inclusionAI/Ring-lite-rl-data huggingface dataset:
Evaluate the limit: \[ \lim_{x\to 0} \left( \frac{1+x\cdot2^x}{1+x\cdot3^x} \right)^{1/x^2} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Given a sequence of independent and identically distributed (i.i.d.) random variables $\{X_i\}$ with mean $0$ and variance $\sigma^2 > 0$, find the limit of the expected value of the squared fraction of the sample mean's absolute value as the number of variables approaches infinity: $$\lim_{n \to \infty} E\left(\left|\frac{\overline X_n}{\overline X_n + \frac{1}{n}}\right|^2\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider a set of points in the plane such that only two distinct Manhattan distances occur between all pairs of distinct points. What is the maximal number of points in such a set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine the number of ring homomorphisms from the polynomial ring \(\mathbb{Z}[X]\) to the integers \(\mathbb{Z}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the value of the fourth derivative of the inverse function \((f^{-1}(y))^{(4)}\) evaluated at \(y=1\), where \(f(x) = \sin{x} + x^2 + 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-140}
For a zebra group of order $2^8p^b$, where $p$ is a prime, determine the maximum number of abelian 2-group stripes that can appear in its chief series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Evaluate the integral \( \int f(x) \, \mathrm{d}x \) where \( f(x) = \lim_{n \to \infty} \left[ 2x + 4x^3 + \cdots + 2nx^{2n-1} \right] \) for \( 0 < x < 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{1 - x^2} + C}
Find the locus of the duals of lines that pass through the point (1, 1) and have variable slopes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x + y + 1 = 0}
Determine the coefficient of the term $x^{60}$ in the power series expansion of $C(x) = \frac{1}{1-x} \cdot \frac{1}{1-x^2} \cdot \frac{1}{1-x^5}$. This coefficient represents the cardinality of the set $A = \{(N, D, Q) \mid 0.05N + 0.1D + 0.25Q = 3 \text{ and } N, D, Q \in \mathbb{N}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{205}
Let \( \phi \) be an automorphism of the quaternion group \( \mathcal{Q}_8 \) such that \( \phi(i) = -i \) and \( \phi(j) = -k \). Determine the order of \( \phi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the lines $l_1: ax+2y-1=0$ and $l_2: 8x+ay+2-a=0$, if $l_1 \parallel l_2$, find the value of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["-4"]}
Let \( q(x) = 1 - (1-x) + (1-x)^2 - \cdots + (1-x)^{10} \). The roots of \( q(x) \) are \( \beta_i, i \in \{1, 2, \ldots, 10\} \). Find the value of \( \sum_{i=1}^{10} \frac{1}{\beta_i} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the number of singular values of a linear map $T: V \to W$ given that $\dim V = n$ and $\dim W = m$. Provide the correct number of singular values for the linear map $T$. \( \text{(Assume $T$ is represented by a matrix in the standard basis.)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\min(m, n)}
Evaluate the limit: $$\lim_{x\to \infty} \int_1^x x\,e^{t^2-x^2} \,dt$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Let $n\geq 2$ be integer. On a plane there are $n+2$ points $O,\ P_{0},\ P_{1},\ \cdots P_{n}$ which satisfy the following conditions as follows.

