inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $A(z_1)$, $B(z_2)$, and $C(z_3)$ be the vertices of a triangle $ABC$ such that $|z_1|=|z_2|=|z_3|=1$. Given that $z_1+z_2\cos\alpha+z_3\sin\alpha=0$ for some $\alpha\in(0,\frac\pi2)$, and $z_2\overline{z_3}=\pm i$, find the length of the longest side of triangle $ABC$. Express your answer in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider a permutation of the numbers $1, 2, 3, \ldots, 2020$ denoted by $a_1, a_2, \ldots, a_{2020}$. Determine the number of permutations such that the absolute difference between each term and its position is constant, i.e., $|a_1 - 1| = |a_2 - 2| = \ldots = |a_{2020} - 2020|$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the curvature of the curve \( c(t) = \int^t_0 r(u) \, du \), where \( r(t) \) is a unit speed curve on a sphere of radius \( R \), in terms of \( R \). Do not use the formula for curvature of a non-unit speed curve.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{R^2}}
$ A$ is a set of points on the plane, $ L$ is a line on the same plane. If $ L$ passes through one of the points in $ A$, then we call that $ L$ passes through $ A$.
Find the biggest integer $ r$, so that if we divide all the rational points on the plane into $ 100$ pairwisely non-intersecting point sets with infinity elements with any method, then there is at least one line that passes through $ r$ sets of the $ 100$ point sets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
r=3
}
Let \( n \) denote the product of the first 2013 primes. Find the sum of all primes
 \( p \) with \( {20} \leq  p \leq  {150} \) such that
 (i) \( \frac{p + 1}{2} \) is even but is not a power of 2,and
 (ii) there exist pairwise distinct positive integers \( a,b,c \) for which
 \( {a}^{n}\left( {a - b}\right) \left( {a - c}\right)  + {b}^{n}\left( {b - c}\right) \left( {b - a}\right)  + {c}^{n}\left( {c - a}\right) \left( {c - b}\right) \)
 is divisible by \( p \) but not \( {p}^{2} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
431
}
Evaluate the line integral \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) where \( \mathbf{F}(x,y,z) = -x^{2}\mathbf{i} + xz\mathbf{j} + yx\mathbf{k} \) and \( C \) is the boundary of the surface \( S \) given by the graph of \( f(x,y) = 4x - 8y + 30 \) over the rectangle \( R = \{ (x,y) \mid -2 < x < 3, 0 < y < 2 \} \), oriented in the counterclockwise direction when viewed from above.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
Determine the minimal possible order of a group $G$ that contains a subset $A \subset G$ with $|A| = 2n$, such that for every $a \in A$, there exists a unique $b \in A$ with $[a, b] \neq e$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6^n}
John folds a rope in half five times then cuts the folded rope with four knife cuts, leaving five stacks of rope segments. How many pieces of rope does he now have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
Determine the angle between the integral curves of the differential equations \( x' = x - t^2 \) and \( x' = 2x + t^2 \) at the point \( M = (1, -2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \( a,b,c,p,q,r > 0 \) such that(a,b,c)is a geometric progression and(p,q,r)is an arithmetic progression. If
 \( {a}^{p}{b}^{q}{c}^{r} = 6\; \) and \( \;{a}^{q}{b}^{r}{c}^{p} = {29} \)
 then compute \( \left\lfloor  {{a}^{r}{b}^{p}{c}^{q}}\right\rfloor \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
29
}
Let $n > 1$ be an odd number. Let $A$ be an $n \times n$ matrix defined as follows:
\[
a_{i, j} = \begin{cases} 1, & \text{for}\ i - j  \equiv \pm 2 \pmod n\\ 2, & \text{for}\ i = j\\ 0, & \text{otherwise}.\end{cases}
\]
Calculate the determinant of matrix $A$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the equation of the circle that is transformed into a straight line by the function \( f(z) = \frac{z-1}{iz+i} \) and passes through the point \( z = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|z| = 1}
