inclusionAI/Ring-lite-rl-data huggingface dataset:
In triangle $ABC$, the vertices are located at the centroids of three houses, and the points are not collinear. Let $N = 2017$. Define the $A$-ntipodes as the points $A_1, A_2, \ldots, A_N$ on segment $BC$ such that $BA_1 = A_1A_2 = \cdots = A_{N-1}A_N = A_NC$. Similarly, define the $B$-ntipodes and $C$-ntipodes. A line $\ell_A$ through $A$ is called a qevian if it passes through an $A$-ntipode, and similarly for qevians through $B$ and $C$. Determine the number of ordered triples $(\ell_A, \ell_B, \ell_C)$ of concurrent qevians through $A$, $B$, and $C$, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6049}
Determine the number of moves required to achieve a reasonably uniform distribution of configurations for a 2x2x2 Rubik's cube.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Find the gradient of the function \( G(Y) = Y^{-3} \), where \( Y \) is an \( m \times m \) matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3 Y^{-4}}
The Great Kingdom is a rectangular island divided into smaller rectangular landscapes. Each landscape has a road running along one of its diagonals. These roads form a closed loop that does not intersect itself. If the Great Kingdom has an odd number of landscapes, what is the minimum number of landscapes it can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $A$ be the number of 2019-digit numbers composed of exactly two different digits. Determine the highest power of 3 that divides $A$. For example, the number $10\underbrace{1...1}_{2016}0$ is such a number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \), where \( S \) is the surface composed of the paraboloid \( y = x^2 + z^2 \) for \( y \in [0,1] \) and the disk \( x^2 + z^2 \leq 1 \) at \( y = 1 \), oriented with outward normal vectors. The vector field is given by \( \mathbf{F} = (0, y, -z) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the values of $v$ for which the rank of the matrix \( \begin{bmatrix} -\sin u + \sin (u+v) & \sin(u+v) \\ \cos u - \cos(u+v) & -\cos(u+v) \\ 1 & 0 \end{bmatrix} \) is equal to 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{v \in \mathbb{R}}
Evaluate \( A^2 \) if \( A = \lim_{n\to\infty} \left(\prod_{r=0}^n \binom{n}{r}\right)^\frac{1}{n(n+1)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
In \( \bigtriangleup {ABC} \) with \( {AB} = {10},{AC} = {13} \) ,and \( \measuredangle {ABC} = {30}^{ \circ  },M \) is the midpoint of \( \overline{BC} \) and the circle with diameter \( \overline{AM} \) meets
\( \overline{CB} \) and \( \overline{CA} \) again at \( D \) and \( E \) ,respectively. The area of \( \bigtriangleup {DEM} \) can be expressed as \( \frac{m}{n} \) for relatively prime positive
 integers \( m,n \) . Compute \( {100m} + n \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{103838}
Given \(z = e^{\frac{2\pi i}{101}}\) and \(w = e^{\frac{2\pi i}{10}}\), evaluate the following product modulo 101:
\[ A = \prod_{a=0}^{9}\prod_{b=0}^{100}\prod_{c=0}^{100}(w^a + z^b + z^c) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Evaluate the limit \( \lim_{(x, y) \to (0, 0)} (1 + x^2 + y^2)^{\frac{1}{x^2 + y^2 + xy^2}} \) using polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Given the companion matrix \(A\) of a monic polynomial \(f(x) = a_0 + a_1 x + \cdots + a_{n-1} x^{n-1} + x^n\), where \(A\) is defined as:

\[A=\begin{bmatrix} 0&\cdots&0&0&-a_{0}\\1&0&\cdots&0&-a_{1}\\ 0&1&\ddots&\vdots&-a_{2}\\ \vdots & \ddots& \ddots&0&\vdots \\ 0& \cdots&0&1&-a_{n-1}\end{bmatrix}\]

and the characteristic polynomial of \(A\) is \(\chi(\lambda)=(-1)^{n}f(\lambda)\). Determine the minimal polynomial of \(A\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x)}
Calculate the integral of the function \( \operatorname{sgn}(\sin(\pi/x)) \) over the interval \((0, 1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - 2 \ln 2}
Calculate the expected value of the integral of the inverse squared norm of a 3-dimensional Brownian motion from 0 to \( t \). Specifically, find \[ \int_0^t \mathbb{E}\left[||(B^1_s,B^2_s,B^3_s)||^{-2}\right]ds. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
How many distinct cards can be constructed by punching holes in two of the small triangles of an equilateral triangle divided into 9 congruent sub-triangles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the coefficient of \(x^{18}\) in the expansion of \((1+x^{3}+x^{5}+x^{7})^{100}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1254792000}
An integer $ n$ is called good if there are $ n \geq 3$ lattice points $ P_1, P_2, \ldots, P_n$ in the coordinate plane satisfying the following conditions: If line segment $ P_iP_j$ has a rational length, then there is $ P_k$ such that both line segments $ P_iP_k$ and $ P_jP_k$ have irrational lengths; and if line segment $ P_iP_j$ has an irrational length, then there is $ P_k$ such that both line segments $ P_iP_k$ and $ P_jP_k$ have rational lengths.