[1] $\angle{P_{k-1}OP_{k}}=\frac{\pi}{n}\ (1\leq k\leq n),\ \angle{OP_{k-1}P_{k}}=\angle{OP_{0}P_{1}}\ (2\leq k\leq n).$
[2] $\overline{OP_{0}}=1,\ \overline{OP_{1}}=1+\frac{1}{n}.$
Find $\lim_{n\to\infty}\sum_{k=1}^{n}\overline{P_{k-1}P_{k}}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
 \((e - 1)\sqrt{\pi^2 + 1}\)}
Evaluate the limit: \[ \lim_{x \to \infty} \frac{((2x)!)^4}{(4x)! ((x+5)!)^2 ((x-5)!)^2}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the range of \( \alpha \in \mathbb{R} \) for which the following statement holds: If a sequence \( \{ x_n \}_{n=1}^{\infty} \) of real numbers satisfies \( \lim_{n \to \infty}{ n^\alpha \,\, |x_n - x_{n+1}|}=0 \), then it is a convergent sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, \infty)}
How many distinct ways can the numbers $1, 2, 3, 4, 5, 6, 7$ be inserted into an empty binary search tree such that the resulting tree has a height of $6$? Note that a tree with a single node has a height of $0$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Calculate the Morley rank of the type $x = x$ in the theory $ACF_{0}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the flow of the vector field \( F(x,y,z) = (xz, yz, -z^2) \) out of the surface \( S = \{(x,y,z) \in \mathbb{R}^3 : z^2 = 1 + x^2 + y^2, 2 \leq z \leq 3\} \) using the definition of surface integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-60\pi}
Determine the number of distinct ways to color a regular hexagon such that each vertex is either Black or White, and each edge is one of the three colors Red, Green, or Blue, considering the symmetries of the dihedral group $D_6$. The group $D_6$ includes rotations by multiples of $\frac{\pi}{3}$ and reflections across axes through opposite vertices or midpoints of opposite edges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4183}
Let $f$ and $g$ be functions of $x$ and $y$. Given that $f(1, 2) = 2$ and $g(1, 2) = 1$. Also at $(1, 2)$, $df = -2dx + 5dy$, $dg = 4dx - 3dy$. $z$ is a function of $(x, y)$ defined implicitly by $z^2f + zg + x - y = 0$ with $z(1, 2) = -1$. Find $\frac{\partial z}{\partial x}(1, 2)$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{5}{3}}
In a triangle $ABC$, given that $r_1 + r_3 + r = r_2$, find the value of $\sec^2 A + \csc^2 B - \cot^2 C$, where $r_1 = \frac{\Delta}{s-a}$, $r_2 = \frac{\Delta}{s-b}$, $r_3 = \frac{\Delta}{s-c}$, and $r = \frac{\Delta}{s}$. Here, $\Delta$ is the area of the triangle and $s$ is the semi-perimeter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A die is rolled repeatedly. Let $X$ be the random variable representing the number of rolls needed to get a 4, and $Y$ be the random variable representing the number of rolls needed to get a 1. Calculate the expected value $E[X|Y=7]$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{86317}{15625}}
Let \( P(x) = 5x^{2015} + x^2 + x + 1 \) be a polynomial with roots \( x_1, x_2, x_3, \ldots, x_{2015} \). Find the value of the expression:
\[ \left(\prod_{i=1}^{2015} (1+x_i)\right) \left( \sum_{i=1}^{2015} \frac{1}{1-x_i} \right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5039}{5}}
Determine the number of distinct equivalence classes for $2 \times 2$ matrices over the field of complex numbers, $\mathbb{C}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Identify the smallest group that is not a complex reflection group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Q_8}
Let $X$ be a set. Define mappings $F, G : \text{Map}(X, X)\times\text{Map}(X, X)\to\text{Map}(X, X)$ such that $F(f, g) = f\circ g$ and $G(f, g) = g\circ f$. What is the necessary and sufficient condition on $X$ that ensures $F=G$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|X| \leq 1}
A wire is shaped as a circle defined by the equation \(x^2 + y^2 = a^2\). The density at any point \((x, y)\) on the wire is given by \(|x| + |y|\). Calculate the moment of inertia of the wire about a diameter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4a^4}
Determine the number of real values of \(x\) that satisfy the equation:
\[ \left| 3 - x \right|^{\log_7(x^2) - 7\log_x(49)} = (3 - x)^3 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine all positive integers \( n \) such that for any sequence \( a_1, a_2, \ldots, a_n \) with each \( a_i \in (0, 1] \), the inequality \( \sum_{i=1}^{n} \frac{\sqrt{1-a_{i}}}{a_{i}} \leq \frac{1}{\prod_{i=1}^{n} a_{i}} \) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the integral \( \int_{0}^{\infty} \frac{\sin{x}}{x^2+1} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2e}}