In a city, 100 graduates applied to 5 different universities. Each university was unable to reach exactly half of its applicants during the first and second waves. Additionally, representatives from at least three universities were unable to reach the graduates. What is the maximum number of graduates who could have been of interest to the military recruitment office?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Compute the Laplacian of the scalar field \( \dfrac{(\mathbf{\mu} \cdot \mathbf{r})}{r^3} \), where \( r = \Vert \mathbf{r} \Vert \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the value of the Ramsey number $R(K_3, C_5)$. Use the hint that $R(3,4) = 9$ to assist in your calculation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate \( \lim_{n\to\infty} I_n \), where \( I_n = \int\int_{r\leq 1} \frac{n}{r\cos \pi r+n^2r^3} \ dxdy \) and \( r = \sqrt{x^2+y^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi^2}
Evaluate the limit of the integral \(\lim_{R\to\infty}\int_C \frac{2z^2-5}{(z^2+1)(z^2+4)}\,dz\) where \(C\) is the upper half-circle \(|z|=R\), \(\operatorname{Im} z>0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of integers  $2 \le n \le 2016$  such that  $n^n-1$  is divisible by  $2$ ,  $3$ ,  $5$ ,  $7$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let $S$ be the vector space of real $n \times n$ skew-symmetric matrices. For a nonsingular matrix $A$, compute the determinant of the linear map $T_A : S \rightarrow S$ where $T_A(X) = AXA^{T}$. What is the determinant of this map?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\det A)^{n-1}}
Evaluate the limit: $$\lim_{r \to \infty} \frac{\int_0^{\pi/2}\left(\frac{x}{\pi/2}\right)^{r-1}\cos x\,dx}{\int_0^{\pi/2}\left(\frac{x}{\pi/2}\right)^r\cos{x}\,dx}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the remainder when \(3^{3^{3^{\cdots}}}\) (with 2020 copies of 3) is divided by 46.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
Determine the supremum of the ratio between the area of a centrally-symmetric convex curve $B$ and the area of an arbitrary planar convex set $A$, where $B$ is the smallest such curve that circumscribes $A$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $ABC$ be an equilateral triangle, and let $D$ be a point inside $ABC$. Let $E$ be a point such that $ADE$ is an equilateral triangle and suppose that segments $DE$ and $AB$ intersect at point $F$. Given that $\angle CAD = 15^o$, compute $\angle DFB$ in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
The focus of a vertically oriented, rotational paraboloid-shaped tall vessel is at a distance of 0.05 meters above the vertex. If a small amount of water is poured into the vessel, what angular velocity $\omega$ is needed to rotate the vessel around its axis so that the water overflows from the top of the vessel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9.9}
$设F为椭圆C：x^2/2+y^2=1的右焦点，过点F且与x轴不重合的直线l交椭圆C于A，B两点。$
$在x轴上是否存在异于F的定点Q，使 \frac{k_{QA}}{k_{QB}}为定值(其中k_{QA},k_{QB}分别为直线QA,QB的斜率)？若存在，求出Q的坐标；若不存在，请说明理由.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(2,0)$}
Compute the derivative of \( \text{tr}(AXB) \) with respect to the matrix \( X \), where \( A, B \in \mathbb{R}^{2 \times 2} \) and \( X \in \mathbb{R}^{2 \times 2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A^\top B^\top}
Find a closed formula for the infinite series \(\sum_{n=1}^{\infty}\frac{n^4}{4^n}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{380}{81}}
In a $2021 \times 2021$ square table, each cell is colored in one of $n$ colors. It is known that for any four cells of the same color located in a single column, there are no cells of the same color to the right of the top cell and to the left of the bottom cell. Determine the minimum value of $n$ for which this coloring is possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{674}
Find the area of the cone defined by the equation \(z=\sqrt{2x^2+2y^2}\) that is inscribed in the sphere \(x^2+y^2+z^2=12^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48\sqrt{3}\pi}
Find the character of the standard representation of the symmetric group $S_3$ on $\mathbb{R}^3$. Express your answer in terms of the conjugacy classes of $S_3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 0, -1)}
