is larger than any integer on the board currently.
$n$ cannot be written as the sum of $2$ distinct integers on the board.
Find the $100$-th integer that you write on the board. Recall that at the beginning, there are already $4$ integers on the board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{388}
Given the functional equation \( f(x) = 3f(x+1) - 3f(x+2) \) and the condition \( f(3) = 3^{1000} \), find the value of \( f(2019) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6561}}
A help center receives students according to a Poisson process with rate $r$. When there are $n \ge 1$ students in the center, the first one to leave does so at a random exponential time with rate $2r$. Calculate the expected time until there are two students in the center, starting from a state with no students.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{r}}
Evaluate the integral \( \int_\Gamma z^{\frac{1}{2}} \, dz \) for the contour \( \Gamma \) in the right half-plane from \( z = -3i \) to \( z = 3i \), choosing appropriate branches of \( \log(z) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2i\sqrt{6}}
Determine the convergence of the complex series \(\sum_{n=1}^\infty\dfrac{n!}{n^n}z^n\) using the Ratio test.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|z| < e}
Consider the differential equation model for population growth given by:

$$ \begin{cases} \dfrac{\mathrm{d}N}{\mathrm{d}t} = rN(t) - r\dfrac{N^2(t)}{kt + b} \\ N(0) = N_0, \end{cases} $$

where $k>0$ and $b<0$. Determine the value of:

$$ \lim_{{t \to \frac{b}{k}}}{N'(t)} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find a differentiable function that serves as a lower bound for \( \min(a,b) \) given \( a, b \geq 0 \) and \( a + b = c \), where \( c \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{ab}{c}}
Two chess players, $P_1$ and $P_2$, are competing in the final of a championship. The series consists of matches, and the winner is the one who has a two-game lead over the other and wins at least 6 games. If $P_2$ wins the first 4 matches, what is the probability that $P_1$ will win the championship? Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{16}}
Determine the maximum number of acute angles possible in a non-convex polygon with $n$ sides.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - 2}
Determine the values of $s$ for which the integral \[ \int_0^\infty \frac{\mathrm{d} x}{x \cdot \ln x \cdot \ln^{(2)} x \cdot \ln^{(3)} x \cdots (\ln^{(k)} x)^s } \] converges, where $\ln^{(k)} x$ denotes the $k$-fold iterated logarithm of $x$. Provide a condition or range for $s$ that ensures convergence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{s > 1}
Determine the nature of the singularity at the point \( z = \pi/2 \) for the function \( f(z) = \tan^2 z \). Specifically, identify if it is a regular point, an essential singularity, or a pole, and if it is a pole, state its degree.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit: \[ \lim_{n \rightarrow \infty} n \sin (2\pi e n!). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Find the minimum order of an ordinary differential equation whose fundamental set of solutions includes the functions $e^x$, $xe^x$, and $\cos 2x$. \( \boxed{\text{Order}} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the triple integral \( \iiint_S {z{\sqrt{x^2+y^2}}}\,dx\,dy\,dz \) over the solid region \( S \) bounded by the cylinder \( x^2+y^2=1 \), the plane \( z=0 \), and the plane \( z=x \) in the first octant (\( z\ge0 \)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{40}}
Determine the injective dimension of \( \mathbb{Z}_n \) as a \( \mathbb{Z} \)-module.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the fundamental group of a Klein bottle with two points removed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z} * \mathbb{Z} * \mathbb{Z}}
Determine $d(A)$ for the set $A = \{f_n:[0,1] \to \mathbb{R} \mid f_n(x) = x^n, n \in \mathbb{N} \}$, where $E = ([0, 1], \mathbb{R})$ is the set of bounded functions equipped with the sup norm and the metric it creates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
All the positive integers that are coprime to $2021$ are grouped in increasing order, such that the nth group contains $2n - 1$ numbers. Hence the first three groups are $\{1\}$, $\{2, 3, 4\}$, $\{5, 6, 7, 8, 9\}$. Suppose that $2022$ belongs to the $k$th group. Find $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Let $a_n$ be a sequence of real numbers such that the series $\sum |a_n|^2$ is convergent. Determine the range of values of $p$ for which the series $\sum |a_n|^p$ is also convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2, \infty)}
There are 2017 turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-turtle. After 2015 seconds, only two turtles remain. What is the probability that there is a turtle that has never been combined with another turtle? Express this probability as a fraction \( \frac{p}{q} \) in simplest form, where \( p \) and \( q \) are relatively prime positive integers. Find \( p + q \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1009}
A palindrome is a whole number that reads the same forwards and backwards. If one neglects the colon, certain times displayed on a digital watch are palindromes. Three examples are: $ \boxed{1:01} $, $ \boxed{12:21} $.How many times during a 12-hour period will be palindromes? 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
57
}
Calculate the area enclosed by the region defined by the equation: \[ \left\lfloor\frac{\left|3x + 4y\right|}{5}\right\rfloor + \left\lfloor\frac{\left|4x - 3y\right|}{5}\right\rfloor = 3 \] where \( \lfloor\cdot\rfloor \) denotes the greatest-integer function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Use Stokes' Theorem to calculate \( \iint_S \text{curl} \mathbf{F} \cdot d\mathbf{S} \) for the surface \( S = \{(x,y,z): x^2+y^2+z^2=2; -1\leq z\leq 1\} \) and the vector field \( \mathbf{F}(x,y,z) = (-x-y+xz^2 , x+z , z+e^{x+y}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate \( \iint_R (x^2+y^2) \, dx \, dy \), where \( R \) is the region in the first quadrant bounded by the curves \( xy=2, \, xy=4, \, x^2-y^2=3, \, x^2-y^2=5. \) Use the transformation \( u=xy \) and \( v=x^2-y^2 \) to simplify the integral. Express \( x^2+y^2 \) in terms of \( u \) and \( v \), and evaluate the integral over the transformed region. Provide the final numerical result.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
There are 2 boys for every 3 girls in Ms. Johnson's math class. If there are 30 students in her class, what percent of them are boys?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40%