Determine the minimum good number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Consider the system\[x+y=z+u,\]\[2xy=zu.\]Find the greatest value of the real constant $m$ such that $m \le \frac{x}{y}$ for any positive integer solution $(x, y, z, u)$ of the system, with $x \ge y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$m = 3 + 2\sqrt{2}$}
Evaluate the integral \( \oint_{\gamma} \frac{z}{\cos z} \, dz \), where \( \gamma \) is the circle \( |z| = 5 \) oriented in the counterclockwise direction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi^2 i}
Determine the maximum number of "almost centers of symmetry" that a finite set of points on the plane can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine all values of $a$ for which the integral \( I = \int_{0}^{+\infty}\frac{dx}{x^2+x^a} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1)}
How many solutions does the equation system\[\dfrac{x-1}{xy-3}=\dfrac{3-x-y}{7-x^2-y^2} = \dfrac{y-2}{xy-4}\]have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2
}
Given the rotation matrix \(K=\ \begin{pmatrix} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix},\) determine the axis of rotation by inspection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}}
Consider a probability game where a target number \( x \) is chosen from the interval \((0, 1)\). A sequence of numbers \( c_1, c_2, \ldots \) is generated such that each \( c_i \) is picked uniformly at random from the interval \((0, c_{i-1})\), starting with \( c_1 \) from \((0, 1)\). The game stops when a number \( c_i \) is picked such that \( c_i < x \). What is the expected number of picks required to stop the game? Express your answer in terms of \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \ln x}
Determine the value of $y_0$ such that the solution to the initial value problem $y'+\frac{2x^2-4xy-y^2}{3x^2}=0, \ x>0, \ y(1)=y_0$ has a limit of $y'(x)$ as $x$ approaches 0 that is not equal to 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
p14. Andrew, Ben, and Clayton are discussing four different songs; for each song, each person either likes or dislikes that song, and each person likes at least one song and dislikes at least one song. As it turns out, Andrew and Ben don't like any of the same songs, but Clayton likes at least one song that Andrew likes and at least one song that Ben likes! How many possible ways could this have happened?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{204}
For which values of \(a \in (0,2)\) must there exist \(x, y \in [0,2]\) such that \(|y - x| = a\) and \(f(x) = f(y)\), given that \(f\) is continuous on \([0,2]\) and \(f(0) = f(2)\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,2)}
Calculate the integral \( \int_{F}^{} \! a \cdot dn \) for the vector field \( a = [-z^2 - 2z, -2xz + 2y^2, -2xz - 2z^2]^T \) over the area \( F \) on the cylinder \( x^2 + z^2 = 4 \) that lies above the ground plane \( z = 0 \), in front of the plane \( x = 0 \), and between the planes \( y = 0 \) and \( y = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{136}{3}}
Find the number of ordered pairs of integers (p,q) satisfying the equation $p^2-q^2+p+q=2014$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In the ring of $2 \times 2$ matrices over the finite field $\Bbb{Z}_p$, consider the set of matrices of the form
$$
\begin{bmatrix} a & b \\ -b & a \\ \end{bmatrix}
$$
where $a, b \in \Bbb{Z}_p$. Determine the order of the centralizer of this set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2}
Determine the maximum possible value of $k$ for which there exist distinct real numbers $x_1, x_2, \ldots, x_k$ greater than 1 such that for all $1 \leq i, j \leq k$, the equation \( x_i^{\lfloor x_j \rfloor} = x_j^{\lfloor x_i \rfloor} \) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Calculate the approximate proportion of positive integers less than 10,000 that are congruent to 11, 13, 17, or 19 modulo 30 and are prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.061}
