inclusionAI/Ring-lite-rl-data huggingface dataset:
Given the unit disk $D^2$ in $\mathbb{R}^2$ and its boundary $S^1$, determine the well-known space that the quotient space $D^2/S^1$ is homeomorphic to.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S^2}
Given the differential equation \(y'(x, y_0) = y^2(x, y_0) + y(x, y_0) \sin(x)\) where \(y_0\) is the initial condition at \(x=0\), find the partial derivative \(\frac{\partial y}{\partial y_0}\) evaluated at \(y_0=0\), i.e., \(\frac{\partial y}{\partial y_0}\bigg|_{y_0=0}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{1 - \cos x}}
Suppose the random variable X has a moment generating function given by \( M(t) = \frac{e^{at}}{1-bt^2} \) for \(-1 < t < 1\). Given that the mean and variance of X are 3 and 2, respectively, find the value of \(a + b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A die is rolled infinitely many times. What is the probability that the sequence (6,6) occurs before the sequence (1,2)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{6}{13}}
Given that \(x + x^{-1} = \frac{1}{2}(1 + \sqrt{5})\), find the value of \(x^{2000} + x^{-2000}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the algebraic degree of \( \tan(\pi/180) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Determine the degree of \( \nu_{n,d}(C) \) for any curve \( C \subset \mathbb{P}^n \) with degree \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a d}
Compute the exterior product \(\phi(X, Y)\) for the 2-form \(\phi = \sum_{j=1}^{m-1} y_j \wedge y_{j+1}\) on \(\mathbb{R}^m\) and the vectors \(X = (2, -1, 3, \dots, 2)\) and \(Y = (0, 2, 0, \dots, 2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the generator of the quotient group \( \frac{\mathbb{Z}[a] \times \mathbb{Z}[b] \times \mathbb{Z}[c]} {\mathbb{Z}[a-c] \times \mathbb{Z}[b+c]} \) in terms of the original generators \(a\), \(b\), and \(c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c}
For a real number $\alpha>0$, consider the infinite real sequence defined by $x_1=1$ and
\( \alpha {x}_{n} = {x}_{1} + {x}_{2} + \cdots  + {x}_{n + 1}\; \) for \( n \geq  1. \)
Determine the smallest $\alpha$ for which all terms of this sequence are positive reals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4
}
Determine the value of \( a \in \mathbb{C} \) such that the function \( f(z) = z^2 + a\overline{z}^2 + 4z\overline{z} + 2z - 8\overline{z} + 1 + 2i \) is not holomorphic at any point in \( \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Determine the values of \( q \in \mathbb{R} \) for which the series \( \sum^{\infty}_{n=3}{\frac{1}{n(\log n) ^ {(\log(\log n))^q}}} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{q > 0}
On a line, there are blue and red points, with at least 5 red points. It is known that on any segment with endpoints at red points that includes at least one red point inside, there are at least 3 blue points. Additionally, on any segment with endpoints at blue points that includes exactly 2 blue points inside, there are at least 2 red points. Determine the maximum number of blue points that can be on a segment with endpoints at red points, not containing other red points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the number of conjugacy classes in a group of order 625, given that its center has an order of 25.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{145}
The distinct symbols $\heartsuit, \diamondsuit, \clubsuit$ and $\spadesuit$ each correlate to one of $+, -, \times , \div$, not necessarily in that given order. Given that$$((((72 \,\, \,\, \diamondsuit \,\, \,\,36) \,\, \,\,\spadesuit \,\, \,\,0 ) \,\, \,\, \diamondsuit \,\, \,\, 32) \,\, \,\, \clubsuit \,\, \,\, 3)\,\, \,\,  \heartsuit \,\, \,\, 2 = \,\, \,\, 6,$$what is the value of$$(((((64 \,\, \,\, \spadesuit \,\, \,\, 8) \heartsuit \,\, \,\, 6) \,\, \,\, \spadesuit \,\, \,\, 5) \,\, \,\, \heartsuit  \,\, \,\, 1) \,\, \,\, \clubsuit \,\, \,\, 7) \,\, \,\, \diamondsuit \,\, \,\, 1?$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{118}