Determine the maximum number of points $P$ in a plane such that all triangles formed by any three points in $P$ are not obtuse. Consider degenerate triangles as well.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Compute the valuation \( v_{\pi_L}(1-\zeta^{a-1}) \) in the cyclotomic field \( L=\mathbb{Q}[\zeta] \), where \( \pi_L=1-\zeta \) is the ideal above \( p \) and \( (a,p)=1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the remainder when the integer $a$, which satisfies the equation \( \frac{1}{1}+\frac{1}{2}+\ldots+\frac{1}{23}=\frac{a}{23!} \), is divided by 13.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Calculate the Lie derivative of the volume form \(\Omega = dx\wedge dy\wedge dz\) on a 3-dimensional orientable manifold along the vector field \( X = (x+y)\partial_x + (x-y)\partial_y+ (x^2+y^2)\partial_{z} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the greatest integer  $A$  for which in any permutation of the numbers  $1, 2, \ldots , 100$  there exist ten consecutive numbers whose sum is at least  $A$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{505}
Calculate the surface area of the portion of the cylinder defined by \(x^2 + y^2 = 2ay\) that is contained within the sphere \(x^2 + y^2 + z^2 = 4a^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16a^2}
Calculate the average squared distance between two points chosen randomly on the disk \( D = \{(x,y) \in \mathbb{R}^2 \mid x^2 + y^2 \leq 1\} \). Use the integral formula: \[ \frac{1}{m(D)^2} \int \int_D \int \int_D ((x_1 - x_2)^2 + (y_1 - y_2)^2) \, dx_1 \, dy_1 \, dx_2 \, dy_2 \] where \( m(D) \) is the Lebesgue measure of \( D \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the line integral \( \oint_C (4x \, dx + 9y \, dy + 3(x^2 + y^2) \, dz) \) where \( C \) is the boundary of the surface \( z = 4 - x^2 - y^2 \) in the first octant (\( x, y, z \ge 0 \)), oriented counterclockwise as viewed from above, using Stokes' Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A sequence $\{a_n\}$ is defined by the recurrence relation $a_n = a_{n-1} + 2a_{n-2} - a_{n-3}$ with initial conditions $a_1 = a_2 = \frac{a_3}{3} = 1$. Find the value of the infinite series $$a_1 + \frac{a_2}{2} + \frac{a_3}{2^2} + \cdots$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the smallest integer \( n \) such that the Lagrange interpolation error \(|f(x) - Q_n(x)|\) for the function \( f(x) = e^x \) on the interval \([-4, 0]\) using equally spaced nodes is less than 0.1. Assume \( M_{n+1} = 1 \) and use the error bound formula \(|f(x) - P_n(x)| \leq \frac{M_{n+1}}{(n+1)!}|(x - x_0)\cdot...\cdot(x - x_n)|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $T$ be a nonzero bounded linear operator in $B(H)$, where $H$ is an infinite dimensional Hilbert space, and let $\|T\|$ be the norm of $T$. Compute the norm $\|I-T\|$, where $I$ is the identity operator.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \|T\|}
Evaluate the sum \( \sum_{k=2}^{\infty}\log{(1-1/k^2)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\ln 2}
A pyramid with a square base, and all its edges of length $2$, is joined to a regular tetrahedron, whose edges are also of length $2$, by gluing together two of the triangular faces. Find the sum of the lengths of the edges of the resulting solid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
22
}
Let  $r$  and  $s$  be positive real numbers, and let  $A=(0,0)$ ,  $B=(1,0)$ ,  $C=(r,s)$ , and  $D=(r+1,s)$  be points on the coordinate plane. Find a point  $P$ , whose coordinates are expressed in terms of  $r$  and  $s$ , with the following property: if  $E$  is any point on the interior of line segment  $AB$ , and  $F$  is the unique point other than  $E$  that lies on the circumcircles of triangles  $BCE$  and  $ADE$ , then  $P$  lies on line  $\overleftrightarrow{EF}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \frac{3r+1}{2}, \frac{s}{2} \right)}
Find the smallest positive integer $n$ such that for every integer $m$,
with $0<m<1993$, there exists an integer $k$ for which
$$\frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3987}
Determine all real numbers $x$ such that the series \( S = \sum_{n=1}^{\infty} \frac{1}{n^2+n} \cdot \tan^2\left(\frac{1}{\sqrt{n}}\right) \cdot n^x \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 2)}
