inclusionAI/Ring-lite-rl-data huggingface dataset:
In how many ways can you rearrange the letters of ‘Alejandro’ such that it contains one of the words ‘ned’ or ‘den’?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
5040
}
Evaluate the limit: \[ \lim_{n\to\infty}\frac{\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}}{\sqrt[n]{n+1}-\sqrt[n]{n}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
$\begin{aligned} & \text{Find the number of integers }n\mathrm{~with~}1\leq n\leq2017\text{ so that }(n-2)(n-0)(n-1)(n-7)\text{is an integer} \\ & \text{multiple of 1001.} \end{aligned}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
99}
Find a divisor of the number $4^{1000} - 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
For what values of \( p \) is the limit comparison test invalid for the sequences \( a_n = 1 - 2^{-1/n} \) and \( b_n = \frac{1}{n^p} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p \neq 1}
Calculate the volume of the solid generated by revolving the region under the first arc of the cycloid, given by the parametric equations \(x(t) = a(t - \sin t)\) and \(y(t) = a(1 - \cos t)\), about the \(x\)-axis. Use the formula \(V = \pi \int [f(y)]^2 \, dy\) to find the volume.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\pi^2 a^3}
Given an integer $k \geq 1$, determine the minimal integer $n$ such that there exist matrices $A_1, A_2, \ldots, A_k \in M_n(\mathbb{C})$ satisfying the conditions $A_i A_j = -A_j A_i$ for all $i \neq j$, and $A_1 A_2 \cdots A_k \neq 0$. Additionally, each $A_i^2$ must be zero. Find this minimal $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^k}
Compute the work done against the force field \( F(x, y) = -\frac{(x, y)}{\sqrt{x^2 + y^2}} \) when a particle is moved from the point \((2a, 0)\) to the origin along the upper half of the circle with the equation \((x - a)^2 + y^2 = a^2\). Use the parametrization \(x = a + a\cos(t)\), \(y = a\sin(t)\) for \(t \in [0, \pi]\) to determine the work done using a line integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2a}
Determine the smallest positive integer \( n \) such that the number of distinct groups of order \( n \) is greater than \( n \) itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Calculate the value of $p + q$ if the integral $$\int_{0}^{\infty} \frac{\log(x)}{x^2 + 2x + 4} \, \mathrm dx$$ equals $\frac{\pi \ln(p)}{\sqrt{q}}$, where $p$ and $q$ are coprime integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Evaluate the limit: \[ \lim_{x\to\infty} \left(\frac{1}{x}\frac{a^x - 1}{a - 1} \right)^\frac{1}{x}, \] where \(0 \leq a\) and \(a \neq 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\max\{a, 1\}}
Determine the ratio of the height to the radius of a cone such that the cone's volume is maximized and its surface area is minimized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Given the functions $f(x)=2x-4,g(x)=-x+4$, then the interval of monotonic increase for the function $y=f(x)g(x)$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty,3]}
Determine the basis or dimension of the vector space $A_k$, consisting of polynomials $f(x_1,x_2,\ldots,x_n)$ in $\mathbb{Q}[x_1,x_2,\ldots,x_n]$ that are symmetric in the first $k$ variables, i.e., $$f(x_{\sigma(1)},\ldots, x_{\sigma(k)},x_{k+1}, \ldots,x_n)=f(x_1,x_2,\ldots,x_n),\,\, \sigma \in S_k.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let \( N \) be the number of ordered pairs of nonempty sets \( \mathcal{A} \) and \( \mathcal{B} \) such that:

- \( \mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\} \),
- \( \mathcal{A} \cap \mathcal{B} = \emptyset \),
- The number of elements in \( \mathcal{A} \) is not an element of \( \mathcal{A} \),
- The number of elements in \( \mathcal{B} \) is not an element of \( \mathcal{B} \).