Determine the dimension of \(\mathcal{L}_F(K)\) as a vector space over \(F\), given that \(K/F\) is a finite-dimensional field extension with \([K:F]=n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2}
Compute the integral \( \oint_{\partial D(3i,5)} \frac{z}{1-e^z} \, dz \) using the Cauchy Integral Formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi^2}
Determine the fundamental group of \( \mathbb{R}^3 \) with the \( z \)-axis removed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Let $G$ be the symmetric group $S_5$ of permutations of five symbols. Consider the set $J$ of subgroups of $G$ that are isomorphic to the non-cyclic group of order $4$. Two subgroups $H$ and $K$ in $J$ are considered equivalent if they are conjugate, meaning there exists an element $g$ such that $gHg^{-1} = K$. How many equivalence classes are there in $J$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given $N$ indistinguishable balls, where only one ball is heavier, and a balance scale that can determine which of two groups of balls is heavier, find the minimal number of weighings required to identify the heaviest ball. Each ball can be weighed at most $K$ times, where $K \geq 1$. For example, if $N = 19$ and $K = 2$, the answer is 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the sum of the infinite series whose general term is given by \( \frac{n^2+2}{5^n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{31}{32}}
Determine the number of integer values of $a$ for which there exists at least one complex number $z$ that satisfies both $|z-ai|=a+4$ and $|z-2|<1$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Compute \( \sup_{f \in \mathcal{F}} |f(0)| \), where \( \mathcal{F} = \{ f \text{ holomorphic in } \mathbb{D} \text{ with } f(\mathbb{D}) \subset \mathbb{D}, f\left(\frac{1}{2}\right) = f'\left(\frac{1}{2}\right) = 0 \} \) and \( \mathbb{D} \) is the unit disc.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
What is the largest positive integer $r$ such that there are infinitely many elliptic curves over the rationals with rank at least $r$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Let \( O \) be the pole in polar coordinates. The hyperbola \(\rho=\frac{a^{2}}{5-6 \sqrt{2} \cos \theta}\) intersects the line \(\rho \cos \left(\theta-\frac{\pi}{4}\right) = a\) at points \( A \) and \( B \). Find the range of values for \( a \) such that the angle \(\angle A O B \leq \frac{2 \pi}{3}\), considering that \(\rho\) can take negative values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-14, 2]}
Calculate \( \lim\limits_{n \rightarrow +\infty} \int_{0}^{1} f_n(x) e^{-x^2} \, dx \), where \( f_n(x) = nxe^{-\sqrt{n}x} \) for \( x \in [0,+\infty) \). The sequence of functions \( f_n(x) \) converges pointwise to the null function on \( [0,+\infty) \) but not uniformly. However, there is uniform convergence on sub-intervals \( [a,+\infty) \) with \( a > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Adam has a box with $15$ pool balls in it, numbered from $1$ to $15$, and picks out $5$ of them. He then sorts them in increasing order, takes the four differences between each pair of adjacent balls, and finds exactly two of these differences are equal to $1$. How many selections of $5$ balls could he have drawn from the box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{990
}
Given that the number of storms in a rainy season follows a Poisson distribution with a parameter \( \lambda \) that is uniformly distributed between 2 and 7, calculate the probability of having at least 5 storms in the season.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.454}
Find the expected value of the maximum of three independent exponential random variables X, Y, and Z, with rate parameters \( \lambda = 1, 2, 3 \) respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{73}{60}}
Evaluate the limit: $$\lim_{x \to \frac{\pi}{2}} \tan \left(\frac{\pi}{4}\sin x\right)^\left({\dfrac 1{\tan(\pi \sin x)}}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{e}}