}
Determine the number of values of \(x\) in the interval \((0, 20)\) such that \(\{x\} + \{x^2\} = 1\), where \(\{\}\) denotes the fractional part of a number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
Compute the surface integral \( \iint_Y F \cdot N\, dS \) using Gauss's theorem, where \( Y = (x-z)^2 + (y-z)^2 = 1 + z^2 \) for \( 0 \leq z \leq 1 \) and \( N \) is the outward-pointing normal vector. The vector field is given by \( F = (y, x, 1 + x^2z) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3\pi}
Determine the sum of all possible surface area of a cube two of whose vertices are $(1,2,0)$ and $(3,3,2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99
}
Given \(\mathbf{m}=(\cos \theta) \mathbf{i}+(\sin \theta) \mathbf{j}\) and \(\mathbf{n}=(\sqrt{2}-\sin \theta) \mathbf{i}+(\cos \theta) \mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors along the \(x\)-axis and \(y\)-axis respectively, and \(\theta \in (\pi, 2\pi)\). If the magnitude of the vector \(\mathbf{m} + \mathbf{n}\) is \(|\mathbf{m} + \mathbf{n}| = \frac{8 \sqrt{2}}{5}\), find the value of \(5 \cos \left(\frac{\theta}{2} + \frac{\pi}{8}\right) + 5\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the integral \( \int_{-\infty}^0 \int_x^\infty \frac{e^{-x^2/2}}{\sqrt{2\pi}} \frac{e^{-y^2/2}}{\sqrt{2\pi}}\, dy\, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{8}}
Determine the greatest value of \( b \) such that any real-valued function \( f \) satisfying the following properties must also satisfy \( f(1) < 5 \):

1) \( f \) is infinitely differentiable on the real numbers.
2) \( f(0) = 1, f'(0) = 1, \) and \( f''(0) = 2 \).
3) \( |f'''(x)| < b \) for all \( x \) in \([0,1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the function \( f(x) \) that satisfies the functional equation \( f(x+1) = x f(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Gamma(x)}
A group of farmers produced fewer than 1000 apples. Each farmer, in turn, took either exactly one-half or exactly one-third of the apples remaining in the collection, without cutting any apples. After all farmers took their shares, the remaining apples were given to charity. What is the greatest number of farmers that could have participated in this apple sharing process?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
An unbiased die is cast seven independent times. Calculate the conditional probability that each side appears at least once, given that side 1 appears exactly twice.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{24}{625}}
Jacob rolls two fair six-sided dice. If the outcomes of these dice rolls are the same, he rolls a third fair six-sided die. Compute the probability that the sum of the outcomes of all the dice he rolls is even.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{12}}
Determine the value of \( n \) in the following limit without using L'Hopital's rule:

