inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $G$ be a finite group and let $x_1, \ldots, x_n$ be an enumeration of its elements. Consider the matrix $(a_{ij})_{1 \le i,j \le n}$, where $a_{ij} = 0$ if $x_i x_j^{-1} = x_j x_i^{-1}$, and $a_{ij} = 1$ otherwise. Determine the parity of the integer $\det(a_{ij})$. \(\text{(Even or Odd)}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Even}
Find the probability density function (pdf) of the random variable \( Y = \frac{T_1}{T_3} \), where \( \{N(t): t \geq 0\} \) is a Poisson process with rate 1, and \( T_1, T_2, \ldots \) denote the event times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(1 - y)}
Find the constant \( c \) such that the random variable \( Y = \frac{c(X_1 + X_2)}{\sqrt{(X_3)^2 + (X_4)^2}} \) has a t-distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the probability that the interval \( \left[\frac{Y}{124.34},\frac{Y}{77.93}\right] \) contains \( \sigma^2 \), where \( Y = \sum_{i=1}^{100}(X_i-\mu)^2 \) and \( \mu \) is known for a normally distributed population of 100 individuals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.90}
Determine the values of $s \in \mathbb{R}$ for which the series \[ I_s = \sum_{n=1}^{\infty} \left(\frac{a_1^{1/s} + a_2^{1/s} + \cdots + a_n^{1/s}}{n}\right)^s \] converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{s > 1}
Find an equivalent expression for the remainder \( R_n = \sum_{k=n+1}^{\infty}{\frac{1}{k^3}} \) of the converging series \( \sum_{n\geqslant1}{\frac{1}{n^3}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2n^2}}
Determine the cardinality of the set \(E(a)\), where \(E(a) = \{(r_1, r_2, \ldots, r_n, \ldots) \mid r_n \in \mathbb{Q} \cap [0,1], r_n \leq r_{n+1}, \forall n \geq 1, \sum_{n=1}^\infty \frac{r_n}{2^n} = a\}\) for any \(a \in (0,1)\). What is \(|E(a)|\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Determine the number of distinct squares that can be formed using 8 congruent triangular tiles, each of which can be colored either white or black. Two squares are considered the same if one can be transformed into the other through rotation or reflection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
Determine the condition on $p$ for which the series \( \sum_{n=1}^\infty \frac{|g(nx)|}{n^p} \) converges almost everywhere, where \( g:\mathbb{R}^m\rightarrow\bar{\mathbb{R}} \) is a Lebesgue integrable function with \( \int |g|>0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p > 1 - m}
Determine the values of \(a \in \mathbb{R}\) for which there exists a non-constant function \(f: (0, 1] \rightarrow \mathbb{R}\) such that \(a + f(x+y-xy) + f(x)f(y) \leq f(x) + f(y)\) holds for all \(x, y \in (0, 1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 0]}
Evaluate the limit:

$$ \lim_{x \to \infty} {\displaystyle{\large\left(1 + x\right)\left(1 + \frac{1}{x}\right)^{2x} - x\,{\rm e}^{2}} \over \displaystyle{\large{\rm e}^{2} - \left(1 + \frac{1}{x}\right)^{2x}}} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
A mouse is placed in one corner cube of a $3 \times 3 \times 3$ grid, and a piece of cheese is placed in the central cube. The mouse moves randomly, stepping into one of the neighboring cubes at each move. On average, how many steps will it take for the mouse to reach the cheese?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Inside regular hexagon $ZUMING$, construct square $FENG$. What fraction of the area of the hexagon is occupied by rectangle $FUME$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{2(3  - \sqrt{3})}{9}$}
What is the probability that five randomly chosen points from a unit square will determine an ellipse when used to form a conic section?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
A rubber band is initially 1 meter long, with one end fixed to a wall. The other end is pulled away from the wall at a constant speed of 1 m/s. A bug starts at the wall and crawls along the rubber band toward the moving end at a speed of 0.001 cm/s. Determine if the bug will ever reach the end of the rubber band, and if so, calculate the time it takes for the bug to reach the end.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{100000} - 1}