Find the values of $x$ such that the recurrence relation defined by \( f_n(x) = f_1(f_{(n-1)}(x)) \) with \( f_1(x) = \frac{2}{3} - \frac{3}{3x+1} \) satisfies \( f_{1001}(x) = x - 3 \) for \( x \ge 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{3}}
Determine the supremum of the expression \( \frac{\cos(2 \pi d t)-\cos(2 \pi (d+1) t)-\cos(4 \pi  t)+\cos(2\pi t)}{4\sin^2( \pi t)} \) as a function of \(d\) and \(t\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the number of trees with vertex set \( \{v_1, v_2, v_3, v_4, v_5, v_6, v_7, v_8\} \) such that at least two vertices have a degree of 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80640}
Determine the values of $p$ for which the function $d(x, y) = |x - y|^p$ defines a metric on the real line $\mathbb{R}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1]}
A particle starts at the origin $(0,0)$ on the Cartesian coordinate plane. Every second, it moves with equal probability to one of the eight nearest lattice points. A square is drawn with vertices at $(2, 2)$, $(-2, 2)$, $(-2, -2)$, and $(2, -2)$. The particle will eventually hit the square for the first time at either a corner or an interior point on a side. If the probability of hitting a corner is $\frac{m}{n}$, where $m$ and $n$ are coprime, find $m+n$. \( \text{Express your answer as a single integer.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
Let $X$ and $Y$ be jointly continuous random variables with the joint density function given by: $$f_{X,Y}(x,y) = \begin{cases} \frac{1}{3}(x+y), & \text{for } x \in (0,1) \text{ and } y \in (0,2) \\ 0, & \text{otherwise}. \end{cases}$$ Compute the expected value $\mathbb{E}[\sin(\pi XY)]$. Express your answer as an integral if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{\pi}}
Solve the equation \( a^2 \cdot \arcsin \left(\frac{4}{a}\right) + 4 \cdot \sqrt{a^2 - 16} = 40 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.53}
Let \( x_1 = 1 \) and define the sequence \( x_{n+1} = x_n + \left\lfloor \frac{x_n}{n} \right\rfloor + 2 \) for \( n = 1, 2, 3, \ldots \). Determine the value of \( x_{1997} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23913}
Evaluate the limit: \[ \lim_{n\to\infty} n^2\int_0^1 \frac{dx}{(1+x^2)^n }. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let $S$ be a set with infinite cardinality $\kappa_1$. Determine the cardinality $k$ of the set of subsets of $S$ with cardinality $\kappa \le \kappa_0 < \kappa_1$. Assume $\kappa_0$ can be either finite or infinite. What is $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\kappa_1^{\kappa_0}}
Determine the order of the algebraic numbers $\sqrt[3]{3}(1+\zeta_4)$ and $\sqrt[3]{3}(1-\zeta_4)$ over the field $\mathbb{Q}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
(b) If we place all of these seven-digit numbers in increasing order, find the seven-digit
 number which appears in the \( {2022}^{\mathrm{{th}}} \) position.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3671542}
Calculate the expected time to failure for a 3-out-of-5 system where each component has an exponentially distributed survival time with a mean of 5 hours.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{47}{12}}
The graph of $r=2+\cos2\theta$ and its reflection over the line $y=x$ bound five regions in the plane. Find the area of the region containing the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9\pi}{2} - 8}
Determine the degree of the field extension $[F(x):F(x^n)]$, where $F(x)$ is the field of fractions of polynomials over a field $F$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Use the divergence theorem to evaluate the surface integral \( \iint_S (2x+2y+z^2) \, dS \), where \( S \) is the sphere defined by \( x^2+y^2+z^2 = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}\pi}
