inclusionAI/Ring-lite-rl-data huggingface dataset:
Evaluate the contour integral \( \oint_{|z|=1} z(z+1) \cosh\left(\frac{1}{z}\right) \, dz \) where the contour is the unit circle \(|z|=1\) oriented counterclockwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi i}
Determine the maximum possible rank of a set such that the ranks of all its finite subsets are bounded by a given ordinal \( \beta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\beta}
In the 22nd century (the years 2101 through 2200, inclusive), find the expected number of years that have either a record low or a record high rainfall, assuming that annual rainfalls are i.i.d. draws from a continuous distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2H_{100} - 1}
Determine the asymptotic behavior of the expression \( n^3 \left ( 1 - \left ( \frac{c \log n}{n} \right )^3 \right )^{n^3} \) as \( n \rightarrow \infty \), where \( c > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine whether the function \( \frac{1}{|x|^\alpha} \) is integrable at infinity in dimensions \( d \geq 2 \) for any \( \alpha \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > d}
Evaluate the integral: $$\int_0^1 A\sin(L+r\cos(2\pi n x +\phi))\cos(2\pi nx + \phi)\ \text{d}x$$ where $n$ is a natural number, and $A$, $L$, $r$, and $\phi$ are real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A J_1(r) \cos L}
Find the derivative of the trace of the product of three matrices, \( \text{tr}(ABC) \), with respect to the matrix \( B \), where \( A \) is an \( m \times n \) matrix, \( B \) is an \( n \times p \) matrix, and \( C \) is a \( p \times q \) matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A^\top C^\top}
Given the force of interest \( \delta_t = \frac{2t}{10+t^2} \) for \( t \geq 0 \), calculate \( \ddot{s}_{\bar{2}|} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{147}{55}}
Determine a generator of the multiplicative group \( \mathbb{Z}^{*}_{242} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine the number of spheres that are tangent to all the faces of a given triangular pyramid $T$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $\mathbb{k}$ be a field, and consider the vector space $\mathbb{k}_{10}[x]$ of polynomials in $x$ with degree at most 10. Define the shift operator $T: \mathbb{k}_{10}[x] \to \mathbb{k}_{10}[x]$ such that $T(f(x)) = f(x+1)$. Determine the minimal polynomial $\mu(t)$ of $T$. \( \boxed{\mu(t)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(t - 1)^{11}}
Find the smallest integer \(x\) such that the absolute difference between \(\pi(x)\) and \(\operatorname{li}(x) - \frac{1}{2}\operatorname{li}(\sqrt{x})\) is greater than the absolute difference between \(\pi(x)\) and \(\operatorname{li}(x)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of elements in the conjugacy class of the permutation $(123)(456)$ in the alternating group $A_7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{280}
Construct a polynomial over the finite field \( \mathbb{Z}_p \) with a degree less than \( p^3(p-1) \) that evaluates to 0 at the origin and 1 at all other points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - (1 - x^{p-1})(1 - y^{p-1})(1 - z^{p-1})}
A point $P$ lies inside equilateral triangle $ABC$. Let $A'$, $B'$, $C'$ be the feet of the perpendiculars from $P$ to $BC, AC, AB$, respectively. Suppose that $PA = 13$, $PB = 14$, and $PC = 15$. Find the area of $A'B'C'$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
Given the function \( f(x) = |1 - 2x| \) for \( x \in [0, 1] \), determine the number of solutions to the equation \( f(f(f(x))) = \frac{1}{2} x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the constant $c_0$ in the power series representation of the shifted exponential function $y = e^{-a(gx-b)^{2}}$. The power series is given by:

$$e^{-a(gx-b)^{2}} = c_{0}\sum_{k=1}^{\infty}\frac{(-2ag)^{k}}{(2k)!!}(gx-b)^{k}$$