Find the position of the string $y(x,t)$ that satisfies the standard wave equation for small amplitudes, given the initial conditions $y(x,0)=\sin x$ and $\frac{\partial y}{\partial t}(x,0)=\cos x$. Provide the solution for $y(x,t)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin(x + t)}
Calculate the minimum distance of a cyclic code of length 9 over $\text{GF}(2)$ with the generator polynomial $g(x) = x^7 + x^6 + x^4 + x^3 + x + 1$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Compute the integral \( \mathcal{J}_2 = \int_{0}^{2\pi} \cos(\cos t)\,dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi J_0(1)}
Determine the smallest positive integer \( m \) for which \( \underbrace{100^{100^{\ldots^{100}}}}_m > \underbrace{3^{3^{\ldots^3}}}_{100} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
Determine the functional derivative \( \frac{\delta \mathfrak{F}}{\delta f} \) of the Fourier transform \( \mathfrak{F} \) given by:
\[ f \mapsto \int_{-\infty}^{\infty} f(x)e^{ix}dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{ix}}
Compute the integral \( \int_{\mathbb{R}} \frac{i \omega - \omega^2}{1+\omega^6} \, d\omega \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{3}}
Find all constants \( K > 0 \) such that for any positive measure space \((X, \Sigma, \mu)\) and any \(\mu\)-integrable function \(f: X \to \mathbb{R}\) satisfying \(\left|\int_E f\,d\mu\right| < K\) for all \(E \in \Sigma\), it follows that \(\|f\|_1 < 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Let \( X \) be an \( n \times (k+1) \) matrix such that \( X^TX \) is invertible. Determine the rank of the matrix \( I - X(X^TX)^{-1}X^T \), where \( I \) is the \( n \times n \) identity matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - k - 1}
Compute the limit \(\lim_{x \to +\infty}\frac{\ln x}{\int_0^x \frac{|\sin t|}{t}\,dt}.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Evaluate the expression \( \frac{\int_0^{\pi/2}\sin^{\sqrt{2}+1}(x) \, dx}{ \int_0^{\pi/2}\sin^{\sqrt{2}-1}(x) \, dx} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 - \sqrt{2}}
Let \( f\left( x\right)  = {x}^{2} - {2x} \) . A set of real numbers \( S \) is valid if it satisfies the following:
 \( \bullet \) If \( x \in  S \) , then \( f\left( x\right)  \in  S \) .
 - If \( x \in  S \) and \( \underset{k \cdot  {f}^{\prime }s}{\underbrace{f(f(\ldots f}}\left( x\right) \ldots )) = x \) for some integer \( k \) ,then \( f\left( x\right)  = x \) .
 Compute the number of 7-element valid sets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
0
}
B and P each deposit $100 in separate bank accounts. B's account has a nominal interest rate convertible semiannually, while P's account has a force of interest \( \delta \). After 7.25 years, each account has grown to $200. Calculate the difference \( i - \delta \), where \( i \) is the nominal interest rate for B's account.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0023}
Let $I$ be the incenter of the triangle $ABC$, et let $A',B',C'$ be the symmetric of $I$ with respect to the lines $BC,CA,AB$ respectively. It is known that $B$ belongs to the circumcircle of $A'B'C'$.
Find $\widehat {ABC}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
60
}
Calculate the integral \( \int_{[0,1]} x^2 \, d\mu_F \), where \( F \) is the Cantor function. Use the following properties of the Cantor function:

- \( F(1-x) = 1 - F(x) \)
- \( F\left(\frac{x}{3}\right) = \frac{F(x)}{2} \) for all \( x \in [0,1] \)
- \( F(0) = 0 \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{8}}
Given a positive integer $k$, there exists a positive integer $n$ such that the sum of the first $n$ positive integers can be obtained by appending $k$ digits to the right of $n$. Determine the remainder when $n$ is divided by 9.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the residue of the sum \(1^{965} + 2^{965} + \ldots + 2018^{965}\) modulo 2021. What is the value of \(x\) such that:
\[
1^{965} + 2^{965} + \ldots + 2018^{965} \equiv x \pmod{2021}?