Given the function $f(x)=x^{3}+x$, for any $m\in[-2,2]$, the inequality $f(mx-2)+f(x) < 0$ always holds true. Determine the range of values for $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2, \dfrac{2}{3}}
The diagram shows a square PQRS with side length 1. W is the center of the square, and U is the midpoint of RS. Line segments TW, UW, and VW split the square into three regions of equal area. Find the length of SV. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{6}}
6. [4 points] Two parallel lines $\ell_{1}$ and $\ell_{2}$ touch the circle $\omega_{1}$ with center $O_{1}$ at points $A$ and $B$ respectively. The circle $\omega_{2}$ with center $O_{2}$ touches the line $\ell_{1}$ at point $D$, intersects the line $\ell_{2}$ at points $B$ and $E$, and intersects the circle $\omega_{1}$ again at point $C$ (with point $O_{2}$ lying between the lines $\ell_{1}$ and $\ell_{2}$). It is known that the ratio of the area of quadrilateral $B O_{1} C O_{2}$ to the area of triangle $O_{2} B E$ is $\frac{6}{5}$. Find the ratio of the radii of the circles $\omega_{2}$ and $\omega_{1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{6}}
Determine the number of $p$-element subsets $A$ of the set $\{1, 2, 3, \ldots, p\}$ such that the sum of the elements in $A$ is divisible by $p$, where $p$ is an odd prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
在7×7的正方形方格纸中，选择k个小方格的中心，使得它们中任意四个都不构成为一个矩形的顶点（矩形的边与原来正方形的边平行），求k的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Find the smallest real number $a$ such that $e^x > x^5$ for all $x > a$. Provide your answer as a single value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.713}
Evaluate the integral \( \int_{|z|=2} \frac{\operatorname{Im} z}{z-\alpha} \, dz \) where \( \alpha \) is a complex number satisfying \( 0<|\alpha|<1 \), and \( \operatorname{Im} z \) denotes the imaginary part of \( z \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \alpha}
How many permutations of the word "arrangement" have exactly two of the following consecutive letter pairs: aa, ee, rr, nn?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{332640}
Evaluate the integral \( \int_{0}^{\infty} e^{-x} f(x) \, dx \), where the function \( f(x) \) is defined by the series \( f(x) = \sum_{n=0}^{\infty} (-1)^n \frac{1}{(2n!)^2} (2015x)^{2n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\cos(2015)}
Consider the fractional order algebraic equation \( x^{\pi} - 1 = 0 \). Let \( U \) be the set of unit magnitude complex numbers defined as \( U = \{ x \mid x \in \mathbb{C}, |x|=1, x^{\pi} \neq 1 \} \). What percentage of the unit circle does the set \( U \) constitute?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Two rectangles are drawn on a sheet of graph paper. The first rectangle has a vertical side shorter than its horizontal side, while the second rectangle has a vertical side longer than its horizontal side. Each rectangle covers more than 2010 but less than 2020 cells. Determine the maximum possible area of their intersection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1764}
Evaluate the limit: $$\lim_{x \to 0}\frac{1-\frac{1}{3} x^3 - \cos\left(\frac{x}{1-x^3}\right)}{x^6}$$ without using L'Hopital's rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Compute the line integral \(\int_C (y^2z^3\,dx + 2xyz^3\,dy + 3xy^2z^2\,dz)\) where \(C\) is the straight line path from \(a = (1,1,1)\) to \(b = (2,2,2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
Evaluate the integral \( \int_1^\infty \frac{1-(-1)^{\lfloor u \rfloor}}{2u^2} \, \mathrm{d}u \) using Abel's Summation formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
Given two ElGamal signatures over the finite field $\mathbb{F}_{2027}$ with the cyclic group $\mathbb{G}$ of order 1013, the signatures are for hash values $h(m_1) = 345$ and $h(m_2) = 567$, given by $(r_1, s_1) = (365, 448)$ and $(r_2, s_2) = (365, 969)$. Compute Alice's secret key $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