\[\lim_{x \to 0} \frac{(e^x+e^{-x})\cos(nx)}{e^x-1}\]

given that \(\lim_{x \to 0} f(x) = 2 - n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the smallest positive integer \( c \) such that for all non-negative integers \( n \), the inequality \( f(n) \leq cn^2 \) holds, where \( f(n) \) is defined recursively by:

- Base case: \( f(0) = 4 \).
- Recursive case: \( f(n+1) = 7f\left(\lfloor \frac{n+1}{3} \rfloor \right) + 5(n+1)^2 \) for \( n > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Given that $Y_1 < Y_2 < \ldots < Y_8$ are the order statistics of $n$ independent observations from a continuous distribution with a 70th percentile $\pi_{0.7} = 27.3$, find the probability $P(Y_7 < 27.3)$. Assume no specific distribution is given.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2553}
Evaluate the integral \( \int_{0}^{+\infty} \frac{M}{a}x \big(1-\exp(-\frac{x}{a})\big)^{M-1}\exp(-\frac{x}{a})dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \sum_{k=1}^{M} \frac{1}{k}}
Determine the smallest number of terms, \( n \), required to approximate the series \( \sum_{k=1}^{\infty} 28 k e^{-0.45 k^2} \) with an accuracy of \( 10^{-7} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
How many odd $3$-digit numbers have exactly four $1$’s in their binary (base $2$) representation? For example, $225_{10} = 11100001_2$ would be valid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{195}
Evaluate the limit: $$\lim_{n\to\infty}\left[\frac{\left(\frac{3n}{e}\right)^{3n}\cdot \sqrt{6\pi n}}{n^n\cdot \left(\frac{n}{e}\right)^{n}\cdot \sqrt{3\pi n}}\right]^{\frac{1}{n}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Calculate the fractional change in the wavelength of the Lyman-alpha line (transition from n = 2 to n = 1) for a hydrogen atom when considering the reduced mass of the electron.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.45 \times 10^{-4}}
How many ways are there to distribute 20 different presents to 4 different children such that no child receives exactly 6 presents?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{523708416736}
Let \( A = k[x,y] \) where \( k \) is an algebraically closed field, and let \( M = A/(xy) \) be an \( A \)-module. Calculate the support of \( M \), denoted as \( \text{Supp}(M) = \{ P \in \text{Spec}(A) : M_P \neq 0 \} \), where \( M_P = S^{-1}M \) and \( S = A \setminus P \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{V(x) \cup V(y)}
Identify a function that grows faster than $n^{kn}$ but slower than $2^{n^2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n^{3/2}}}
Evaluate the expression: \(\cos\left(\frac{\pi}{11}\right)\cos\left(\frac{2\pi}{11}\right)\cos\left(\frac{3\pi}{11}\right)\cdots\cos\left(\frac{10\pi}{11}\right) = -2^n \). Find the value of \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-10}
Determine the codimension of the variety defined by the zero set of the determinants of all $m \times m$ submatrices of a symbolic $m \times n$ matrix, where $m \leq n$. Express your answer in terms of $m$ and $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - m + 1}
Given a square matrix $A = [a_{ij}] \in \mathbb{R}^{n \times n}$ with entries $\pm 1$ and balanced columns (i.e., $\sum_{i=1}^n a_{ij}=0$ for each column $j$), determine the maximum possible spectral norm of $A$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \((x-y)(f(x)+f(y))=(x+y)f(x-y)\), given that \(f(0)=0\) and \(f\) is an odd function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = kx}
Determine the maximum value of \( C \) such that for any continuously differentiable function \( f \) on the interval \([0,1]\) with \( f(0) = f(1) = 0 \), the inequality \( \max_{x \in [0,1]} |f'(x)| \geq C \int_0^1 |f(x)| \, dx \) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}