Find \( N \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{772}
Determine the angle between the planes represented by the factors of the expression \(6x^2+4y^2-10z^2+3yz+4zx-11xy\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
In the Cartesian coordinate system \(xOy\), the set of points \(K=\{(x, y) \mid x, y=-1,0,1\}\). Three points are randomly selected from \(K\). What is the probability that the distance between any two of these three points does not exceed 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["5/14"]}
Find the greatest integer \( n \) such that, regardless of the 100 distinct integers written by Peter, Basil can fill a \( 100 \times 100 \) table with integers so that each number from 1 to \( n \) appears at least once, while ensuring the sum in each \( 1 \times 3 \) rectangle (either vertical or horizontal) equals one of Peter's numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Determine the range of the real number $a$ such that the inequality \(x^{2} + 25 + \left|x^{3} - 5 x^{2}\right| \geq a x\) holds for all \(x \in [1,12]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 10]}
Let $z = 5 + 12i$ be a complex number and $w$ be a complex number such that $|w| = 13$. What is the probability that $|z - w| > 13\sqrt{3}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Find the minimum value of \( \sum_{1\le i<j\le 6}[a_{i}+a_{j}] \) where \( a_{1},a_{2},\cdots,a_{6} \) are real numbers such that \( a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}=2014 \) and \( [x] \) denotes the greatest integer less than or equal to \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10060}
Determine the norm of the differential operator \( \frac{d}{dx} : P^n[0,1] \rightarrow P^{n-1}[0,1] \) in the space of polynomials of degree \( \leq n \) on \([0,1]\), equipped with the sup norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n^2}
Let $\mathcal{A} = A_0A_1A_2A_3\cdots A_{2013}A_{2014}$ be a regular 2014-simplex, meaning the 2015 vertices of $\mathcal{A}$ lie in 2014-dimensional Euclidean space and there exists a constant $c > 0$ such that $A_iA_j = c$ for any $0 \leq i < j \leq 2014$. Let $O = (0, 0, 0, \ldots, 0)$, $A_0 = (1, 0, 0, \ldots, 0)$, and suppose $A_iO$ has length 1 for $i = 0, 1, \ldots, 2014$. Set $P = (20, 14, 20, 14, \ldots, 20, 14)$. Find the remainder when
$$PA_0^2 + PA_1^2 + \cdots + PA_{2014}^2$$
is divided by $10^6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
348595
}
Find the interval of convergence for the Maclaurin series of the function \( f(x) = \ln(1+x) \) using the formula for the remainder term in Taylor's Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1]}
Find the fundamental group \( \pi_1(X) \) of the topological space \( X = (S^1 \times S^2) \setminus (A \cup B) \), where \( A = \{(0,1)\} \times S^2 \) and \( B = S^1 \times \{(0,0,1)\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral \( \int_0^\infty \frac{e^{-x^2}}{(x^2+1/2)^2} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\pi}}
Calculate the limit: $$\lim_{x\to 0}\dfrac{1}{1-\cos(x^2)}{\sum_{n=4}^\infty {n^5 x^n} }$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2048}
Find the supremum of the function \( \frac{f(x)}{g(x)} \), where
\[
f(x) = \dfrac{\beta \exp(-\beta x) \int\limits_{-\infty}^{x} \exp\left(-\frac{(x-\mu)^2}{2 \sigma^2}\right) \,\mathrm{d}x}{\int\limits_{-\infty}^{\infty}\beta \exp(-\beta x) \,\mathrm{d}x \int\limits_{-\infty}^{x} \exp\left(-\frac{(x-\mu)^2}{2 \sigma^2}\right) \,\mathrm{d}x}
\]
and
\[
g(x)=e^{-x} \;.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the number of ways to divide 12 people into groups such that person A and person B are never in the same group, considering all possible group configurations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3535027}
Calculate the limit of the sequence \( b_n = \frac{(n+2)!}{\sqrt{(2(n+2)+1)!}} \cdot \frac{\sqrt{(2n+1)!}}{n!} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
