inclusionAI/Ring-lite-rl-data huggingface dataset:
Four particles are situated at the corners of a square with side length 'a' and move at a constant speed 'v'. Each particle moves towards the particle at the next corner. The time when all particles will meet is given by \( \frac{2a}{\sqrt{n} v} \). Find the value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Approximate $\sqrt{f(t)}$ given that the Taylor expansion of $f(t)$ at $t=1$ is $f(t) = c^2(t-1)^2 + o(t-1)^2$, where $c$ is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|c| |t - 1| + o(|t - 1|)}
In a triangle $ABC$, it is given that $\cos 3A + \cos 3B + \cos 3C = 1$. Determine the measure of any one angle in the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
In a party attended by 2015 guests, among any 5 guests, at most 6 handshakes have been exchanged. Determine the maximal possible number of handshakes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1015056}
Compute the correlation coefficient \( r(X_{(1)},X_{(3)}) \) for the order statistics of three independent random variables \( X_1, X_2, \) and \( X_3 \), each following an \( \text{Exp}(1) \) distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{7}}
Evaluate the integral \( \int_{0}^{1}x^{m-1}\exp(-x)\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\gamma(m, 1)}
Determine the maximum number of people that can be arranged in a 2D plane such that no more than 3 people get wet when each person sprays the person closest to them.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Evaluate the limit: \[ \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{\infty}|\ln n-\ln k|\left(1-\frac{1}{n}\right)^k \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the largest known natural number $n$ such that every sequence of length $n$ appears in the base 10 expansion of $\pi$. Provide the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate the covariant derivative of the scalar density of weight 1, \( \rho=\sqrt{|g|} \), without using the formula for the Christoffel symbols.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider the differential form \( a = p_1 dq_1 + p_2 dq_2 - p_1 p_2 dt \) in the space \( \mathbb{R}^5 \) with coordinates \((p_1, p_2, q_1, q_2, t)\). Compute \( da \) and \( da \wedge da \). Evaluate the integral \( \int_S t \, da \wedge da \) where \( S \) is the 4-dimensional surface (with boundary) in \( \mathbb{R}^5 \) defined by the relations: \( p_1^2 + p_2^2 \le 1, \ q_1^2 + q_2^2 \le 2 \) and \( t = 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-12\pi^2}
Compute the line integral \( \int_C ze^{\sqrt{x^2+y^2}} \, ds \) where \( C \) is the curve of intersection of the sphere \( x^2 + y^2 + z^2 = a^2 \) and the plane \( x + y = 0 \), with \( a > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
How many $4$ element subsets of $\{0,1,2,\dots,20\}$ contain their sum modulo $21$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$1152$
}
Calculate the integral \( \int_A \left(\frac{xe^{x^2+y^2}}{x^2+y^2}-1\right)\mathrm{d}x\mathrm{d}y \) over the region \( A=\{(x,y)\in \mathbb{R}^2: 1 \le x^2+y^2\le 4 \} \) using polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3\pi}
Determine the sub-differential of the function \( f(x) = -\sqrt{x} \) at \( x = 0 \), where \( f: [0, 1] \rightarrow \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Compute $\left\lfloor \sum_{k=0}^{10}\limits  \left(3+2\cos\left(\frac{2k\pi}{11}\right)\right)^{10}\right\rfloor \pmod{100}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