Let \(x\) be a nonnegative real number. The function \(f_n\) is defined by: \[ f_n(x) = 3x^n e^{-x^2} - 1. \] Let \(v_n\) be a real number for all \(n > 1\) such that \(v_n > 1\) and \(f_n(v_n) = 0\). Calculate the limit of \(v_n\) as \(n\) approaches infinity, i.e., find \(\ell\) such that \(\lim\limits_{n \to \infty} v_n = \ell.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the limit of the expression obtained by substituting \( n=2 \) in the equation \( r^{n-1}w'+\frac{1}{2}r^nw=a \), as \( r \) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a}
Determine the smallest degree $n$ of a Taylor polynomial centered at $0$ for the function $f(x) = (1-x)^{-1}$, such that the error in the interval $[0, 0.5]$ is no greater than $10^{-8}$. Use the remainder formula:
\[ \frac{|f^{(n+1)}(c)|}{(n+1)!} \cdot |(x-\bar{x})^{(n+1)}| \le 10^{-8} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Determine the number of possible solutions for the equation \( \int_{0}^{x} (t^2-8t+13)\,\mathrm{d}t = x\sin\left(\frac{a}{x}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the order of magnitude of \( \log h(n) \) as \( n \to \infty \), given that \( \lim_{n \to \infty} n^n h(n) = \int_{\mathbb{R}} e^{-e^x} g(x) \, dx \) for some function \( g(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-n \log n}
Determine the interval for the value of $k \in (0, 2\pi]$ such that for the function $f(x) = \tan\left(\frac{3x - \pi}{2}\right)$, $f(x) = f(x+k)$ for all $x$ in the domain of $f$. What is the interval over which $k$ can be found?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\pi}{3}}
In a circle with center $A$, point $E$ is the tangent point, and $[BA] \perp [AC]$. Given that $\angle ABD = \angle DCB$ and $\angle DBC = \angle DCA$, with $D$ on the circle $\omega$, and $|EC| = x$, find the area of triangle $BCD$ in terms of the radius $r$ of the circle and $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{r x}{2}}
Find the boundary of the manifold $D^2 \times S^1$, where $D^2$ is the closed unit disk and $S^1$ is the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S^1 \times S^1}
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{x \sin (\Omega x)}{x^2(x^2+1)} \, dx \) for \( \Omega \gg 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
For which values of \(a \in \mathbb{R}\) does the second-order matrix differential equation \(\frac{d^2}{dt^2}x = \begin{pmatrix}1 & 1 \\ 0 & a\end{pmatrix}x\) have periodic solutions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a < 0}
Let $W_t$ be a standard Brownian motion and define the process $$X_t = X_0 e^{t(1-\sigma^2/2) + \sigma W_t}$$ where $\sigma$ has an exponential distribution given by $$ P(\sigma \leq x) = 1-e^{-x}$$ for $x \geq 0$. Determine the probability $$P\left(\lim_{t \to \infty}X_0 e^{t(1-\sigma^2/2) + \sigma W_t}=0\right).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-\sqrt{2}}}
Find the spectrum of the operator \( T \) defined on the space of double-sided sequences \( \ell_2(\mathbb{Z}) = \{ \{ a_n \}_{n = -\infty}^{\infty} ; \sum_{n= -\infty} ^{\infty} | a_n |^2 < \infty \} \) by \( Tx = y \) where \( y_n = -x_{n-1} + x_n - x_{n+1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 3]}
What is the number of ways the numbers from $1$ to $14$ can be split into $7$ pairs such that for each pair, the greater number is at least $2$ times the smaller number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
144
}
Find the radius of convergence of the series \( \sum_{n=0}^{\infty} 2^{n} z^{n!} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of elements of order 5 in a group of order 100.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
A point mass is attracted towards a center by a force given by \( \frac{m k^2}{r^3} \), where \( r \) is the distance from the center. If the point starts at a distance \( r = a \) and is initially at rest, determine the time required for the point to reach the center. Note that the mass \( m \) can be eliminated from the equation \( m r'' = -\frac{m k^2}{r^3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{a^2}{k}}