A teacher has 2010 exam questions and wants to divide them into three folders, each containing 670 questions. Each folder is to be given to a student who has solved all 670 questions in that folder. Determine the minimum number of students required in the class, given that for any question, at most two students did not solve it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let \( n \geq 3 \) and consider the linear transformation \( T : p(x) \rightarrow 6p(x) + p'(x) - x^2p''(x) \) on the vector space \( \mathbb{R}[x]_n \). Determine the nullity of \( T \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the expected value of the ratio between the second largest and the largest number in a set of $n$ independent, identically distributed random variables, each following a uniform distribution from $0$ to $1$. Specifically, given $X_1, X_2, \ldots, X_n \sim U[0,1]$, where $x_p = \max_i X_i$ is the largest number and $x_q = \max_{i \neq p} X_i$ is the second largest number, calculate $\mathbb{E}\left[\frac{x_q}{x_p}\right]$. \(n\) is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n-1}{n}}
Given the function \( f(n) \) defined as follows: \( f(n) = n - 2 \) for \( n > 3000 \) and \( f(n) = f(f(n+5)) \) for \( n \leq 3000 \), find the value of \( f(2022) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3001}
Evaluate the integral: \[ \int_0^{\ln 2}\left(\frac{1}{e^x-1} - \left\lfloor\frac{1}{e^x-1}\right\rfloor\right)dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \ln 2}
How many ways are there to divide 30 objects, consisting of 5 pears, 5 apples, 5 doughnuts, 5 lollipops, 5 chocolate cats, and 5 candy rocks, into two unordered piles of 15 objects each?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2166}
Calculate the limit: \[ \lim_{k\to\infty}\int_{\mathbb{R}^2}\frac{(x^2+y^2)^{k/2}}{1+(x^2+y^2)^{(k+3)/2}}\,dx\,dy \] without using the change of variable theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Determine an upper bound on the number of iterations required for the recurrence relation \( R(0) = n \) and \( R(t) = R(t-1) - \max\left(1, \left\lfloor\sqrt{R(t-1)}\right\rfloor\right) \) to terminate, where the recurrence stops when \( R(\cdot)=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{n}}
Calculate the inverse Fourier transform of \( \frac{1}{a+\mathrm{j}w} \) using the definition:
\[
f(t)=\frac{1}{2\pi}\int_{-\infty}^\infty\frac{1}{a+\mathrm{j}w}\mathrm{e}^{\mathrm{j}wt}\,\mathrm{d}w.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-at} u(t)}
Determine the dimension of the smallest faithful irreducible representation of the direct product $(C_3)^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the omega limit set of a trajectory in the dynamical system given by \( \dot x = -y, \dot y = x, \dot z = -x^2-y^2 \) that lies on a helix and travels downward at a constant speed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Let \( p = 101 \). The sum \( \sum_{k=1}^{10}\frac{1}{\binom{p}{k}} \) can be expressed as a fraction of the form \( \frac{a}{p!} \), where \( a \) is a positive integer. Compute \( a \mod p \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In an isosceles trapezoid with bases \( AD \) and \( BC \) where \( AD > BC \), a circle is inscribed such that it touches side \( CD \) at point \( M \). If segment \( AM \) intersects the circle at point \( N \) and the ratio \( AN : NM = k \), determine the ratio of \( AD \) to \( BC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8k - 1}
Find the larger of the two positive integers \(n \ge 3\) for which the expression \(\frac{\log 2 \times \log 3 \times \log 4 \times \cdots \times \log(n-1)}{10^n}\) takes its smallest value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22027}
You are at location \((0,0)\) and know the treasure is within 100 meters of you. Person A claims that they walked 108 meters to the treasure, with their distance calculated by \(|x| + |y|\). Person B claims that they walked 105 meters to the treasure, with their distance calculated by \(\frac{|x|}{\sqrt{3}} + \max\left\{\frac{|x|}{\sqrt{3}}, |y|\right\}\). Determine the number of possible locations for the treasure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the kernel of the ring homomorphism \( f: k[x,y] \to k[t] \) defined by \( f(x) = t \) and \( f(y) = q(t) \), where \( q(t) \in k[t] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(y - q(x))}