In an equilateral triangular prism \( S-ABC \) with height \( SO = 3 \) and a base edge length of 6, a perpendicular is drawn from point \( A \) to the opposite face \( SBC \), with the foot of the perpendicular at \( O' \). On \( AO' \), a point \( P \) is chosen such that \( AP:PO' = 8 \). Calculate the area of the cross-section passing through point \( P \) and parallel to the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Determine the radius of convergence for the function \( f(z) = \frac{z - z^3}{\sin(\pi z)} \) at the point \( z=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral: $$\int_{0}^{1} \ln (1-x^2)\ dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \ln 2 - 2}
Determine the values of \(a\) such that the function \(g(u,v) = u_1v_1 + 5u_2v_2 + au_3v_3 + 2(u_1v_2+u_2v_1) + 3(u_1v_3+u_3v_1) + 4(u_2v_3+u_3v_2)\) defines a scalar product in \(\mathbb{R}^3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > 13}
Determine the number of solutions to the equation \( x^{10} = \begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix} \) in the space of 2x2 real matrices, \( M_{2}(\mathbb{R}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the number of 3x3 matrices over the field of three elements that are conjugate to a diagonal matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2109}
Determine the values of $b$ for which the integral \( \int_{1}^{\infty} \frac{\ln x}{x^b} \cos x \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b > 0}
Determine the limiting value of the expression: $$\sqrt{2 + 3 \sqrt{2 + 4 \sqrt{2 + \cdots \sqrt{2 + (n - 1) \sqrt{2 + n}}}}}$$ as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
What is the probability that a randomly selected subset of the interval [0, 1] is measurable according to the Lebesgue measure?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $L$ be a list of unique elements. Consider a modified algorithm for finding the maximum value in $L$ where we initialize the current maximum to be the first element of $L$. Calculate the expected number of times the current maximum will be reset when we randomly permute the elements of $L$ before running the modified algorithm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H_n - 1}
Given the integral \( w = \int_{xy}^{2x-3y} \frac{du}{\ln(u)} \, du \), find \( \frac{\partial y}{\partial x} \) at \( x = 3, y = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Find the maximum value of \(|z|\) given that \(|z^2 + 2z \cos \alpha| \leq 1\) for all real values of \(\alpha\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\sqrt{5} - 2}}
Find the value of $1006 \sin \frac{\pi}{1006}$. Approximating directly by $\pi=3.1415 \ldots$ is worth only 3 points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.1415875473}
Find the volume of the solid obtained by revolving the region bounded by \( y = \sin\left(\frac{\pi x}{12}\right) \) and the x-axis about the line \( x = -1 \) using the method of cylindrical shells.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{336}
Determine the range of \( \alpha \) for which the improper double integral \[ I = \iint_{x^2+y^2>0} \frac{\sin(x^2+y^2)}{(x^2+4y^2)^{\alpha}}\, dx\, dy \] is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 < \alpha < 2}
Find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(x) = f(x^2 + C) \) for all \( x \in \mathbb{R} \), where \( C > 0 \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = k}
Magnus and Viswanathan play a series of three games with the following conditions:

- Each player's score in each game is a positive integer, and the scores across all games are six different integers.
- The winner's score in each game is at least 25 points. If the winner scores 25, the opponent's score is at most 23. If the winner scores more than 25, the opponent's score is exactly 2 less than the winner's score.
- Viswanathan wins either the first or the second game, but not both.
- Viswanathan wins the third game with a score of 25.
- Each player's score in the second game is the average of their scores in the first and third games.