Determine the number of independent components of the Riemann curvature tensor $R_{iklm}$ in a 4-dimensional spacetime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Consider a projection valued measure $E$ from the complex plane into the projections on a Hilbert space $\mathcal{H}$, with compact support $K \subseteq \mathbb{C}$. Determine the spectrum of the normal operator $\int_K z \, dE(z)$ in $\mathcal{B}(\mathcal{H})$. \( \mathcal{B}(\mathcal{H}) \) denotes the bounded linear operators on $\mathcal{H}$. \( K \) is the compact support of the measure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{K}
Given the transfer function \( G(s) = \frac{1}{(s+7)(s+12)} \) and the input \( x(t) = \left(1 + \frac{1}{t}\right)^t \), calculate the limit of the output \( y(t) \) as \( t \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{e}{84}}
Find the unique polynomial \( r(x) \) such that for all \( p(x) \in \mathbb{P}^2 \), the functional \( f(p(x)) = p'(-15) + 8p(-1) \) can be expressed as \( \langle p, r \rangle = \int_0^1 p(x)r(x)dx \). Here, \( \mathbb{P}^2 \) is the space of polynomials of degree less than two.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-132x + 74}
Let $(B_t)_{t \ge 0}$ be a standard one-dimensional Brownian motion, independent of $W_1$. Calculate the probability that $W_1 + B_t > 0$ for all $t \in (0, 1)$. Provide your answer as a probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Evaluate the integral \( \int_{-\infty}^\infty \frac{\sin(xt)}{1+t^2} \, dt \) using the residue theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A piece of paper in the shape of an isosceles triangle is folded such that the vertex $A$ of its base is brought to the midpoint of side $BC$. As a result, one endpoint of the crease (folding line) divides the side $AC$ in a ratio of $p: q$. Determine the ratio in which the other endpoint of the crease divides the base $AB$. Provide your answer in the form of a ratio.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p : (2p - q)}
Let $f:[0,1]\rightarrow [0,1]$ be a continuous function such that $f(0)=f(1)$. Define the set \( A = \{ (t,s) \in [0,1] \times [0,1] \mid t \neq s \text{ and } f(t)=f(s) \} \). Determine the number of elements in the set $A$. \( \text{(Express your answer as a number or state if it is infinite.)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine all real numbers \( \alpha \) such that the set \( \{(x,y)\in\mathbb{R}^2: \sin(x+y)=\alpha y\} \) represents the graph of a differentiable function \( g:\mathbb{R}\rightarrow\mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -1) \cup (1, \infty)}
Calculate the product: \[ \prod_{k=0}^{2001} \left( 1 + \left( \frac{1+i}{2} \right)^{2^k} \right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1 + i)\left(1 - 2^{-2^{2001}}\right)}
During the softball season, Judy had 35 hits. Among her hits were 1 home
 run, 1 triple and 5 doubles. The rest of her hits were single. What percent of
 her hits were single?
 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
80%
}
In preparation for the World Cup, the coach of country $F$ evaluates players $A_{1}, A_{2}, \ldots, A_{7}$ in 3 training matches, each lasting 90 minutes. Only one player is on the field at any time. Players $A_{1}, A_{2}, A_{3},$ and $A_{4}$ each spend a total time on the field that is divisible by 7 minutes. Players $A_{5}, A_{6},$ and $A_{7}$ each spend a total time on the field that is divisible by 13 minutes. With unlimited substitutions allowed, how many different scenarios are possible for the total time each player spends on the field?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{208806}
What is the best known bound on the index of an abelian subgroup in a discrete subgroup of the Euclidean isometry group \( \isom(\mathbb{R}^n) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n n!}
Given that $a$ and $b$ are positive integers where $b$ is a multiple of $a$, and \[ \log\left(\left(\frac{b}{a}\right)^{\frac{b}{3}}\right) + \log\left(\left(\sqrt[3]{\frac{a}{b}}\right)^{9a}\right) = 1, \] find the value of $a^2 + b^2$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{909}
Find \( \lim_{x \to 0} \frac{e^{\arcsin x} - e^{\sin x}}{e^{\arctan x} - e^{\tan x}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}