Evaluate the triple integral \( \int_{-2}^{-1} \int_{-2}^{-1} \int_{-2}^{-1} \frac{x^2}{x^2+y^2+z^2} \, dx \, dy \, dz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Calculate the limit: $$\lim_{z\to 0}\frac{d^2}{dz^2}\left(\frac{e^zz}{\sin z}\right)$$ without using l'Hôpital's rule or Taylor series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Simplify the expression \((\lambda y. z)((\lambda x. x x)(\lambda x. x x))\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{z}
Evaluate the integral: $$\int\limits_{\frac{1}{e}}^{1} \frac{dx}{x\sqrt{\ln{(x)}}} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the spectrum of the C*-algebra \( B = \bigoplus_{k=1}^\infty \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{N}}
Determine the degree of the expression $\rho_7+\rho_7^2+\rho_7^4$ over $\mathbb{Q}$, where $\rho_7$ is a primitive seventh root of unity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A mother with blood type O and a father with blood type AB have twins, both sons, with blood type B. Given that 32% of all twins have different genders, calculate the probability that the twins are identical.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{13}}
Evaluate the improper integral: $$\int_0^{\infty}\frac{1}{x(1+x^2)}\,dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Express the truncated Riemann zeta function in terms of generalized harmonic numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H_N^{(s)}}
In a scalene triangle $ABC$, the angle bisector $AD$ intersects the circumcircle of the triangle at point $P$. Point $I$ is the incenter of the triangle $ABC$. Given that $ID = DP$, determine the ratio $AI: ID$. (20 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral \( \int_{0}^{\pi/2}\ln(4\sin^2\frac{u}{2})\,du \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2G}
Determine all real values of \( \alpha \) such that the integral \( \int_{0}^{\infty} x(x^2/2 + 1)^{\alpha} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -1)}
Evaluate the integral \( \int_{-\pi/2}^{\pi/2}\log(\cos(\theta))\cos(\theta)\,\mathrm{d}\theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \ln 2 - 2}
Evaluate the limit: \[\lim_{n\to +\infty}\left(\frac{f(a+\frac{1}{n})}{f(a)}\right)^{\frac{1}{n}}\] where \(f: \mathbb{R} \to \mathbb{R}\) is differentiable at \(a \in \mathbb{R}\) with \(f(a) > 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The price of an article is cut \( {10}\% \) . To restore it to its former value, the new price must be
 increased by:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11\frac{1}{9}\%}
Given an $11\times 11$ diagonal matrix $A$ with the characteristic polynomial $(x-3)^4(x-4)(x-7)^4(x-9)^2$, find the dimension of the space $V$ of $11\times 11$ matrices $B$ such that $AB = BA$. Determine the dimension of $V$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Compute the Lebesgue integral \( \int_{0}^{\pi} f(x) \, dx \) where \( f(x) = \begin{cases} \sin x, & \text{if } x \in \mathbb{I} \\ \cos x, & \text{if } x \in \mathbb{Q} \end{cases} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the sign of the integral \( \int_{0}^{2\pi} \frac{\sin x}{x} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{positive}
Let $ p = 2017 $. If $ A $ is an $ n \times n $ matrix composed of residues $ (\mod p) $ such that $ \det A \not\equiv 0 \pmod{p} $ then let $ \text{ord}(A) $ be the minimum integer $ d > 0 $ such that $ A^d \equiv I \pmod{p} $, where $ I $ is the $ n \times n $ identity matrix. Let the maximum such order be $ a_n $ for every positive integer $ n $. Compute the sum of the digits when $ \sum\limits_{k=1}^{p+1} a_k $ is expressed in base $ p $.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
6048
}
There are 2010 lemmings. At each step, we may separate the lemmings into groups of 5 and purge the remainder, separate them into groups of 3 and purge the remainder, or pick one lemming and purge it. Find the smallest number of steps necessary to remove all 2010 lemmings.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
804
}
Evaluate the limit: $$\lim_{x\to 0}\dfrac{1}{1-\cos(x^2)}{\sum_{n=4}^\infty {n^5 x^n} }$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2048}