Find the expectation of the random variable \( T \), where \( T = \min\{n \geq 1 \mid X_1 + X_2 + \ldots + X_n > 1\} \) and \( X_1, X_2, \ldots \) are i.i.d. standard uniform random variables on \((0, 1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
On an island, there are 12 blue chameleons, 15 green chameleons, and 7 red chameleons. When two chameleons of different colors meet, they both change into the third color. What is the minimum number of meetings required to convert all the chameleons into the same color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the smallest number of vertices in a graph that is vertex-transitive but neither edge-transitive nor edge-flip-invariant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
For which real number $a$ does the series \( \sum_{n=0}^\infty \frac{1}{u_n} \) converge, where the sequence \( \{u_n\} \) is defined recursively by \( u_0 = 2 \) and \( \int_{u_n}^{u_{n+1}} (\ln u)^a \, du = 1 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a < -1}
Simplify the polynomial \( p_n(x) = a_0 \binom{n}{0} (1-x)^n + a_1 \binom{n}{1} x(1-x)^{n-1} + a_2 \binom{n}{2} x^2 (1-x)^{n-2} + \cdots + a_n \binom{n}{n} x^n \), where \( \{a_n\} \) is an arithmetic sequence with common difference \( d \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a_0 + d n x}
Given a convex $n$-gon with distinct letters at each vertex, determine a formula in terms of $n$ for the maximum number of distinct $n$-letter words that can be read by an observer standing outside the polygon. Each word must include all $n$ letters, with no letter hidden behind another from the observer's viewpoint.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n}
Calculate the percentage of prime numbers among all numbers with 100 decimal digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.43\%}
Evaluate the integral: \[ \int_{-\infty}^\infty e^{-x^{-2}} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
A casino patron will continue to make $5 bets on red in roulette until she has won 4 of these bets. Calculate her expected winnings when she stops.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{20}{9}}
Find \( \lim_{t \to \infty} P (X>t+ \frac{x}{t} | X > t) \) where \( X \sim N(0,1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-x}}
Find a single generator for the kernel of the homomorphism \( \varphi : \mathbb{Z}[i] \to \mathbb{F}_{13} \) defined by \( \varphi(a+bi) = a + 5b \pmod{13} \). Given that \( \ker(\varphi) \) is the ideal generated by \( 13 \) and \( i-5 \), determine a single generator for \( \ker(\varphi) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 + 2i}
Determine the asymptotic bound for the expression \( \sum_{i=0}^{\log(n)} 2^{i} \sum_{k=0}^{\frac{n}{2^{i}}} (k+2)^{2} + \theta(n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n^3)}
Determine the number of homomorphisms from the dihedral group $D_{21}$ to the dihedral group $D_{15}$. Assume that $\phi(ab) = \phi(a)\phi(b)$, where $s$ and $r$ are the standard generators of $D_n$ with $s$ having order 2 and $r$ having order $n$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
Let \(X_1, X_2, \ldots, X_n\) be a random sample from a Poisson distribution with parameter \(\lambda = 1\). Define the statistic \(T = \bar{X}(n - \bar{X})\), where \(\bar{X}\) is the sample mean. Find an upper bound for the probability \(P(T = 0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2e^{-n}}
Suppose that \( {ABCDE} \) is a convex pentagon with
 \( \angle A = {90}^{o},\angle B = {105}^{o},\angle C = {90}^{o} \) and \( {AB} = 2,{BC} = {CD} = {DE} = \sqrt{2} \) . If the
 length of \( {AE} \) is \( \sqrt{a} - b \) where \( a,b \) are integers,what is the value of \( a + b \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4}
Let \(\beta\) be an element in \(\mathbb{F}_{16}\) with a minimal polynomial over \(\mathbb{F}_{2}\) given by \(x^4+x+1\). Determine the minimal polynomial of \(\beta^7\) over \(\mathbb{F}_{2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^4 + x^3 + 1}
Find the value of the limit \( L = \lim\limits_{n\to\infty} \dfrac{2^{n^{1.1}}}{n!} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Compute the double integral \( \iint_R xe^y \, dx \, dy \) where \( R \) is the region inside \( x^2+y^2 = 1 \) but outside \( x^2+y^2 = 2y \) with \( x \ge 0 \) and \( y \ge 0 \). Use the change of variables \( u = x^2 + y^2 \) and \( v = x^2 + y^2 - 2y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{e} - \dfrac{3}{2}}