Given the bases $B_1 = \{1+x,1-x,1+x^2\}$ and $B_2 = \{1+2x,x,1+x+x^2\}$ for the vector space $P_2(\Bbb R)$, and the linear transformation $T$ represented by the matrix $M(T,B_1,B_2) = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 2 & 1 \\ 1 & 1 & -1 \\ \end{pmatrix}$, determine $T(1-x+3x^2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7 - 7x - 5x^2}
Evaluate the integral \( \int_0^{2\pi} \cot(e^{it})ie^{it}dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
Determine the asymptotic growth rate of the sequence defined by the recurrence relation: \[ a(n+1) = a(n) + \frac{1}{1+2^{a(n)}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log_2 n}
Determine the cardinality of the set \([0, \Omega) \setminus (\bigcup \{ U_\lambda : \lambda < \Omega \text{ is limit} \})\), where \(\Omega\) is the first uncountable ordinal with the order topology, and for each limit ordinal \(\lambda < \Omega\), \(U_\lambda\) is an open neighborhood of \(\lambda\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
Evaluate the triple summation \( \sum_{1 \le i < j \le k \le 20} a_i a_j a_k \) where \( a_i = 1 \) for \( 1 \le i \le 10 \) and \( a_i = 2 \) for \( 11 \le i \le 20 \). Find the value of the summation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4585}
Find the value of \( \alpha \in \mathbb{R} \) such that the change of coordinates \( \varphi(p,q) \rightarrow (P,Q) \) is symplectic, where \( Q = q^2 + \alpha\sqrt{q^2+p} \) and \( P = q + \sqrt{q^2+p} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Compute the line integral \( \int -4x^2y \, dx -(x^3+y^3) \, dy \) along the closed curve that starts at \((-2,0)\), follows the curve \(y=4-x^2\) to \((2,0)\), and returns to \((-2,0)\) along the curve \(y=x^2-4\), using Green's Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{256}{15}}
Determine the rank of the elliptic curve given by the equation \( y^2 = x^3 + p^3x \) where \( p \) is a prime number such that \( p \equiv 7 \pmod{16} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit: $$\lim_{x\to\infty}(x-\ln\cosh x)$$ where $$\cosh x=\frac{e^x+e^{-x}}{2}."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
Two equally strong tennis players play against each other until one player wins three games in a row. Each game is independent, and each player has a probability of \(\frac{1}{2}\) of winning any given game. What is the expected number of games they will play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given the functional equation \(f(x)f(y) = f(x+y)\) and the condition \(f(0) = 1\), find the derivative \(f'(0)\) if \(f\) is differentiable and continuous at \(0\), and there exists a function \(g(x)\) such that \(f(x) = 1 + \sin(3x)g(x)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the Bayes estimator for the parameter \( \theta \) given the loss distribution \( l(a,\theta)=\frac{(a-\theta)^2}{\theta(1-\theta)} \), a prior distribution \( \pi(\theta) \) that is Uniform from 0 to 1, and a likelihood function \( f(x|\theta) \) for a sequence of \( n \) independent Bernoulli trials, each with a success probability \( \theta \). The total number of ones in the sequence is \( X \sim \text{Bin}(n, \theta) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{X}{n}}
Evaluate the surface integral \(\int_M(x-y^2+z^3)dS\) over the portion of the cylinder \(x^2+y^2=a^2\) bounded by the planes \(x-z=0\) and \(x+z=0\), where \(a>0\), without using vector calculus theorems.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{8}{3} a^4}
Calculate the variance of the random variable $X$, which represents the number of different digits that appear in two randomly and independently drawn 5-digit pins from a set where all digits must be unique. Each pin is selected from the set $A=\{(d_1, d_2, d_3, d_4, d_5): d_i \in \{0, \ldots, 9\}, i \in \{1, \ldots, 5\}, \forall (i \neq j, i, j \in \{1, 2, 3, 4, 5\}) d_i \neq d_j\}$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{25}{36}}