Find \( \lim_{n \rightarrow \infty} \left( \frac{1}{\sqrt{n}} \sum_{k=1}^n \left| \frac{a_k}{\sqrt{k}} \right| \right) \), where \(a_n\) converges to \(a \in \mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2|a|}
Let \( z = \frac{1}{2}(\sqrt{2} + i\sqrt{2}) \). Evaluate the sum \( \sum_{k = 0}^{13} \frac{1}{1 - ze^{k \cdot \frac{i\pi}{7}}} \) and express it in the form \( a - bi \). Find \( a + b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given a graph $T$ that is a spanning tree of a graph $G$ with a diameter of $x$, determine the diameter of the graph $$(T \cup T^c) + (T \cup T^c)$$ where $T^c$ is the complement of $T$, and $T$ and $T^c$ are different copies of $T$. What is the diameter of this graph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( f(y) = \sum_{n=0}^{\infty} y^{b_n} \) where \( b_n \) is a strictly increasing sequence of positive integers. Given that \( f(1/2) = 5 \), find \( \lim_{y \rightarrow 1^-} f(y) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the integral \( \int_{0}^{1} C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \, dy \), where \( C(\alpha) \) is the coefficient of \( x^{1992} \) in the power series expansion of \( (1 + x)^{\alpha} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1992}
What is the value of\[\tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {3\pi}{16} + \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {5\pi}{16}+\tan^2 \frac {3\pi}{16} \cdot \tan^2 \frac {7\pi}{16}+\tan^2 \frac {5\pi}{16} \cdot \tan^2 \frac {7\pi}{16}?\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
68
}
Determine the number of zeroes of the polynomial \( p(z) = z^8 - 20z^4 + 7z^3 + 1 \) that lie within the disk \( D(0,2) \), given that the polynomial has 4 real roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \( {ABCD} \) be a convex quadrilateral such that the triangle \( {ABD} \) is equilateral and the triangle \( {BCD} \) is isosceles, with
 \( \angle C = {90}^{o} \) . If \( E \) is the midpoint of the side \( {AD} \) , determine the measure of the angle \( \angle {CED} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
45°
}
Evaluate the line integral \( \int_C F \cdot ds \) for the vector field \( F(x, y, z) = \left(4x\left(1+\frac{z^2}{8}\right), \frac{\pi}{3}\cos\left(\frac{\pi}{3}y\right), \frac{x^2z}{2}-e^{\frac{z}{3}}\right) \) along the curve \( C \) parameterized by \( \gamma(t) = (e^{t(t-3)}, t, 2t) \). Consider using a theorem from multivariable calculus to simplify the computation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12 - 3e^2}
Find the minimum distance from the origin to the surface defined by the equations: \( xyza = 1 \) and \( x + y + z + a = 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral: $$ \int_{0}^{2 \pi} \sqrt{(-5\sin t-10\cos t\sin t)^2+(-5\sin^2t+5\cos^2t+5\cos t)^2} \, dt $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Determine the region of convergence in \(\mathbb{R}^2\) for the power series \[ \lim\limits_{n \to +\infty} \sum_{m , n = 0}^n \frac{m^3 u^m v^n}{n!}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1) \times \mathbb{R}}
The fraction\[\dfrac1{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},\]where $n$ is the length of the period of the repeating decimal expansion. What is the sum $b_0+b_1+\cdots+b_{n-1}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
883
}
Let $v \in \mathbb{C}^n$ be a complex unit norm vector, and let $G$ be an $m \times n$ matrix with i.i.d. $\mathcal{CN}(0,1)$ elements. Compute the expectation $\mathbb{E}\{G v v^H G^H\}$, where $v^H$ denotes the conjugate transpose of $v$. This expectation is a $m \times m$ matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I_m}
Determine the value of \( c \) that makes the Riemannian metric \( g = (1-x^2-y^2-z^2)^{2c}(dx \otimes dx + dy \otimes dy + dz \otimes dz) \) geodesically complete on the open unit ball \( B = \{ (x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2 <1 \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Evaluate the triple integral \( \iiint_R (r^2) \, \mathrm{d}V \), where \( R \) is the region in cylindrical coordinates bounded by \( r = \theta \), \( z = 2 - r \), and \( z = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{16}{15}}