Determine Magnus's score in the third game.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Let \( \mathbf{F} \) be a vector field defined on \( \mathbb{R}^2 \setminus\{(0,0)\} \) by \( \mathbf{F} (x,y)=\frac{y}{x^2+y^2}i-\frac{x}{x^2+y^2}j \). Consider the curves \( \gamma(t) = (8\cos 2\pi t, 17\sin 2\pi t) \) and \( \alpha(t) = (26\cos 2\pi t, -10\sin 2\pi t) \) for \( t \in [0,1] \). If \( 3\int_{\alpha} \mathbf{F \cdot dr} - 4 \int_{\gamma} \mathbf{F \cdot dr} = 2m\pi \), find the value of \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Evaluate
\[\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\sum_{n=0}^{\infty}\frac{1}{k2^n+1}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Avik holds six identical ropes with their midpoints in his fist, and the ropes' ends lie on opposite sides. Kamrul randomly pairs and ties the ends together from one side and then the other. If the probability of creating a loop by tying all six ropes is \(\frac{a}{b}\), where \(a\) and \(b\) are coprime, find the value of \(a+b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Alice and Bob are playing a coin-flipping game. Alice wins when the sequence "h, h, h" appears, and Bob wins when the sequence "t, h, t" appears. What is the expected number of coin flips until one of them wins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{35}{6}}
Evaluate \( \bigtriangleup ^{6} p(x) \big| _{x=0} \), where \( p(x) = \sum_{k=0}^{10} k^{2} x^{ \underline{k} } \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25920}
Given the probability matrix of a Markov chain:
\[ P = \begin{bmatrix}0&1/2&1/2\\1/3&0&2/3\\1/2&1/2&0\end{bmatrix} \]
find the limit:
\[ \lim_{n \to \infty} P(X_n = 2 \mid X_0 = 1) \] analytically.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
How many ways are there to cover a 3 × 8 rectangle with 12 identical dominoes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{153
}
Find the remainder when $99{,}999^{99}$ is divided by $999{,}999$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{123579}
Let $f$ be an analytic function on the annulus $\{z : 1/2 < |z|<2\}$ except for a simple pole at $z=1$ with residue 1. The Laurent expansions of $f$ on the annuli $\{ z: 1/2<|z|<1\}$ and $\{z: 1<|z|<2\}$ are given by $\sum a_n z^n$ and $\sum b_n z^n$, respectively. Compute $b_n - a_n$ for every integer $n$. Provide your answer as a single expression or value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the asymptotic bound in terms of $n$ for the summation:
\[ \sum_{i=0}^{n}{\frac{P(n, i)}{n^i}} = \sum_{i=0}^{n}{\frac{n!}{(n-i)! \cdot n^i}}. \] Is it $O(n)$, $O(\log n)$, or another bound?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(\sqrt{n})}
Find the smallest positive integer \( k \) such that for any \( k \) vectors \( v_1, v_2, \ldots, v_k \in \mathbb{R}^n \), there exist real numbers \( a_1, a_2, \ldots, a_k \), not all zero, satisfying \( a_1v_1 + a_2v_2 + \cdots + a_kv_k = 0 \) and \( a_1 + a_2 + \cdots + a_k = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + 2}
For any set $S$, let $P(S)$ be its power set, the set of all of its subsets. Over all sets $A$ of $2015$ arbitrary finite sets, let $N$ be the maximum possible number of ordered pairs $(S,T)$ such that $S \\in P(A), T \\in P(P(A))$, $S \\in T$, and $S \\subseteq T$. (Note that by convention, a set may never contain itself.) Find the remainder when $N$ is divided by $1000.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
872
}
Find the uniformly most powerful (UMP) test at significance level \( \alpha = 0.05 \) for the hypothesis test concerning a geometric distribution with probability mass function \( P(X=k) = p^k(1-p) \), where \( 0 < p < 1 \). The null hypothesis is \( H_0: p \leq 0.95 \) and the alternative hypothesis is \( H_1: p > 0.95 \). Determine the rejection region for this test.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X \geq 59}
Determine the values of $n$ for which the binomial expansion of $(A+B)^n$ implies that $A$ and $B$ commute.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 2}
Determine the number of elements in the module \(A = (\mathbb{Z}[i]/I^{2014})\otimes_{\mathbb{Z}[i]}(\mathbb{Z}[i]/J^{2014})\), where \(I\) and \(J\) are the principal ideals of \(\mathbb{Z}[i]\) generated by \(7-i\) and \(6i-7\), respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5^{2014}}
Consider the region $V$ bounded by the paraboloid $z=5-4x^2-4y^2$ and the $xy$-plane. Given that the surface integral of the vector field \( \vec{F}=\nabla\times \vec{G}=2\vec{i}+2y^2\vec{j}+z\vec{k} \) over the circle in the $xy$-plane is 15, find the value of the surface integral over the paraboloid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Find the center of the polynomial ring $R[x, y]$, where $R$ is a non-commutative ring.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Z(R)[x, y]}
Given a rectangular box with a diagonal of length 10 units, determine the maximum possible perimeter of the box.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40\sqrt{3}}
Let $ n$ and $ k$ be positive integers with $ k \geq n$ and $ k - n$ an even number. Let $ 2n$ lamps labelled $ 1$, $ 2$, ..., $ 2n$ be given, each of which can be either on or off. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on).