Find the interval of convergence for the series \(\sum_{n=1}^\infty \frac{\ln(1+nx)}{n x^n}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, \infty)}
A coin is flipped 100 times. Let $X$ be the number of heads in the first 70 flips and $Y$ be the number of heads in the last 50 flips. Compute the correlation of $X$ and $Y$. Provide your answer as a decimal rounded to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.34}
Evaluate the limit \( \lim_{x\to0^{-}}\frac { f(-x){ x }^{ 2 } }{ \left\{ \frac { 1-\cos(x) }{ [f(x)] } \right\} } \). Given that \( f(x) \) is a real-valued continuous function except at \( x=0 \), with \( f'(x)<0 \) for negative \( x \) and \( f'(x)>0 \) for positive \( x \). Also, \( \lim_{x\to0^{+}} f(x)=2 \), \( \lim_{x\to0^{-}} f(x)=3 \), and \( f(0)=4 \). Here, \([\cdot]\) denotes the greatest integer function, and \(\{\cdot\}\) denotes the fractional part function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the minimal polynomial of \(\alpha^{21}\) over \(\mathbb{Z}_2\), given that \(\alpha\) satisfies the equation \(\alpha^6 + \alpha + 1 = 0\) over \(\mathbb{Z}_2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 + x + 1}
Find the solution to the second-order ordinary differential equation \(2xx''-3(x')^2=4x^2\) with the initial conditions \(x(0) = 1\) and \(x'(0) = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sec^2 t}
A cube with side length \( n \) (\( n \geq 3 \)) is divided into unit cubes by partitions. Determine the minimum number of partitions that must be removed so that it is possible to reach the boundary of the cube from any unit cube.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n - 2)^3}
Evaluate the limit: \[ \lim_{x\to 0}\frac{1}{1-\cos(x^2)}{\sum_{n=4}^\infty {n^5x^n} } \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2048}
Calculate \( \liminf_{n\to\infty} \left( \frac{p_{n}}{p_{n}+p_{n+1}} \right) \), where \( p_{n} \) is the \( n \)-th prime number and \( p_{n+1} \) is the next prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the range of the function \( f(x) = \sqrt{\frac{-x^2 + 2x + 3}{18x - 3x^3}} \) without using limits or derivatives.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, \infty)}
In how many ways can the letters of the word "INDIVISIBILITY" be rearranged such that no two 'I's are adjacent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3386880}
Given an $n$-dimensional vector space $V$ with a basis $\{e_1, \ldots, e_n\}$, consider partitions of this basis into two subsets of $k$ and $n-k$ elements, which define subspaces $P$ and $Q$. Determine the maximum number of partitions for which a given $k$-dimensional subspace $W$ has a trivial intersection with $Q$. What is this maximum number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dbinom{n}{k}}
If Natalie cuts a round pizza with $4$ straight cuts, what is the maximum number of pieces that she can get? Note: Assume that all the cuts are vertical (perpendicular to the surface of the pizza). She cannot move the pizza pieces until she finishes cutting.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
11
}
Evaluate the limit \( \lim_{x \to 0} \left( x^2 \left( 1 + 2 + \cdots + \left[ \frac{1}{|x|} \right] \right) \right) \), where \([a]\) denotes the greatest integer less than or equal to \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Calculate the limit as \( n \) approaches infinity: \[ \lim_{n\to\infty}{\dfrac{1}{n}\sum_{k=1}^{n}{\left(\dfrac{n}{k}-\left\lfloor\dfrac{n}{k}\right\rfloor\right)}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \gamma}
Find the Bayes estimator of \(\theta\) given the following information:

- Loss function: \(l(a,\theta)=\frac{(a-\theta)^2}{\theta(1-\theta)}\)
- Prior distribution: \(\pi(\theta)\) is uniformly distributed from 0 to 1.
- Likelihood function: Bernoulli, where \(X\) is the total number of ones in a sequence of \(n\) independent Bernoulli trials, each with sample space \(\{0,1\}\), so \(f(x|\theta) = \theta^x(1-\theta)^{1-x}\), \(x=0,1\).