where the coefficients $c_{2k}$ are defined by the recurrence relation $c_{2k} = \frac{(-2ag)^{k}}{(2k)!!}c_{0}$ for $k = 1, 2, 3, \dots$. What is the appropriate value for $c_0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the periodicity of the entire function $g(z)$, given that $g(iy)$ and $g(2+iy)$ are real for all $y \in \mathbb{R}$. Provide your answer as the period of the function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the value of \( t \) such that the test \( \delta = \chi[\bar{X}< a - t\sqrt{\frac{\sigma^2}{n}}] + \chi[\bar{X}> b + t\sqrt{\frac{\sigma^2}{n}}] \) for testing \( H_0: \theta \in [a, b] \) versus \( H_1: \theta \not\in [a, b] \) has a size of 0.05.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.96}
Find the isomorphism group of the directed graph that becomes isomorphic to the complete bipartite graph $K_{3,3}$ when all its edges are made undirected.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_3 \times S_3}
Determine the convergence of the series \( \sum\limits_{n=1}^{\infty} \frac{2n+1}{3n^2+1} x^n \) in terms of \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 1)}
Let \( \xi = \cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5} \) be a complex fifth root of unity. Define \( a = 20\xi^2 + 13 \xi \), \( b = 20\xi^4 + 13\xi^2 \), \( c = 20\xi^3 + 13\xi^4 \), and \( d = 20\xi + 13\xi^3 \). Calculate \( a^3 + b^3 + c^3 + d^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42063}
Given the function $f: \mathbb{N}^+ \rightarrow \mathbb{N}$ defined by the rules $f(2^n) = n^2$ and $f(3n) = f(2n) + 5$, find the value of $f(3^{100})$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10500}
Given the equations:
\[ \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} = 9, \]
\[ \frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b} = 32, \]
\[ \frac{a^3}{b+c} + \frac{b^3}{c+a} + \frac{c^3}{a+b} = 122, \]
find the value of \(abc\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Find the sum of the series \( \sum_{k=0}^{n/2} C(n-k, k) \), where \( C(n, k) \) denotes the number of combinations of \( n \) items taken \( k \) at a time.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F_{n+1}}
Determine the rank of the first homology group of the double branched cover of a knot in $S^3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the metric $d(A,B)$ for the sets $A = \{ x \mid x^2 \leq x + 1 \}$ and $B = [-1, 1]$ in the measure space $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \lambda)$. The metric should be a real-valued non-negative function, symmetric, and satisfy the triangle inequality. What is the value of $d(A,B)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the area of the sphere defined by \(x^2 + y^2 + z^2 = a^2\) that lies outside the regions defined by the cylinders \(x^2 + y^2 - ax = 0\) and \(x^2 + y^2 + ax = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8a^2}
Given a polynomial $f$ of degree $1007$ such that $f(k) = 2^k$ for $0 \le k \le 1007$, determine the value of $f(2015)$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2014}}
Compute the expected value \( E[e^{X^2Y}] \) where \( X \sim \text{Exp}(2) \) and \( Y \sim \text{Uniform}(-1, 1) \) are independent random variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given a linear transformation \( T: \mathbb{R}^2 \to \mathbb{R}_2[x] \) defined by \( T(1,0) = 1+x \) and \( T(1,1) = 1-x \), determine \( T^{-1}(2x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, -1)}
In how many distinct ways can 5 identical red beads, 3 identical green beads, and 2 identical blue beads be arranged in a circular necklace?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}
Find the radius of the circle formed by the intersection of the plane $x + 2y - z = 4$ and the sphere $x^2 + y^2 + z^2 - x + z - 2 = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the expression \( \frac{J_{6,3}}{J_{5,2}} \), where \( J_{m, n} = \int_{0}^{\pi/2} \cos^m x \cos nx \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Evaluate the improper integral expression: \[ \int_{-\infty}^{e} \ln x \,dx - \int_{-\infty}^{e} \frac{x}{e} \, dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{e}{2}}
Cindy cuts regular hexagon $ABCDEF$ out of a sheet of paper. She folds $B$ over $AC$, resulting in a pentagon. Then, she folds $A$ over $CF$, resulting in a quadrilateral. The area of $ABCDEF$ is $k$ times the area of the resulting folded shape. Find $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2}
