inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the remainder when the smallest positive multiple of 15, consisting of exactly 3k digits with k 0's, k 3's, and k 8's, is divided by 11.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine all complex numbers $z \in \mathbb{C}$ for which the series \( \sum_{n=1}^\infty \frac{1}{n^3}(z+i)^{n^2} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \{ z \in \mathbb{C} \mid |z + i| \leq 1 \} }
In the Cartesian coordinate system, there are 50 different parabolas of the form \( y = ax^2 + bx + c \) and another 50 different parabolas of the form \( x = a_1 y^2 + b_1 y + c_1 \). Determine the maximum number of regions these 100 parabolas can divide the coordinate plane into.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15001}
There are 2022 distinct integer points on the plane. Let \( I \) be the number of pairs among these points with exactly1unit apart.
 Find the maximum possible value of \( I \) .
 (Note. An integer point is a point with integer coordinates.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3954
}
Find the area of the region enclosed by the curve parameterized by \( g(t) = (t^4 - t^2, t^6 - t^2) \), where \( 0 \leq t \leq 1 \), using Green's Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{60}}
Let $x=3^{a_1}5^{a_2}7^{a_3}$ and $y=3^{b_1}5^{b_2}7^{b_3}$ where $a_i,b_i$ are nonnegative integers such that $a_k + b_k \le 5$ for all $k$. Find the sum $$\sum \gcd(x,y),$$ where the sum is over all possible $x,y$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1477773}
Let \((x_n)\) be a sequence such that \(\lim\limits_{n\to\infty} x_n = \infty\) and \(\lim\limits_{n\to\infty} (x_{n+1} - x_n) = 0\). Define \(v_n = x_n - [x_n]\), where \([x_n]\) is the integer part of \(x_n\). Determine the set of adherence values of the sequence \((v_n)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 1]}
Find the sum of all integer values of \(a\) in the interval \([-10, 10]\) such that for each \(a\), the inequality \(5 \leq x \leq 10\) implies \(a x + 3a^{2} - 12a + 12 > a^{2} \sqrt{x-1}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-47}
Given the equations:
\[ 2\sin^2 A - 2\tan A \sin A + 2 - \sin^2 A - \cos^2 B = 0 \]
\[ 2\cos^2 B - 2\tan A \cos B + 2 - \sin^2 A - \cos^2 B = 0 \]
If \( \sin A \neq \cos B \), find the value of \( \sin^2 A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Let \( f(x) \) be a continuous function on \( \mathbf{R} \) such that \( f(0)=1 \) and \( f(x+y) \geqslant f(x) f(y) \) for all real numbers \( x \) and \( y \). Determine the function \( f(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{kx}}
Find the Jacobson radical of the ring \( R = \left\{ \frac{a}{b} : a, b \in \mathbb{Z}, b \neq 0 \text{ and } q \nmid b \right\} \), where \( q \) is a prime different from \( p \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{qR}
Given a frequency response $H(f)$, find the inverse Fourier transform of $|H(f)|^2$ in terms of a convolution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{h(t) * h^*(-t)}
Evaluate the limit: $$\lim_{n \to \infty} \left\{ \frac{n!}{\prod_{k=1}^{n} H_k} \right\}^{\frac{1}{n}}$$ where $$H_k = \sum_{i=1}^{k} \frac{1}{i}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the region in the complex plane where the series \( \sum_{n=0}^{\infty} \frac{z^2}{(1+z^2)^n} \) is uniformly convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{ z \in \mathbb{C} \mid |1 + z^2| > 1 \}}
You have two fair six-sided dice, one labeled $1$ to $6$, and for the other one, each face is labeled $1$, $2$, $3$, or $4$ (not necessarily all numbers are used). Let $p$ be the probability that when the two dice are rolled, the number on the special die is smaller than the number on the normal die. Given that $p = 1/2$, how many distinct combinations of $1$, $2$, $3$, $4$ can appear on the special die? The arrangement of the numbers on the die does not matter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine the degree of the splitting field of an irreducible Eisenstein polynomial $g$ with $p = 5$, given that one of its roots is $\sqrt{\frac{5}{2}+\frac{\sqrt{-5}}{2}}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Evaluate the double integral \( \iint_D \arctan e^{xy} \, dy \, dx \), where the region \( D \) is defined by \( \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \leq 4x\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi^2}
Find the largest real number $\alpha$ such that, for all non-negative real numbers $x$, $y$ and $z$, the following inequality holds:
\[
(x+y+z)^3 + \alpha (x^2z + y^2x + z^2y) \geq  \alpha (x^2y + y^2z + z^2x).
