inclusionAI/Ring-lite-rl-data huggingface dataset:
In trapezoid \(ABCD\), point \(X\) is located on base \(AD\) such that segments \(XB\) and \(XC\) divide the trapezoid into three similar, yet pairwise unequal, non-isosceles triangles. Given that side \(AB\) has a length of 6, determine the value of \(AX \cdot DX\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Given that the area of triangle \( QGH \) is \( 4 \sqrt{2} \) and the area of triangle \( FGH \) is greater than 16, find the angle \( GFH \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
Express the recurrence relation \( t(2^{2^{k}}) = 1 + t(\sqrt{2^{2^k}}) \) in terms of \( k \), i.e., find \( t(k) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k}
Find the supremum of \(|f(1)|\) for all analytic functions \(f\) mapping from \(\mathbb{C} \setminus \{0\}\) to \(\mathbb{C}\) such that \(|f(z)| \leq 5|z|^{-\sqrt{2}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a group of order 120, determine the number of elements of order 12, given that there is at least one Sylow 5-subgroup.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the sum of the series: $$\sum_{n=-\infty}^{\infty} \sin \left(8n+\frac{2\pi}{3}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the contour integral \( \oint_{\gamma}{\Re\left(z\right) \over z - a}\,\mathrm{d}z \) in terms of \(a\), where \(\gamma\) is the unit circle of radius 3 oriented in the positive sense and \(0 < \left\vert a\right\vert < 2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi i a}
Find the smallest value of $P(A) = P(B) = P(C)$ such that $P(A \cap B \cap C) > 0.95$, given that the relationship between the events $A$, $B$, and $C$ is unknown.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{59}{60}}
Determine the number of solutions to the equation \(6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018\), where \(\{x\}\) denotes the fractional part of \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the maximum value of the function \( f(x) \), where \( f(x) \) is a twice differentiable function satisfying the differential equation \( f(x) + f''(x) = -x|\sin x|f'(x) \) for \( x \geq 0 \). Given that \( f(0) = -3 \) and \( f'(0) = 4 \), determine the maximum value of \( f(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the limit: \[ \lim_{x \to \infty} \frac{\exp(2x)}{\exp(s\log(\cosh(\frac{2x}{s})))} \] for \( s > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^s}
A square with side length $2n$ is divided into $4n^2$ smaller squares of equal size. A circle with diameter $2n-1$ is inscribed in the larger square. Determine the number of smaller squares that intersect the circumference of the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8n - 4}
Evaluate the limit: $$\lim_{x\to 0}\frac{\cos^2(1-\cos^2(1-\cos^2(1-\cos^2(\ldots 1\ldots \cos^2 x))\ldots))}{\sin\left(\pi\left(\frac{\sqrt{x+4}-2}{x}\right)\right)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Let $\mathbb{P}$ be the uniform distribution on $\Omega = \{1, 2, \ldots, 10\}$. An event $A \subseteq \Omega$ is said to be non-trivial if $0 < P(A) < 1$. Determine the largest integer $k$ such that there exist non-trivial events $A_1, A_2, \ldots, A_k \subseteq \Omega$ that are independent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
We say that a four-digit number $\overline{abcd}$ is slippery if the number $a^4+b^3+c^2+d$ is equal to the two-digit number $\overline{cd}$. For example, $2023$ slippery, since $2^4 + 0^3 + 2 ^2 + 3 = 23$. How many slippery numbers are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Determine an asymptotically tight bound for the summation \(\sum_{k=1}^n k^u \cdot v^k\), where \(u\) and \(v\) are constants with \(u \geq 0\) and \(v > 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n^u v^n)}
In how many ways can 12 students be assigned to 6 pairs of seats (one in front, one behind) such that students in each pair receive the same exam variant, there are 3 variants available, and no two adjacent pairs receive the same variant?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12! \times 96}
Calculate the sum \( \sum_{k=1}^{2018} a_{k} \), where \( a_{k} \) is the number of integer terms in the sequence \( \log_{2} k, \log_{3} k, \ldots, \log_{2018} k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4102}
Evaluate the contour integral \( \oint_{C}\frac{e^{\frac{1}{z}}}{z-1}dz \) where \( C: |z-1|=\frac{3}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
1. Find the term that does not contain $x$ in the expansion of $\left(\frac{1}{x}+3 x\right)^{n}$, if the coefficient of the power of $x$ in the tenth term is the largest.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{673596}
How many two digit numbers are the product of two distinct prime numbers ending in the same digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the function $f(x)=x+\sqrt{1-x}$, determine the minimum value of $f(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{5}{4}"]}
Calculate the expected time of failure for a machine with two components, X and Y, where Y starts working only after X fails, and the machine fails when Y fails. The joint density function is given by \( f_{X,Y}(x,y)=4e^{-2x}e^{-y} \) for \( 0<x<y< \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{16}{9}}
Determine the values of $a$ for which the series \( \sum_{n=2}^\infty \left( 1+\frac{1}{3} + \dotsb + \frac{1}{n-1} \right) \frac{\sin (na)}{n} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Determine all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the inequality \( f(c)-b^2 \leq f(a+b)-f(a)+f(c) \leq b^2+f(c) \) for all real numbers \( a, b, \) and \( c. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = k}