Jefferson Middle School has the same number of boys and girls. Three-fourths of the girls and two-thirds of the boys went on a field trip. What fraction of the students were girls?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\(\frac{9}{17}\)
}
Evaluate the integral: \[ \int_{0}^{\infty}\text{sech}^2(x+\tan(x))\,dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given real numbers \(a\), \(b\), and \(c\) such that:
\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 4 \left( \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a} \right) = \frac{c}{a+b} + \frac{a}{b+c} + \frac{b}{c+a} = 4 \]
Determine the value of \(abc\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{49}{23}}
Evaluate the improper integral: $$I=\int_{-1}^1 \left(\frac{1}{x}+1-\frac{\sqrt{1-x^2}}{x}\right )\arctan\left(\frac{2}{x^2}\right)dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \arctan 2 - \ln 5}
Calculate \( \lim_{x \rightarrow 0}{\frac{\int_{0}^{x} \frac{\sin(t)}{t}-\tan(x)}{2x(1-\cos(x))}} \) using Maclaurin polynomials.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{7}{18}}
Given an ellipse with a known semi-major axis and a circle with a known radius, calculate the semi-minor axis of the ellipse if it is tangent to the circle at one point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{a r}}
How many of the rearrangements of the digits $123456$ have the property that for each digit, no more than two digits smaller than that digit appear to the right of that digit? For example, the rearrangement $315426$ has this property because digits $1$ and $2$ are the only digits smaller than $3$ which follow $3,$ digits $2$ and $4$ are the only digits smaller than $5$ which follow $5,$ and digit $2$ is the only digit smaller than $4$ which follows $4.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{162

}
Determine the number of possible values of \(\theta\), where \(0 < \theta < \pi\), for which the system of equations
\[ (y + z)\cos 3\theta = xyz\sin 3\theta \]
\[ x\sin 3\theta = \frac{2\cos 3\theta}{y} + \frac{2\sin 3\theta}{z} \]
\[ xyz\sin 3\theta = (y + 2z)\cos 3\theta + y\sin 3\theta \]
has a solution \((x_0, y_0, z_0)\) with \(y_0z_0 \neq 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In triangle \( \triangle ABC \), the sides are \( AB = 30 \), \( BC = 40 \), and \( CA = 50 \). Squares \( A_1A_2BC \), \( B_1B_2AC \), and \( C_1C_2AB \) are constructed externally on the sides of \( \triangle ABC \). The lines \( A_1A_2 \), \( B_1B_2 \), and \( C_1C_2 \) intersect pairwise at points \( P \), \( Q \), and \( R \). Determine the length of the shortest altitude of triangle \( \triangle PQR \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{124}
What is the proportion of prime numbers \(p\) such that the Legendre symbols \(\left(\frac{2}{p}\right) = 1\) and \(\left(\frac{-5}{p}\right) = -1\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
In a right triangle \(ABC\) with hypotenuse \(AB\), a point \(P\) is marked on leg \(AC\). Let \(D\) be the foot of the perpendicular from \(A\) to line \(BP\), and \(E\) be the foot of the perpendicular from \(P\) to side \(AB\). A point \(T\) is chosen such that lines \(TA\) and \(TP\) are tangent to the circumcircle of triangle \(PAB\), with \(O\) as the center of this circumcircle. The perpendicular from \(T\) to line \(DE\) intersects line \(BC\) at point \(Q\). The line through \(C\) parallel to \(OQ\) intersects segment \(BO\) at point \(K\). Determine the measure of angle \(\angle OKE\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Find the product of all possible values of $a$ such that the system of equations \(|2x| - y = 5\) and \(x - |2y + 2| = a\) has an odd number of distinct solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Determine the number of distinct compact Hausdorff spaces, up to homeomorphism, that exist with a given uncountable cardinality \( \kappa \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^\kappa}