Let $ N$ be the number of such sequences consisting of $ k$ steps and resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n + 1$ through $ 2n$ are all off.

Let $ M$ be number of such sequences consisting of $ k$ steps, resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n + 1$ through $ 2n$ are all off, but where none of the lamps $ n + 1$ through $ 2n$ is ever switched on.

Determine $ \frac {N}{M}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
The answer is \( 2^{k - n} \).

}
If $G$ is a non-abelian group of order 27, what is the order of its commutator subgroup $G'$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the number of common normals to the curves given by the equations $x^2 + (y-1)^2 = 1$ and $y^2 = 4x$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given real numbers \(x, y, z\), solve the equation \(4^{\sqrt{5x+9y+4z}} - 68 \cdot 2^{\sqrt{5x+9y+4z}} + 256 = 0\). Find the product of the minimum and maximum values of \(x+y+z\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Consider the topological space $[0,\infty]$ with the order topology. Let $E_n = [n, \infty)$. Determine the limit of $\mu(E_n)$ as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the value of \( A = \lim_{x\to +\infty}\left(\frac{x}{x+1}\right)^x + \lim_{x\to +\infty}\left[\frac{x}{x+1}\right]^x \), where \([x]\) is the floor function of \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{e}}
Let \( f: \mathbb{R} \to \mathbb{R} \) be a function satisfying the equation \( f(2xy) + f(f(x + y)) + f(x + y) = xf(y) + yf(x) \) for all real numbers \( x \) and \( y \). Determine the sum of all possible values of \(|f(48)|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{186}
Let ${(a_n)}_{n=0}^{\infty}$ and ${(b_n)}_{n=0}^{\infty}$ 
be real squences such that $a_0=40$, $b_0=41$ and for all $n\geq 0$ the given equalities hold.
$$a_{n+1}=a_n+\frac{1}{b_n} \hspace{0.5 cm} \text{and} \hspace{0.5 cm} b_{n+1}=b_n+\frac{1}{a_n}$$
Find the least possible positive integer value of $k$ such that the value of $a_k$ is strictly bigger than $80$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2460
}
Find a non-zero function \( f(x) \) such that \( f(2x) = f'(x) \cdot f''(x) \), where \( f' \) and \( f'' \) are the first and second derivatives of \( f \), respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^x}
Find the limit of the ratio of consecutive terms in the sequence $b_n$, where $b_n$ is the number of alternating sequences of length $n$ that do not start or end in 0, as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the triple integral \( \iiint_W (x^2 + y^2 + z^2) \, dV \), where \( W \) is the region bounded by the planes \( x = 0 \), \( y = 0 \), \( z = 0 \), \( x + y = 1 \), and \( z = x + y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{5}}
Find an upper bound for the limit: $$\lim_{n\to\infty} \frac{1}{\log(n)}\sum _{k=2}^{2n} \frac{\sin (\cos (2 \pi \log (k)))}{k}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $A$ be a $\mathbb{C}$-algebra and $V$ be an irreducible $A$-module with countable dimension. Determine the dimension of $\text{End}(V)$ as an $A$-module.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $S$ be the set of all permutations of $\\{1, 2, 3, 4, 5\\}$. For $s = (a_1, a_2,a_3,a_4,a_5) \\in S$, define $\\text{nimo}(s)$ to be the sum of all indices $i \\in \\{1, 2, 3, 4\\}$ for which $a_i \u003E a_{i+1}$. For instance, if $s=(2,3,1,5,4)$, then $\\text{nimo}(s)=2+4=6$. Compute \[\sum_{s\in S}2^{\text{nimo}(s)}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
9765
}