Determine the minimum sample size \( n \) such that the probability that the sample mean \( \frac{Y_1 + Y_2 + \cdots + Y_n}{n} \) is within two standard deviations of the mean \( \mu \) is at least 98%, given that \( Y_1, Y_2, \ldots, Y_n \) are independent and identically distributed random variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Compute the second-order mixed partial derivative \( \frac{\partial^2 f}{\partial y \partial x} \) at the point \((0,0)\) for the function \( f(x,y) = \frac{xy(x^2-y^2)}{x^2+y^2} \) when \((x,y) \neq (0,0)\), and \( f = 0 \) otherwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Fiona has an open chain with 2018 links. She can cut out links one by one, each time creating 2 or 3 new chains. What is the minimum number of links she needs to remove to be able to pay for any amount from 1 to 2018 links using some combination of her chains?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Evaluate the integral: \[ \int_{0}^{\infty}\dfrac{1-e^{-2t}}{t}\sin{2t}\operatorname d\!t \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Consider a tetrahedron inscribed in a cube. Determine the number of rotation matrices that correspond to rotations of the tetrahedron fixing two of its vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the dimension of the $R$-module $N$ over the ring $R = \mathbb{R}[T]$, where $N$ is generated by elements $u$ and $v$ with the relations $Tu = v$ and $T^{3}u = v$. What is the dimension of $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
For how many integers $n\neq1$ does $\left(n-1\right)^3$ divide $n^{2018\left(n-1\right)}-1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Diagonalize the quadratic form \( q(x_1, x_2, x_3, x_4) = x_1x_2 + x_2x_3 + x_3x_4 \) and determine its signature.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 2)}
Determine the values of $m$ for which the equation \((a_{m-1}+a_1)+\ldots+ 10^{2m-2}(a_{m-1}+a_1) = \frac{10^{2m}-1}{9}\) has a solution, where $a_1$ and $a_{m-1}$ are digits from 0 to 9.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( n \) be the least positive integer for which \( {149}^{n} - {2}^{n} \) is divisible by \( {3}^{3} \cdot  {5}^{5} \cdot  {7}^{7} \) . Find the number of positive divisors of \( n \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
270
}
Find the real value(s) of \(a\) (where \(a \neq -1\)) for which the limit \( \lim_{ n \to \infty} \frac{ 1^a+2^a+\cdots+n^a}{(n+1)^{a-1}[ (na+1)+(na+2)+\cdots+(na+n)]} = \frac{1}{60} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Calculate the Shapley-Shubik power index of player 1 in a weighted majority game with weights \([q;w_1,1,\dots,1]\), where \(q\) and \(w_1\) are whole numbers satisfying \(w_1\leq q\leq n-1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{w_1}{n}}
In a medical diagnostic scenario, consider the following probabilities: $\Pr(C)$ (condition present), $\Pr(N)$ (condition absent), $\Pr(+)$ (test positive), and $\Pr(-)$ (test negative). Additionally, there are eight conditional probabilities related to these events. Given the cyclical pattern in these conditional probabilities, determine the number of degrees of freedom in this framework. How many probabilities must be specified to determine the rest?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the minimum number of distinct non-trivial solutions for the Diophantine equation \(\frac{1}{a} = \frac{1}{x_1} + \cdots + \frac{1}{x_n}\) for any positive integers \(a\) and \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the dimension of a maximal commutative vector subspace of $M_3(\mathbb{C})$. A vector subspace $V$ of $M_3(\mathbb{C})$ is commutative if for every $A, B \in V$, $AB = BA$. Additionally, $V$ is maximal if any larger subspace $W$ containing $V$ has some $A, B \in W$ such that $AB \neq BA$. What is $\dim V$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Compute the discriminant of the number field $K = \mathbb{Q}(\alpha)$, where $\alpha$ is a root of the minimal polynomial $x^3 - x - 1$. The polynomial has degree 3 over $\mathbb{Q}$, with 2 complex roots and 1 real root.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Evaluate the limit: \[ \lim_{x\rightarrow0}\frac{\int_{\cos(\sqrt{x})}^{1}\frac{\cos(t)}{1-t}dt}{\sqrt{x}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}