Find the value of \( a \) such that the point on the curve \( f(x) = e^x \) closest to the line \( y = \frac{x}{a} - 1 \) has a \( y \)-coordinate of \( \frac{1}{e} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Determine the smallest integer $j$ such that it is possible to fill the cells of a $10 \times 10$ table with numbers from 1 to 100, ensuring that every set of 10 consecutive numbers lies within some $j \times j$ sub-square of the table.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let \( \text{Revolution} (x) = x^3 + Ux^2 + Sx + A \), where \( U \), \( S \), and \( A \) are integers such that \( U + S + A + 1 = 1773 \). Given that \( \text{Revolution} \) has exactly two distinct nonzero integer roots \( G \) and \( B \), find the minimum value of \( |GB| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{392}
In a group \( G \) of individuals, each person can be a friend of another. Determine the smallest number of individuals that must be chosen as subgroup leaders so that every person in the group has at least one friend who is a leader. Express your answer in terms of graph theory, where this problem involves finding the minimum number of radial-subgraphs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\gamma(G)}
Find the sum of the two positive integers $n$ such that when $1^3+2^3+3^3+\cdots +n^3$ is divided by $n+5$, the remainder is $17$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{239}
Find the order of the group given by the presentation: $$\langle a, b\mid a^3, b^3, [a, [b, a^{-1}]], [b, [b, a^{-1}]]\rangle.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Given a projective curve \( C \) in \( \mathbb{P}_\mathbb{C}^2 \), if a point \( p \in C \) has multiplicity \( m \), and \( \mathcal{O}_p(C) \) is the local ring of \( C \) at \( p \) with its maximal ideal \( \mathfrak{m} \), find the dimension of \( \mathfrak{m}^k/\mathfrak{m}^{k+1} \) over \( \mathbb{C} \) for \( k < m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k + 1}
Determine the sign of the permutation represented by the matrix: \[ \begin{pmatrix} 1 & \dots & p & p+1 & \dots & p+q \\ q+1 & \dots & p+q & 1 & \dots & q \end{pmatrix} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1)^{pq}}
In a triangulation of a polygonal domain $\Omega$ in $\mathbb{R}^3$, suppose $z$ is an interior vertex shared by $m$ tetrahedra. Determine the number of interior faces adjacent to $z$ in terms of $m$. Express your answer using $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3m}{2}}
Determine the number of zeros of the function \( y = \tan(2015x) - \tan(2016x) + \tan(2017x) \) in the interval \([0, \pi]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
Determine the cardinality of the set $G$ of all entire functions $g$ such that $g(x) = x^k$ for all $x \in (-1, 1)$, where $k$ is an odd integer $\geq 1$. \( K \) is given as an odd integer $\geq 1$. \( G \) is the set of all entire functions $g$ such that $g(x) = x^k$ for all $x \in (-1, 1)$. What is the cardinality of $G$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $G$ be a group of order $p^3$, where $p$ is a prime number. If $|Z(G)|=p$, determine the number of conjugacy classes of $G$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2 + p - 1}
Consider a 4 × 4 grid of squares, where each square can be colored either black or white. Two colorings are considered identical if one can be transformed into the other by any combination of flipping, rotating, or swapping the two colors (changing all black squares to white and vice versa). Determine the number of non-identical colorings of the grid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4324}
Determine the minimum degree $n$ of the Taylor polynomial $T_n(x)$ for the function \( f(x) = \int_{\frac{\pi}{2}}^{x} \frac{\cos(t)}{t - \frac{\pi}{2}}dt \) around \( a=\frac{\pi}{2} \) such that \( \mid f(\pi) - T_n(\pi) \mid \leq \frac{1}{100} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the number of connected components of the set \( L \cap \Delta \), where \( L = \alpha b \) is a line in \( \mathbb{R}^n \) with \( \alpha \in \mathbb{R} \) and \( b \in \mathbb{R}^n \) is fixed, and \( \Delta = \{ x \in \mathbb{R}^n : \pi(x) \text{ has roots in the open unit disk of } \mathbb{C} \} \). Assume \( n > 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the reproducing kernel Hilbert space (RKHS) induced by the kernel $k: X \times X \to \{0, 1\}$, where $X$ is an uncountably infinite set such as $X = [0, 1]$ or $X = \mathbb{R}$. The kernel is defined as $k(x, y) = \delta_{x, y}$, where $\delta_{x, y} = 1$ if $x = y$ and $0$ otherwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ell^2(X)}