Calculate the Bayes estimator of \(\theta\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{X}{n}}
In a plane, there are \( n+4 \) points, four of which are located at the vertices of a square, and the remaining \( n \) lie inside this square. You are allowed to connect any of these points with segments, provided that no segment contains any of the marked points other than its endpoints, and no two segments share any points other than their endpoints. What is the maximum number of segments that can be constructed in this way?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3n + 5}
Determine the radius of convergence of the power series $$\sum (n^2 \sin(2/n^2))^{n^2+2} x^{n^2}$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
The function $f(x)$ is defined on the reals such that

$$f\left(\frac{1-4x}{4-x}\right) = 4-xf(x)$$
for all $x \ne 4$. There exists two distinct real numbers $a, b \ne 4$ such that $f(a) = f(b) = \frac{5}{2}$. $a+b$ can be represented as $\frac{p}{q}$ where $p, q$ are relatively prime positive integers. Find $10p + q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180
}
Determine the number of incongruent triangles that can be formed with integer side lengths and a perimeter of 1994.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{82834}
Evaluate the limit of the series:

$$\lim_{x\to\infty} \sum ^\infty_{n=3} \frac{n\cdot x^{n-2}\cdot(-1)^n\cdot(n-1)}{(2n)!}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{12}}
Given three distinct points \( L, M, \) and \( N \) on a straight line, with \( M \) between \( L \) and \( N \) and \( LN \neq MN \), semicircles are constructed on diameters \( LM, MN, \) and \( LN \). The centers of these semicircles are points \( A, B, \) and \( C \) respectively, with \( C \) on one side of the line \( LN \) and \( A \) and \( B \) on the opposite side. Determine the ratio of the area of the region enclosed by these three semicircles to the area of triangle \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Find the positive time $T$, in years, such that the amount in Fund A, which accumulates at a rate of $3\%$ convertible monthly, equals the amount in Fund B, which accumulates with a force of interest given by $\delta_t = t^{20}$ for all $t$. At time $t = 0$, $\$1 is deposited in each fund. Calculate $T$. \( \left(1+\frac{0.03}{12}\right)^{12T} = e^{\int_0^T t^{20} \, dt} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.977}
159 The area enclosed by the curve $(|x|-1)^{2}+(|y|-2)^{2}=5$ is $=$

The area enclosed by the curve $(|x|-1)^{2}+(|y|-2)^{2}=5$ is $=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10\pi+16}
Evaluate the integral \( \int_{B(0,r)} \frac{1}{|x|^{n-1}} \, dx \) and determine if it can be expressed as a constant times the radius \( r \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\pi^{n/2}}{\Gamma(n/2)} r}
Determine the limit of the sequence \( \{x_n\} \) where \( x_n = \frac{M_n + a}{M_n(M_n + b)} \), given that \( M_n = o_p(\sqrt{n}) \), \( M_n > 0 \), and \( \mathbb{P}(\liminf\{M_n > K\}) = 1 \) for any integer \( K \). Here, \( a \) and \( b \) are positive constants.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral \( \int_0^\pi \int_0^\pi |\cos(x+y)| \, dx \, dy \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Determine the order of growth for the error function, defined as \( \text{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt \), when expanded into its Taylor series and considered as an entire function in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In the binomial expansion of \((1+\sqrt{11})^{208}\), determine the value of \(k\) for which the term \(C_{208}^{k}(\sqrt{11})^{k}\) is maximized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
Determine the number of surjective functions from the set $A = \{1,2,3,4,5,6\}$ to the set $B = \{1,2,3,4,5,6\}$, given that $f(1) = 2$ and $f(i) \neq i$ for all $i \in A$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
Identify the topological group that is isomorphic to the quotient group $\mathbb{R}/\mathbb{Z}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S^1}
Given a $2\pi$ periodic function defined as $f(t) = t$ for $0 \leq t < \pi$ and $f(t) = 0$ for $\pi \leq t < 2\pi$, find the value of its complex Fourier series representation at $t = \pi$. The complex Fourier series is given by:
\[ f(t) = \frac{\pi}{4} + \sum_{k \neq 0} \frac{1}{2\pi} \left( \frac{(-1)^k - 1}{k^2} \right) e^{ikt}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Calculate the Euler characteristic of the Grassmannian $G(3, 7)$ by counting the fixed points of a torus action.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
An integer $x$ is selected at random between 1 and $2011!$ inclusive. The probability that $x^x - 1$ is divisible by $2011$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1197}
Given the sequence $a_n$ where \(\sum_{n=1}^\infty \left(na_n-\frac{n^2+1}{2n+1} \right)=3,\) determine the limit \(\lim_{n\rightarrow\infty}(a_n^2+2a_n+2).\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{4}}
Let \( \sigma \left( n\right) \) be the sum of all the positive divisors of \( n \) . Let \( a \) be the smallest positive integer greater than or equal to 2015 for which there exists some positive integer \( n \) satisfying \( \sigma \left( n\right)  = a \) . Finally,let \( b \) be the largest such value of \( n \) . Compute \( a + b \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3943}
What is the sum of all ordinals up to \(\omega\), where \(\omega\) is the first transfinite ordinal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\omega \cdot 2}
Find the distance between the point \( p = (1,0,0,1) \) and the subspace \( L = (\operatorname{span}{(1,0,2,1),(2,1,2,3),(0,1,-2,1)})^\perp \) in \( \mathbb{R}^4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{14}}{3}}
In a box, there are eight real coins and two fake coins, with the fake coins having a different weight than the real ones. You randomly select a pair of coins without replacement from the ten coins, and then select another pair from the remaining eight coins. What is the probability that all four coins selected are real, given that the combined weight of the first pair is the same as the second pair?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{15}{19}}
Evaluate \( \Delta z \) along the path \( y = x^2 \) from the point \((0,0)\) to \((1,1)\). Given the differential \( dz = xydx + xydy \), find the change in \( z \) as you move from \((0,0)\) to \((1,1)\) along the path \( y = x^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{20}}
A motorist arrives at a toll bridge with a single toll booth in operation and finds 50 cars ahead of them. The toll collection time for each vehicle follows an exponential distribution with a probability density function given by \( f(x) = 2\exp(-2x) \), where \( x \) is in minutes. Using the central limit theorem, calculate the approximate probability that the motorist will have to wait more than 25 minutes to reach the toll booth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5}
Evaluate the integral \( \frac{1}{2\pi} \int_{0}^{2\pi} e^{\cos(\theta)}\cos(n\theta) \, d\theta \) for \( n \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I_n(1)}
Find the smallest positive integer \( n \) such that for any integer \( p \geq n \), a square can be partitioned into \( p \) non-congruent smaller squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
In an \( 8 \times  8 \) grid of squares, each square was colored black or white so that no \( 2 \times  2 \) square has all its squares in the same
 color. A sequence of distinct squares \( {x}_{1},\ldots ,{x}_{m} \) is called a snake of length \( m \) if for each \( 1 \leq  i < m \) the squares \( {x}_{i},{x}_{i} \) .
 are adjacent and are of different colors. What is the maximum \( m \) for which there must exist a snake of length \( m \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8
}
Calculate the volume of the region between the surfaces $z=0$ and $z=\frac{1}{(x^2+y^2)^9}$, and outside the cylinder $x^2+y^2=1$, using a triple integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{8}}
Given a metric on \(\mathbb{R}^2\) in polar coordinates of the form \(dr^2 + f(r,\theta)d\theta^2\), and a curve \(\gamma(t) = (t, \theta_0)\) where \(\theta_0\) is a constant, find the covariant derivative \(\nabla_{\gamma'(t)}\gamma'(t)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $n > 1$ be an integer that is divisible by exactly two distinct primes, $p$ and $q$. If $G$ is a group with exactly $n$ elements of order $n$, determine the maximum possible number of cyclic subgroups of order $n$ in $G$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}