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$6\sqrt{3}$}
Determine the values of $x$ for which the series \( \sum_{n=2}^\infty \Bigg|\frac{x^n}{n(\log n)^2}\Bigg| \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 1]}
Determine the asymptotic behavior of the function \( f(x) = \sum_{i_{u-1}=1}^{x}\sum_{i_{u-2}=1}^{i_{u-1}} \cdots \sum_{i_2=1}^{i_3} \sum_{i_1=1}^{i_2}i_1 \) as \( x \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{x^u}{u!}}
In a convex quadrilateral \(ABCD\), diagonals \(AC\) and \(BD\) intersect at point \(P\). Given that \(PA = 1\), \(PB = 2\), \(PC = 6\), and \(PD = 3\), and that \(O\) is the circumcenter of \(\triangle PBC\). If \(OA\) is perpendicular to \(AD\), determine the circumradius of \(\triangle PBC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $X_1, X_2, \ldots, X_{10}$ be independent and identically distributed random variables, each uniformly distributed on $(-5,5)$. Determine the distribution of the random variable \( Y = -2\sum\limits_{i=1}^{10}\log\left(\frac{|X_i|}{5}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\chi^2(20)}
Evaluate the limit: $$ \lim_{n \to \infty} \frac{2^n + \left(1 + \dfrac{1}{n^2}\right)^{n^3} + \dfrac{4^n}{n^4}}{\dfrac{4^n}{n^4} + n^3 \cdot 3^n} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $A$ be a set with cardinality $a$. Determine the cardinality of the set of countably infinite subsets of $A$. Express your answer in terms of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^{\aleph_0}}
Let $ f(x)=\left(1+\frac{1}{x}\right)^{x}\ (x>0)$.

Find $ \lim\limits_{n\to\infty}\left\{f\left(\frac{1}{n}\right)f\left(\frac{2}{n}\right)f\left(\frac{3}{n}\right)\cdots\cdots f\left(\frac{n}{n}\right)\right\}^{\frac{1}{n}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \sqrt{e} \]}
Given elements $a, b, c$ in a unital non-commutative ring, where $c$ is an inverse of $1-ab$ such that $c(1-ab) = 1$, find an inverse for $1-ba$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + bca}
Let \( p \) be a prime such that \( p^n \equiv 5 \pmod{6} \) for each odd positive integer \( n \). Determine the value of \( p^n \) modulo 36, i.e., find \( p^n \equiv ? \pmod{36}. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Vasya has a triangle with vertices labeled 1, 2, and 3. Rotating the triangle 15 times clockwise around vertex 1 by the angle at this vertex returns it to its original position. Similarly, rotating the triangle 6 times clockwise around vertex 2 by the angle at this vertex also returns it to its original position. Determine the minimal number of times, n, that Vasya must rotate the triangle clockwise around vertex 3 by the angle at this vertex to return it to its original position.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the number of digits in the number \(2^{2^{22}}\) using number theory techniques, without employing logarithms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1262612}
Let \( {ABCD} \) be a parallelogram. Point \( P \) is selected in its interior such that the distance from \( P \) to \( {BC} \) is exactly6times
 the distance from \( P \) to \( {AD} \) , and \( \angle {APB} = \angle {CPD} = {90}^{ \circ  } \) . Given that \( {AP} = 2 \) and \( {CP} = 9 \) , the area of \( {ABCD} \) can be
 expressed as \( m\sqrt{n} \) where \( m \) and \( n \) are positive integers and \( n \) is not divisible by the square of any prime. Find \( m + n \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
Find the five-digit number \(\overline{abcde}\) such that for the two-digit numbers \(\overline{ab}, \overline{bc}, \overline{cd}, \overline{de}\), the equation \((\overline{ab} + \overline{bc})(\overline{bc} + \overline{cd})(\overline{cd} + \overline{de}) = 157605\) holds. Note that multi-digit numbers cannot start with zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12345}
Given a probability density function (PDF) with a minimum value of $243.483K$ and a maximum value of $308.05K$, calculate the probability that the temperature lies between $281K$ and $291K$. Assume the PDF is defined over the range from $243.483K$ to $308.05K$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.155}