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
Evaluate the expression \(2\sqrt{2^3\sqrt{2^4\sqrt{2^5\sqrt{2\ldots}}}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Compute the following limit involving the hypergeometric function \(_2F_1(1,b;b+1;z)\) around the branch point at \(z=1\):
\[ \lim_{\epsilon\rightarrow 0^+} \ _2F_1(1,b;b+1;1+\epsilon)-\ _2F_1(1,b;b+1;1-\epsilon) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-i \pi b}
Evaluate the integral \( \iiint_V xe^{-(x^2+y^2+z^2)^2}\,\mathrm{d} V \) using spherical coordinates, where \( V \) is the solid that lies between \( y=0 \) and the hemisphere \( x^2+y^2+z^2=4 \) in the right half-space \( (y>0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the process \(Y_t = Y_{t-1} + X_t - \phi^2 X_{t-2}\), where \(X_t = \phi X_{t-1} + Z_t + Z_{t-2}\) and \(Z_t\) is white noise with mean 0 and variance \(\sigma^2\), identify the parameters \((p,d,q)\) of the ARIMA process that describes \(Y_t\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1, 3)}
Compute the limit: \[ \lim_{n\to\infty}\left(\prod_{k=1}^{n} \binom{n}{k}\right)^\frac{1}{n} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the image of the Lemniscate defined by the polar equation \(r^2 = \cos(2\theta)\) under the inversion transformation \(f(z) = \frac{1}{z}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 - y^2 = 1}
Determine the minimum number of nails required to fix any convex polygon on a table, where nails can touch the boundary but not pass through the polygon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the conditional probability $P(X_2 > X_3 > X_4 > X_5 | X_1 < X_2)$ for independent and identically distributed continuous random variables $X_1, X_2, X_3, X_4, X_5$ with a continuous distribution function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{15}}
You have 100 boxes, and C of them contain a gift. You can open up to 16 boxes. What is the smallest number of C that will give you a probability greater than 0.5 of finding at least one gift? Assume that once you find a gift, you stop opening boxes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the power to which the root \( x_0 \) of the equation \( x^{11} + x^{7} + x^{3} = 1 \) must be raised to obtain the value \( x_0^{4} + x_0^{3} - 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the maximum power of 2 that can be expressed as a sum of terms, where each term is a power of 2 multiplied by a power of 10, using only the digits 0, 1, 2, 4, and 8.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2048}
Solve for \(x\) in the equation involving the Gaussian error function: \[ \frac{2}{\sqrt{\pi}}\int\limits_{0}^{x}e^{-t^2}dt = 0.7 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.733}
Given the joint density function of two random variables \(X\) and \(Y\):
\[
f_{X,Y}(x,y)=\begin{cases}0.0009x(1+y), & \text{if } (x,y)\in \Omega \\
0, & \text{else}\end{cases}
\]
and the marginal distributions:
\[
f_X(x)=0.0036(21x-2x^2-x\sqrt{x}), \quad \text{and} \quad f_Y(y)=0.0009(1+y)\frac{y^4}{512}
\]
where
\[
\Omega=\{(x,y):4\sqrt{x}<y \quad \text{and} \quad 0<y<12\}
\]
Calculate the conditional probability \(P(X<4|Y>4)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.42}
Two particles are positioned at opposite vertices of the same face of a cube. At any given time $t$, if the particles occupy the same edge, they remain there until time $t + 1$. Otherwise, one particle is randomly selected with equal probability and moves to one of its three neighboring vertices with equal probability. What is the expected value for the least amount of time it will take for the particles to be on the same edge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For varieties $X$ and $Y$ over an algebraically closed field and a morphism $f: X \rightarrow Y$, determine the minimum dimension of an irreducible component of the fiber $f^{-1}(y)$ for a closed point $y \in Y$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dim X - \dim Y}
Given a context-free grammar with $n$ non-terminals and a maximum production rule length of $k$, determine the maximum length of the shortest word that can be generated by a grammar with $n+1$ non-terminals and a maximum production rule length of $2k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2k)^n}