In a sequence of natural numbers from 1 to 37, each number, starting from the second, divides the sum of all preceding numbers. The number 37 is placed first. Determine the number that occupies the third position in this sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Joe has 1729 unit cubes, which are initially unpainted. He randomly forms two cubes of side lengths 9 and 10 or 1 and 12, and dips them into white paint. This process is repeated, with the side lengths being chosen randomly each time, until every side of every unit cube is painted. After how many repetitions is the expected number of painted faces closest to half of the total possible painted faces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Evaluate the integral \( \int_1^3 e^{2x}\left(1+\{2x\}+\{2x\}^2\right)dx \) using the properties of the fractional part function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^6 - e^2}
Determine the number of sequences \((a, b, c)\) where \(a, b, c \in \{1, 2, \ldots, 2n\}\) that are called *joke* sequences. A sequence is a joke if its shortest term is odd and if only that smallest term, or no term, is repeated. Express your answer in terms of \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4n^3}
Erin the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\dots$, right after arriving at the point $p_n$, if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1}$, he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}$. Thus the sequence of points continues $p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)$, and so on in a counterclockwise spiral pattern. What is $p_{2015}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(13, -22)$

}
On a chessboard with 2 rows and 2019 columns, each cell contains a real number. The numbers in the first row are all distinct, and the numbers in the second row are a permutation of the numbers in the first row. In each column, the sum of the two numbers is rational, and the two numbers are different. What is the maximum number of irrational numbers that can be placed on the chessboard?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4032}
Calculate \( \mathbb{E}(N(1,4] \mid N(3,10] = 7) \) where \((N_t, t \geq 0)\) is a Poisson counting process with rate \(\lambda = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Consider the set $E$ of all positive integers $n$ such that when divided by $9, 10, 11$ respectively, the remainders (in that order) are all greater than 1 and form a non-constant geometric progression. If $N$ is the largest element of $E$, find the sum of the digits of $N$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Evaluate the integral \( \int_{|z|=3} \frac{\mathrm{e}^{1/(1-z)}}{z} \,\mathrm{d}z \) using the Residue Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
A desert expedition camps at the border of the desert, and has to provide one liter of drinking water for another member of the expedition, residing on the distance of $n$ days of walking from the camp, under the following conditions:
$(i)$ Each member of the expedition can pick up at most $3$ liters of water.
$(ii)$ Each member must drink one liter of water every day spent in the desert.
$(iii)$ All the members must return to the camp.
How much water do they need (at least) in order to do that?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$3^n$
}
Given two sequences of sets \(\{A_{n}\}\) and \(\{B_{n}\}\) defined as follows:

- \(A_{1} = \varnothing\)
- \(B_{1} = \{0\}\)
- \(A_{n+1} = \{x + 1 \mid x \in B_{n}\}\)
- \(B_{n+1} = (A_{n} \cup B_{n}) \backslash (A_{n} \cap B_{n})\)

Find all positive integers \(n\) such that \(B_{n} = \{0\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^k}
Determine the number of solutions for \( x \in \{1, 2, \ldots, 1000\} \) that satisfy the congruence \( x^2(x+1)^2 \equiv 0 \pmod{1000} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Calculate the variance of the stochastic integral \( \int_0^4 \Phi(t) \, dW(t) \), where the stochastic process \( \Phi(t) \) is defined as follows:
\[
\Phi(t) = \begin{cases} 
2, & \text{if } t \in [0,1] \\
W(1), & \text{if } t \in (1,2] \\
W(1.5), & \text{if } t \in (2,3] \\
0, & \text{if } t \in (3,4]
\end{cases}
\]
Here, \( W(t) \) represents a Brownian motion. Use Ito isometry to find the variance.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{2}}
Find the smallest positive integer \( k \) such that \( k! \cdot S_{2016} \) is an integer, where \( S_n \) is the sum of the reciprocals of the non-zero digits of all positive integers up to and including \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Use the binomial series to find the value of $X+Y-Z$ in the equation: $$\sum_{n=1}^{\infty}{\frac{\prod_{r=0}^{n-1}{(2-r)}}{4^nn!}}=\left(\frac{X}{Y}\right)^2-Z$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The sequence \(\{\alpha_{n}\}\) is an arithmetic sequence with common difference \(\beta\). The sequence \(\{\sin \alpha_{n}\}\) is a geometric sequence with common ratio \(q\). Given that \(\alpha_{1}\) and \(\beta\) are real numbers, determine the value of \(q\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm 1}
Calculate the limit \( I_{\infty}(N) = \lim_{x\to\infty} \int \prod_{k=1}^N \left(k-\frac{k}{x}\right) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
$k$ is a positive integer,
$R_{n}={-k, -(k-1),..., -1, 1,..., k-1, k}$ for $n=2k$
$R_{n}={-k, -(k-1),..., -1, 0, 1,..., k-1, k}$ for $n=2k+1$.