Compute the integral \( \int_{2-i\infty}^{2+i\infty} \frac{1}{t} \, \mathrm{d}t \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i\pi}
Let $z \in \mathbb{C}$ be a root of the equation $z^2 - 2z + 3 = 0$. Find the modulus of the complex number $f(z) = z^{17} - z^{15} + 6z^{14} + 3z^2 - 5z + 9$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
$240$ students are participating a big performance show. They stand in a row and face to their coach. The coach askes them to count numbers from left to right, starting from $1$ . (Of course their counts be like $1,2,3,...$)The coach askes them to remember their number and do the following action:
First, if your number is divisible by $3$  then turn around.
Then, if your number is divisible by $5$  then turn around.
Finally, if your number is divisible by $7$  then turn around.
How many students are face to coach now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{136
}
Given that $a, b, c$ are the roots of the cubic equation $f(x) = x^3 + 3x^2 - 7x + 1$, calculate the value of $f'(a)f'(b)f'(c)$. Provide your answer in a simplified form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1300}
Find the limit of the sequence of polynomials \( \{f_n\} \) defined recursively by \( f_0(x) = 0 \) and \( f_{n+1}(x) = f_n(x) + \frac{x^2 - f_n^2(x)}{2} \) for \( n = 0, 1, 2, \ldots \), in the supremum norm on the interval \([-1, 1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|x|}
Determine the maximum number of solutions for the equation \( g(x) = e^x - (px^3 + qx^2 + rx + s) \), where \( p, q, r, \) and \( s \) are constants.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given a row of n lamps, where n is a fixed positive integer and n ≥ 1, some lamps are on and some are off. In one move, you can choose a positive integer i (1 ≤ i ≤ n) and switch the state of the first i lamps from the left. Determine the smallest number k such that you can turn all the lamps on in at most k moves, regardless of the initial configuration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Determine the set of points in the complex plane where the function \( f(z) = \sum_{n=0}^{\infty} \frac{z^{2n}}{(5n)^n} \) is holomorphic.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{C}}
Identify the group defined by the presentation \( \langle x, y: xy=yx, x^5=y^3 \rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Determine the asymptotic relationship between the functions \( f(n) = \frac{\log n}{\log\log n} \) and \( g(n) = \log(n - \log n) \). Specifically, find whether \( f(n) = K(g(n)) \) for some asymptotic notation \( K \), such as \( \Omega, \Theta, \) or \( O \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O}
Evaluate the sum \( \sum_{n=-\infty}^{\infty} \text{sinc}(\pi(2t-n)) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the measure of the set \( F = \left\{x\in [-1, 1]: \text{there exist finitely many }p,q \in\mathbb{N} \text{ such that }\left|x -\frac{p}{q}\right|> \frac{1}{q^{2}} \right\} \). Denote this measure as \( m(F) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the product \( \prod_{k=0}^{2^{1999}}\left(4\sin^2\left(\frac{k\pi}{2^{2000}}\right)-3\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the rate of convergence for the limit \( \lim_{k \rightarrow \infty} \left(1+\frac{2}{k}\right)^k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O\left(\frac{1}{k}\right)}
Let $n \geq 2$ be an integer and consider the vector space $V=F^{2n}$ with the standard basis $e_1,\ldots,e_{2n}$. The second exterior power $\Lambda^2(V)$ contains the element $$\omega=e_1 \wedge e_2 + e_3 \wedge e_4 + \cdots + e_{2n-1} \wedge e_{2n}.$$ Since $\Lambda^{2n}(V)$ is spanned by $x := e_1 \wedge \cdots \wedge e_{2n}$, the $n$-fold wedge product $\omega \wedge \cdots \wedge \omega$ is a scalar multiple of $x$. Compute this scalar.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