Evaluate the limit: \[ L = \lim_{x\to \infty} \frac{\int_{1}^{x} t^{t-1} ( t + t\ln (t) +1 ) dt}{x^x} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the remainder when \( \prod_{i=1}^{1903} (2^i + 5) \) is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{931}
Determine the radius of convergence for the power series \( f(x) = \sum_{n=1}^\infty \left(1+\frac{(-1)^n}{n}\right)^{n^2} \cdot \frac{(2x+1)^n}{n}.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2e}}
Charlie has a golden ticket that is a rectangle with side lengths in the ratio of the golden ratio \( \varphi = \frac{1+\sqrt{5}}{2} \). He folds the ticket so that two opposite corners coincide, creating a sharp crease. The area of the resulting shape can be expressed as \( a + b \varphi \). Find the value of \( \frac{b}{a} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( \phi \left( n\right) \) be the number of positive integers \( k < n \) which are relatively prime to \( n \) . For how many distinct values of \( n \) is \( \phi \left( n\right) \)
 equal to 12 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In an equilateral triangle with side length 1, 8 points are placed either on the boundary or inside the triangle. Determine the maximum possible value for the shortest distance between any two of these points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
What is the genus of the two-component unlink?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the values of \( \alpha \in \mathbb{R} \) for which the double integral \( \iint_{\mathbb{R}^2} \frac{\sin(x^2+y^2)}{(x^2+y^2+1)^\alpha} \, dx \, dy \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > 0}
Evaluate the limit: $$\lim_{n\to \infty} \frac{(n + 2){{\ln }^2}(n + 2) - (n+1){{\ln }^2}(n+1)}{\ln n}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the integral: $$\int_{-\infty}^{\infty} \left|2Af_0 \frac{\sin(2\pi f_0 t)}{2\pi f_0 t}\right|\,dt$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the function $f(x)$ such that its Maclaurin series has coefficients $f(0)=1$, $f'(0)=1$, $f''(0)=2$, $f^{(3)}(0)=3$, and so on. What is $f(x)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + x e^x}
Determine the minimum number of table arrangements required for 20 people, seated in groups of 4, so that everyone meets each other at least once.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given the group $G = \mathbb{Z}_6 \times \mathbb{Z}_3$, an element $a = (4, 1)$ in $G$, the subgroup $H = \langle a \rangle$, and the quotient group $G_1 = G/H$. Let $k$ denote the order of the coset of $(3, 1)$ in $G_1$, and let $\ell$ represent the order of the coset of $(2, 2)$ in $G_1$. Calculate $(k + \ell) \mod{5}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the largest integer \( m \) such that \( a^{12} \equiv 1 \pmod{m} \) for all integers \( a \) that are coprime to \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65520}
Let $H$ be a subgroup of a group $G$. Determine the stabilizer of the coset $aH$ for the action of $G$ on the set $G/H$ by left multiplication.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{aHa^{-1}}
Compute the Galois group of the splitting field of the polynomial \( t^4 - 3t^2 + 4 \) over \( \mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{V_4}
Determine the radius of convergence for the power series \( \sum_{k=0}^\infty 3^k \cdot z^{2^k} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the triple integral \( \iiint_W \sqrt{x^2 + y^2 + z^2} \, dx \, dy \, dz \) over the region \( W = \{ (x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 \leq x \} \) using an appropriate change of variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{10}}
Evaluate the double integral \( \iint_Q \frac{5y-x}{y^2(y-2)^2} \, dA \), where \( Q \) is the region enclosed by the quadrilateral with vertices \((2, 4)\), \((6, 3)\), \((8, 4)\), and \((4, 8)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{4}}
Determine whether the series \( \sum_{n=1}^\infty \left(\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}\right) x^n \) converges using the Cauchy Root Test.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|x| < 1}
Calculate the volume of the solid in $xyz$-space bounded by the surfaces \( z=\frac{1}{x^2+y^2+1} \) and \( z=\frac{1}{x^2+y^2+4} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi \ln 2}