The sum of ages of some children born at the intervals of 3 years each is 60 years. The age of the youngest child is 6 years. How many children are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
An $m\times n\times p$ rectangular box has half the volume of an $(m+2)\times(n+2)\times(p+2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\le n\le p.$ What is the largest possible value of $p$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
130
}
Determine the operator norm of the linear operator $A$ defined by \( Ax=(x_1+x_2, x_2+x_3, \ldots) \) for all $x \in \ell_2$. Given that the operator is bounded with \( \sum_{i=0}^\infty|x_i+x_{i+1}|^2 \leq 6\sum_{i=0}^\infty|x_i|^2 \), find the norm of $A$ by evaluating the supremum of $\|Ax\|$ for $\|x\|=1$. \( \| \cdot \| \) denotes the $\ell_2$ norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a regular triangular prism \(P-ABC\), a hemisphere is inscribed such that its base coincides with the base of the prism, and all three lateral faces of the prism are tangent to the hemisphere. Given that the radius of the hemisphere is 1, determine the height of the prism that minimizes its volume.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the integral \( \int_{[0,1]} x^3 \, d\mu_F \), where \( F \) is the Cantor function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{16}}
Find all complex numbers $z$ such that $iz + \sqrt{1-z^2} \le 0$, where the square root is defined as follows: for $z = re^{i\theta}$ with $-\pi < \theta \le \pi$, $\sqrt{z} = \sqrt{r}e^{\frac{1}{2}i\theta}$ and $\sqrt{r} \ge 0$. Determine the set of all such $z \in \mathbb{C}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Determine the minimum diameter that can be ensured when a planar set with a unit diameter is divided into $k$ pieces.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin\left(\frac{\pi}{k}\right)}
Determine the number of roots of the equation \( \overbrace{f(f(\ldots f}^{10 \text{ times } f}(x) \ldots)) + \frac{1}{2} = 0 \), where \( f(x) = |x| - 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Determine the real rank of the $C^*$-algebra $C_0(\mathbb{R})$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of real solutions for the equation \(3^{x+2}=\frac{1}{x^2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the arithmetic mean of all positive rational numbers less than 1 that can be expressed with a 10-digit period in decimal representation, where all 10 digits are distinct.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
In a standard deck of 52 cards numbered from 1 to 52, 26 cards are selected at random without replacement. Determine the number of ways to select these 26 cards such that the sum of the reciprocals of their ranks equals exactly 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the order of subgroups that appears most frequently in the symmetric group $S_n$. Specifically, find the order $k$ for which the number of subgroups of $S_n$ with order $k$ is maximized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the degree of accuracy for an $n$-point Gauss-Legendre quadrature when applied to the multidimensional integral \( \int_{a}^{b}{\int_{p}^{q}{f(x,y) \,dx \,dy}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n - 1}
Determine the values of $x$ in the interval $(-3, 17)$ for which the series \( \sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n[\log (n+1)]^2} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 1]}
Given a list of numbers \([a_1, a_2, a_3, \ldots, a_n]\) and a target value \(t\), determine the number of non-negative integer solutions to the equation \(x_1a_1 + x_2a_2 + \ldots + x_na_n = t\), where each \(x_i\) is a non-negative integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[z^t] \prod_{i=1}^{n} \frac{1}{1 - z^{a_i}}}