Calculate the result of the convolution \( \delta_{im} \epsilon_{ijk} \) using Einstein summation notation, considering that two indices coincide.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider a sphere of radius $r$ centered at the origin. Two great circles of the sphere lie on the xy-plane and xz-plane. A part of the sphere is removed such that the cross-section of the remaining part, perpendicular to the x-axis, is a square with vertices on the great circles. Compute the volume of the remaining part of the sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{3} r^3}
Compute the dimension of the space of cubic polynomials over $\mathbb{P}^5$ that vanish on $\mathbb{P}^2$, where $\mathbb{P}^2$ is embedded in $\mathbb{P}^5$ via the Veronese embedding.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Determine the number of distinct bracelets that can be made using six identical white beads, three identical blue beads, and five identical red beads, considering rotational and reflectional symmetries.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6036}
Identify the smallest number system that fulfills the requirements of geometry and applied mathematics, excluding any numbers that are only used for mathematical convenience.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Compute the product of the following integrals: \( \int_0^1 \frac{\mathrm{d}x}{\sqrt{1-x^4}} \) and \( \int_0^1 \frac{x^2\mathrm{d}x}{\sqrt{1-x^4}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Calculate the expected value of the length of the longest increasing subsequence of a random permutation of \(n\) elements, divided by \(\sqrt{n}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find a polynomial \( f \in \mathbb{Z}[x] \) that is irreducible over \( \mathbb{Q} \) but whose reduction modulo \( p \) is reducible for the primes \( p = 2, 3, \) and \( 5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^4 + 1}
In a game played within the unit disc $\mathcal{D}$, player $\mathcal{A}$ selects an initial point $p_0 \in \mathcal{D}$. At each step $n$, player $\mathcal{B}$ draws a line $\ell_n$ through $p_{n-1}$, and player $\mathcal{A}$ chooses a new point $p_n$ from the intersection of $\mathcal{D}$ and $\ell_n$. Player $\mathcal{B}$ wins if the sequence $(p_n)$ converges, while player $\mathcal{A}$ wins if it does not. Given that $\mathcal{B}$ has a winning strategy, determine the probability that $\mathcal{B}$ wins when both players make random choices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the asymptotic behavior of the function \(a(n) = \sum_{i=1}^n d(i)\), where \(d(n)\) is the number of divisors of the positive integer \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \log n + (2\gamma - 1)n}
Determine the minimum number of values needed to describe the orientation of a 3-dimensional rigid body in n-dimensional space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3n - 6}
Determine the limit \(\lim_{r\to \infty}\frac{f(r)}{\pi r^2}\), where \(f(r)\) represents the number of integral points inside a circle of radius \(r\) centered at the origin. Given that the formula for the number of lattice points inside the boundary of such a circle is \(f(r)=1+4\lfloor r\rfloor+4\sum_{i=1}^{\lfloor r\rfloor}\lfloor \sqrt{r^2-i^2}\rfloor\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the conditions on the real numbers \(a\) and \(b\) such that the sequence \(X_n = \mathrm{e}^{a S_n - b n^2}\), where \(S_n = \sum_{k=1}^n \xi_k\) and \(\xi_1, \xi_2, \ldots\) are iid standard normal random variables, converges to 0 in \(\mathcal{L}^r\) for \(r \geq 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b > 0}
Calculate the covariance \( \text{cov}(5W_7 + 6W_9, W_7) \) where \( W_t \) is a standard Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77}
Calculate the flux of the vector field \( F=(x^3, y^3, z^3) \) through the surface of the sphere defined by \( x^2+y^2+z^2=x \) using the divergence theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{5}}