Determine the maximum number of distinct seating plans for 15 people in a row of 15 chairs, where two seating plans are considered the same if they share the same adjacent quadruples. For example, if the people are seated in the order 1, 2, 3, 4, 5, ..., 15, then the quadruples 1234, 2345, 3456, ..., 12131415 cannot be used in other plans. What is the maximum number of distinct seating plans possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2730}
There are infinitely many boxes - initially one of them contains $n$ balls and all others are empty. On a single move we take some balls from a non-empty box and put them into an empty box and on a sheet of paper we write down the product of the resulting amount of balls in the two boxes. After some moves, the sum of all numbers on the sheet of paper became $2023$. What is the smallest possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
65
}
Calculate the sum of the series \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{5}}\) to an accuracy of \(10^{-3}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.972}
Evaluate the limit \( \lim_{n\to \infty} \int_0^n (1+x/n)^{-n}\log(2+\cos(x/n))\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log 3}
Determine the smallest number of sets required to generate the power set of a finite set $A$ where $|A| = n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lceil \log_2 n \rceil}
Find the kernel of the adjoint representation \( P: SU_2 \to \text{Aut}_\mathbb{R}(E) \), where \( P(A) \) maps \( X \) to \( AXA^{-1} \). Here, \( E \) is the space of traceless Hermitian matrices of the form:
\[
\begin{bmatrix}x_1 & x_2 + ix_3 \\ x_2 - ix_3 & -x_1\end{bmatrix}
\]
with \( x_1, x_2, x_3 \in \mathbb{R} \). Determine the matrices \( A \) such that \( AXA^{-1} = X \) for all \( X \) in \( E \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{\pm I\}}
Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=1}^{n}\frac{2015^{k}}{\sum_{i=0}^{k-1}2015^i\sum_{l=0}^k2015^l} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the number of all $6$-digit natural numbers such that the sum of their digits is $10$ and each of the digits $0,1,2,3$ occurs at least once in them.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
490
}
Let $f:\mathbb{N}\to\mathbb{N}$ be a function such that\[f(n) =\sum_{abc = n | a,b,c \in \mathbb{N}} ab + bc + ca.\]For example, $f(5) = (1 \cdot 1 + 1 \cdot 5 + 5 \cdot 1) + (1 \cdot 5 + 5 \cdot 1 + 1 \cdot 1) + (5 \cdot 1 + 1 \cdot 1 + 1 \cdot 5) = 33$, where we are summing over the triples $(a,b,c) = (1,1,5), (1,5,1)$, and $(5,1,1)$. Find the last three digits of $f(30^{3})$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{164}
Evaluate the surface integral \( \iint_S \mathbf{A} \cdot \mathbf{n} \, \mathrm{d}S \) using the divergence theorem, where \( \mathbf{A} = y\mathbf{i} + 2x\mathbf{j} - z\mathbf{k} \) and \( S \) is the surface of the plane \( 2x + y = 6 \) in the first octant, cut off by the plane \( z = 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Find the sum of the series \( \sum_{n=1}^{\infty} a_n \), where \( a_n = \frac{n^2 + 1}{4n^2 - 3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the derivative of the expression \( \text{trace} (\Sigma^{\frac{1}{2}} S^{-1}\Sigma^{\frac{1}{2}}) \) with respect to the matrix \( \Sigma \), where both \( \Sigma \) and \( S \) are positive definite matrices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S^{-1}}
Calculate the Lie derivative of the metric tensor \(g_{ab} = \begin{pmatrix} 1 & 0 \\ 0 & x^2 \end{pmatrix}\) along the vector field \(X = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\begin{pmatrix} 0 & 0 \\ 0 & 2x \end{pmatrix}}
Given that $T \sim t_7$, find the value of $t$ such that:
\[ P(-t < T) + P(0 < T) + P(t < T) + P(2t < T) = 1.75. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.356}
Find the sum of the roots of the equation \(\cos(12x)=5\sin(3x)+9\tan^2(x)+\frac{1}{\tan^2(x)}\) for \(0\le x\le 2\pi\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\pi}