Evaluate the integral \( \int_{C}{xy^{2}dy-yx^{2}dx} \) where \( C \) is the circle \( x^{2}+(y-1)^{2}=1 \) using Green's theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{2}}
A fly moves through the point \((\sqrt{3}, 1, 2)\) at time \(t = 0\) along the curve of intersection of the surfaces \(z = x^2 - y^2\) and \(z^2 = x^2 + y^2\). The temperature in the room is given by \(T(x,y,z) = \frac{1}{2}(4x^2 + y^2 + z^2)\). If the fly's vertical velocity at this time is 5, what is the rate of change of temperature \(\frac{dT}{dt}\) experienced by the fly at \(t = 0\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{155}{4}}
Solve the recurrence relation given by:
\[ T(n) = \frac{1}{n}(T(0) + T(1) + \ldots + T(n-1)) + 5n \]
with the initial condition \( T(0) = 0 \). Find a closed-form expression for \( T(n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10n - 5H_n}
Evaluate the integral \( \frac{1}{2\pi i}\int_{\gamma} \frac{1}{z^2(e^{z}-1)}dz \), where \( \gamma=\{z: z=\frac{1}{2}e^{2\pi i t},0\leq t\leq1 \} \). Note that \( z=0 \) is a pole singularity of order 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
Evaluate the limit using Taylor series expansion: $$\lim\limits_{h \to 0^-}{\int\limits_{0}^2{\frac{h}{h^2+(x-1)^2}}g(x)\:dx}$$ where $g(x)$ is a continuous function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi g(1)}
Given the function \( f(x,y) = \frac{xy \sin x}{x^2+y^2} \) for \((x,y) \neq (0,0)\) and \( f(0,0) = 0 \), find the directional derivative of \( f \) at the origin in the direction of the unit vector \( \phi = (\phi_1, \phi_2) \). Calculate \( \frac{\partial f(0,0)}{\partial \phi} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\phi_1^2 \phi_2}
Find the asymptotic expression for the average number of prime factors of a number as the number of digits goes to infinity. More formally, determine the asymptotic behavior as $N \to \infty$ of
\[ \frac{\sum_{1\le k\le N} M(k)}{N} \]
where
\[ M(p_1^{d_1}p_2^{d_2}\cdots p_k^{d_k}) = d_1+d_2+\cdots+d_k \]
For example, $M(24) = M(2^3\cdot3) = 4$. Provide your answer in terms of $N$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log \log N}
In a $2009 \times 2009$ grid, each small square contains a real number with an absolute value not exceeding 1. Additionally, for every $2 \times 2$ square, the sum of the 4 real numbers is 0. Determine the maximum possible sum of all the numbers in the grid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2009}
In a group \((G, \cdot)\) with order \(n\), where \(p\) is a prime divisor of \(n\), consider the set \(M\) defined as:
\[M=\left\{(g_1,g_2,\dots,g_p)\in G^p:g_1\cdot g_2\cdots g_p=e_G\right\}\]
Determine the cardinality of the set \(M\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^{p-1}}
In triangle $ABC$, $F$ is on segment $AB$ such that $CF$ bisects $\\angle ACB$. Points $D$ and $E$ are on line $CF$ such that lines $AD,BE$ are perpendicular to $CF$. $M$ is the midpoint of $AB$. If $ME=13$, $AD=15$, and $BE=25$, find $AC+CB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{104}
Suppose $g$ is a function from the positive integers to the positive integers that satisfies $g(n+1) > g(n)$ and $g(g(n)) = 4n$ for all $n$. Find $g(10)$. 

Provide your answer in the form \( \boxed{} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Vertices \( A, B, C \) of an equilateral triangle with side length 1 are on the surface of a sphere with radius 1 and center \( O \). Let \( D \) be the orthogonal projection of \( A \) onto the plane \( \alpha \) determined by points \( B, C, O \). Let \( N \) be one of the intersections of the line perpendicular to \( \alpha \) passing through \( O \) with the sphere. Find the angle \( \angle DNO \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
Two kings are placed on the same square of a 3x3 chessboard. Each king moves independently and randomly to any adjacent square (including diagonally) with equal probability. Determine the probability that, after a large number of moves, both kings will be on the same square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{8}}
Evaluate the integral \( \int\limits_0^\infty e^{-1/x} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}