Determine the smallest positive integer \( n \) such that in any two-coloring of the complete graph on \( n \) vertices \( K_n \), there always exist \( m \) lines of the same color, no two of which share a common endpoint.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3m - 1}
A group of $100$ students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students $A$ and $B$, student $A$ speaks some language that student $B$ does not speak, and student $B$ speaks some language that student $A$ does not speak. What is the least possible total number of languages spoken by all the students?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
9
}
Evaluate the limit: \[ \lim_{x\to\infty}\sum_{k=1}^{\infty} \frac{(-1)^{k+1} (2^k-1)x^k}{k k!} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
Let \( {ABC} \) a scalene triangle and \( {AD},{BE},{CF} \) your angle
 bisectors, with \( D \) in the segment \( {BC},E \) in the segment \( {AC} \)
 and \( F \) in the segment \( {AB} \) . If \( \angle {AFE} = \angle {ADC} \) .
 Determine \( \angle {BCA} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \angle BCA = 120^\circ \]}
Find all positive integers \( n \) such that \( 2n^3 + 5 \mid n^4 + n + 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Let \( K \) be a field, and consider the ideal \( I = (XY, (X-Y)Z) \subseteq K[X,Y,Z] \). Determine the radical of \( I \), denoted by \( \sqrt{I} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(XY, XZ, YZ)}
A collection of $n$ positive numbers, where repeats are allowed, adds to $500$. They can be split into $20$ groups each adding to $25$, and can also be split into $25$ groups each adding to $20$. (A group is allowed to contain any amount of integers, even just one integer.) What is the least possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
40
}
Consider an infinite sequence of independent, fair coin tosses. Let $H_i$ denote the event that the $i$th coin toss results in heads. Determine the probability \( \Bbb{P}\left(\bigcap_{i=1}^{\log_2(n)} H_{n+i} \text{ i.o.}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In the floating point number system \( F(10,10,-98,100) \), calculate the number of significant decimal digits lost when subtracting \( \sqrt{1001} \) from \( \sqrt{1002} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the integral \[ \int_{(-\infty,\infty)^n} \frac{\prod_{k=1}^n \sin(a_k x_k)}{\prod_{k=1}^n x_k} \frac{\sin(\sum_{k=1}^n a_k x_k)}{\sum_{k=1}^n a_k x_k} \, dx_1 \cdots dx_n \] where \( a_1, \cdots, a_n > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi^n}
Evaluate the integral: \[ \int_{-\infty}^{\infty} \frac{e^{ix}}{x} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i\pi}
Find the smallest multiple of 2019 that cannot be expressed as the sum of a perfect square and the product of two consecutive natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8076}
Determine the order of the group of homomorphisms from the dihedral group $D_{2n}$ to the multiplicative group of complex numbers $\mathbb{C}^*$, where $n$ is an odd integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $a,b,c$ be the roots of the polynomial $x^3 - 20x^2 + 22.$ Find\[\frac{bc}{a^2}+\frac{ac}{b^2}+\frac{ab}{c^2}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$-3$
}
Given that $xf(x)=e^x-1$, calculate $\lim\limits_{n \to \infty}{nf^{(n)}}(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^x}
Determine the minimum rank of a matrix in $M(n \times n)$, where each entry is a distinct prime number, for $n > 3$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the smallest integer \( n \) such that the symmetric group \( S_n \) contains an even permutation of order 312.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}