A mechanism consists of some marbles and white/red ropes that connects some marble pairs. If each one of the marbles are written on some numbers from $R_{n}$ with the property that any two connected marbles have different numbers on them, we call it nice labeling. If each one of the marbles are written on some numbers from $R_{n}$ with the properties that any two connected marbles with a white rope have different numbers on them and any two connected marbles with a red rope have two numbers with sum not equal to $0$, we call it precise labeling.
$n\geq{3}$, if every mechanism that is labeled nicely with $R_{n}$, could be labeled precisely with $R_{m}$, what is the minimal value of $m$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
The minimal value of \( m \) is \( 2n - 1 \).

}
Let an ordered pair of positive integers $(m, n)$ be called regimented if for all nonnegative integers $k$, the numbers $m^k$ and $n^k$ have the same number of positive integer divisors. Let $N$ be the smallest positive integer such that $\left(2016^{2016}, N\right)$ is regimented. Compute the largest positive integer $v$ such that $2^v$ divides the difference $2016^{2016}-N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
10087
}
Let $\{W_t\}_{t\ge 0}$ be a standard 2D Brownian motion starting at $(1,1)\in\Bbb R^2$. Determine the probability that $W_t$ hits the positive half of the $x$-axis before it hits the negative part.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
Evaluate the integral \( \int\int_{R} y^2 \, dx \, dy \) using polar coordinates, where \( R \) is the region in the first quadrant bounded by the curves \( x^2 + y^2 = 16 \) and \( x^2 + y^2 = 4x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14\pi}
Determine the number of 2-3 trees of depth 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6563711232}
Let $V$ be a finite-dimensional vector space over $\mathbb{C}$, and let $T: V \to V$ be a linear transformation with minimal polynomial $x^{13}(x-1)$. Determine the minimal polynomial of the transformation $T^4$. \( \text{Provide your answer as a polynomial.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^4(x - 1)}
Find the imaginary part of the holomorphic function \( f(z) = z\overline{z} + iv(x,y) = x^2 + y^2 + iv(x,y) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2xy}
What is the maximum number of prime implicants for a conditional statement in a Karnaugh map with \( n \) variables?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-1}}
Determine the number of Sylow 2-subgroups of the symmetric group $S_5$. Given that $|S_5| = 120 = 2^3 \cdot 3 \cdot 5$, use the Sylow theorems to find the number of Sylow 2-subgroups.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A box contains 674 apples, 674 oranges, and 674 pears. Determine the number of ways to pack these fruits into two boxes such that each box contains all three types of fruits and the product of the number of apples, oranges, and pears in each box is the same.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
A square $ABCD$ with a side length of $2$ is rotated around $BC$ to form a cylinder. Find the surface area of the cylinder.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16\pi}
Given the function \( F(x, y) = 2x^4 + 3y^3 + 5xy \) and the condition \( F(x, y) = 10 \), find the derivative \( y'(x) \) at the point \( x = 1 \) and \( y = 1 \). Assume a small change in \( x \) from 1 to 0.999. Use the Implicit Function Theorem to determine how \( y \) changes with respect to \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{13}{14}}
Compute
$$\int_0^13x^2dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A board game is played on a hexagonal grid of 19 tiles. A 'traveler' token starts on the center tile. Each turn, a die is rolled to determine which neighboring tile the traveler moves to, with all six directions being equally likely. The game ends on the turn that the traveler leaves the board. What is the expected number of turns until the game ends?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{55}{9}}
Determine the values of $k$ for which the set of vectors $S = \{1+kx, k + x + x^2 + x^3, kx^2 + x^3 \}$ is linearly independent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k \neq 1}
Consider the set $L$ of binary strings of length less than or equal to $9$, and for a string $w$ define $w^{+}$ to be the set $\{w,w^2,w^3,\ldots\}$ where $w^k$ represents $w$ concatenated to itself $k$ times. How many ways are there to pick an ordered pair of (not necessarily distinct) elements $x,y\in L$ such that $x^{+}\cap y^{+}\neq \varnothing$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1250
}