A set of distinct positive integers has a sum of 1987. Determine the maximum possible value of the expression \(3n + k\), where \(n\) is the total number of integers in the set and \(k\) is the number of odd integers in the set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{221}
Let $ABC$ be a triangle with $AB=3$, $BC=4$, and $CA=5$. Points $A_1$, $B_1$, and $C_1$ are chosen on its incircle. Compute the maximum possible sum of the areas of triangles $A_1BC$, $AB_1C$, and $ABC_1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
12

}
Evaluate the integral \( \int_0^\infty \frac{t^2 - \sin^2 t}{t^4} \, dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
Given a Markov chain with state space \( \{0, 1, 2\} \), where state 0 represents making no money, state 1 represents making a small amount of money, and state 2 represents making a large amount of money, the transition matrix is:
\[
\begin{bmatrix} 
0.1 & 0.6 & 0.3 \\
0.4 & 0.4 & 0.2 \\
0.7 & 0.3 & 0 
\end{bmatrix}
\]
Calculate \( P(X_5 = 2 \mid X_3 = 0, X_1 = 1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.15}
Determine the number of non-isomorphic groups of order 42, given that there are 6 different homomorphisms from \( \mathbb{Z}_6 \) to \( \text{Aut}(\mathbb{Z}_7) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In square \(ABCD\), lines \(EF\) and \(GH\) divide the square into four rectangles. Line \(EF\) is parallel to \(AB\), and \(GH\) is parallel to \(BC\). Given that \(\angle BAF = 18^\circ\) and the area of rectangle \(PFCH\) is twice that of rectangle \(AGPE\), find the nearest integer to the value of \(\angle FAH\) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Determine the number of individual equations needed to solve the geodesic equation in three dimensions on a spherical surface with coordinates $(r, \theta, \phi)$. The geodesic equation in Einstein notation is given by: $$\frac{d^2u^k}{d\lambda ^2}+\Gamma^k_{ij}\frac{du^i}{d\lambda}\frac{du^j}{d\lambda}=0$$ How many equations are there to solve for the geodesic paths on this surface?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the function \( f(X)=\exp\left(\frac{-\gamma^2}{a^2X+b^2}\right) \) where \( X \sim \mathrm{Binomial}(n,p) \), use the central limit theorem to find an approximate expression for the value of \( \gamma^2 \) that maximizes the variance of \( f(X) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^2 np + b^2}
Let \(X_1, X_2, \dots, X_n\) be a random sample from a distribution with probability density function (p.d.f.): \[ f(x, \theta) = e^{-(x-\theta)}; \quad \theta < x < \infty; \quad -\infty < \theta < \infty. \] Find the sufficient statistic for the parameter \(\theta\) using the factorization theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X_{(1)}}
Let \( S = \{1, 2, 3, 4\} \). A sequence \( a_{1}, a_{2}, \ldots, a_{n} \) of \( n \) terms has the property that for any non-empty subset \( B \) of \( S \), there exists a sequence of \( |B| \) consecutive terms in the sequence that exactly forms the set \( B \). Determine the minimum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In a pyramid with a square base ABCD and apex S, the height of the pyramid is 8 units, and the angle between lines SA and SC is 48 degrees. Calculate the angle between the height SO (where O is the center of the square base) and the face SBC of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17.5^\circ}
Calculate
 \( {\oint }_{\left| z\right|  = 2}\frac{dz}{\sqrt{1 + {z}^{10}}} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
0}
When tossing 3 coins 400 times, you score a point for 3 heads, lose a point for 3 tails, and get no points for any other outcome. The total score \( S \) is the difference between the number of 'triple heads' \( X \) and 'triple tails' \( Y \), both following a binomial distribution with parameters \( n=400 \) and \( p=1/8 \). Given that \( E(S) = 0 \), find the variance \( \text{Var}(S) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Find a closed form for the infinite series:
\[ \sum_{i=0}^{\infty}\left[\frac{1}{i!}\left(\frac{e^2 -1}{2} \right)^i \prod_{j=0}^{i}(x-2j) \right]\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x e^{x - 2}}