Determine the measure of the set \(S = \{w \in \{0,1\}^\omega \mid \text{for every } N \in \mathbb{N}, w \text{ has a prefix of the form } pp \text{ with } |p| \ge N \}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of right-angled triangles with integer sides that have an inradius of 2009.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Find the minimum value of \( n \) such that the Chebyshev interpolation of \( f(x) = e^x \) on the interval \([-1, 1]\) has an error bound less than \(10^{-10}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine the radius of convergence of the power series \( \sum_{n=-\infty}^{\infty} 3^{-|n|} z^{2n} \), where \( z \in \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Given that $f(k)$ is a strictly increasing function from $\mathbb{Z}^+$ to $\mathbb{Z}^+$ satisfying $f(f(k)) = 3k$, find the sum $f(1) + f(9) + f(99)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
A necklace consists of 100 beads, which are colored red, blue, and green. In any sequence of five consecutive beads, there is at least one blue bead, and in any sequence of seven consecutive beads, there is at least one red bead. Given that the beads are arranged cyclically, what is the maximum number of green beads that can be in this necklace?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Evaluate the sum: $$\frac{1}{2^{99}}\sum_{k=1}^{50} \sum_{r=0}^{k-1}\binom{50}{k}\binom{49}{r}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Calculate the expected time $T_n$ for a bacterium, which divides exponentially with rate $\lambda = 1$, to reach a population of $n$ bacteria. Assume the bacterium initially divides into two at time zero, and each subsequent division time is an exponential random variable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H_{n-1} - 1}
Find the least square approximation of the function \( p(x) = x^{1/3} \) in the subspace \( W = \text{span}\{1, x, x^2 - \frac{1}{3}\} \) of the inner product space \( V = \mathcal{C}[-1, 1] \) with the inner product \( \langle f, g \rangle = \int_{-1}^{1} f(x)g(x) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{7}x}
Evaluate the limit: $$\lim_{n\to \infty} \frac{1}{n}\left(\sqrt[n]{\int_0^{\pi/4}e^{nx}(\tan^{n-1}x+\tan^nx+\tan^{n+1}x)dx}-1\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
There are relatively prime positive integers $m$ and $n$ so that\[\dfrac{\dfrac{1}{2}}{\dfrac{\dfrac{1}{3}}{\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{4}}+\dfrac{\dfrac{1}{3}}{\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{4}}}=\dfrac{m}{n}.\]Find $m+2n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
41}
Consider an $n \times n$ matrix with elements randomly chosen from a uniform distribution over $[-1,1]$. Determine the probability that this matrix is diagonalizable over $\mathbb{R}$ as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $U$ be the open unit disk, and let $f: U \rightarrow U$ be a holomorphic function such that $f(1/2) = 1/3$. Determine the maximum value of $|f'(1/2)|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{32}{27}}
Evaluate the limit \( \lim_{n \to \infty} \frac{\sqrt[n]{a_2a_3\cdots a_{n+1}}}{a_{n+1}} \), where \( \alpha > 1 \) and \( a_n \) is the largest natural number \( k \) such that \( k! \leq \alpha^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{e}}
Evaluate the limit \( \lim\limits_{x\rightarrow\infty} \frac{e^{x/e^x}-1}{\frac{x}{e^x}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The temperature outside a building is stable at $0^{\circ}$ Celsius. At time $t = 0$, the temperature inside the building is $18^{\circ}$ Celsius. The building has a time constant of 4 hours. At $t = \frac{1}{3}$ hour, a heat source is turned on, generating heat at an intensity of 60000 BTU/hr for 1/2 hour. Given that the heat capacity $\gamma$ of the building is $\frac{1}{6}$ degree per thousand BTU, find the temperature $T(t)$ inside the building at $t = 1$ hour. Use the equation $\frac{dT(t)}{dt} = K(M(t) - T(t)) + \gamma H(t)$, where $H(t)$ describes the production of heat, and express $H(t)$ using unit step functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18.53}
Determine the order of growth of the sum \(\sum_{k=1}^n\frac{1}{\sqrt{k+2}}\) as \(n\) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(\sqrt{n})}
Find the greatest common divisor (GCD) of the complex numbers \(4 + 22i\) and \(17 + i\) in the Gaussian integers \(\mathbb{Z}[i]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + 3i}
Use the summation convention to simplify the expression \( g^{1}_{11} + g^{1}_{12} + g^{1}_{21} + g^{1}_{22} + g^{2}_{11} + g^{2}_{12} + g^{2}_{21} + g^{2}_{22} \) and determine the value of \( n \) necessary for this simplification.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $a$ be an element in $GF(2^9)$ such that $a^9 + a^8 = -1$. Determine the order of the subgroup $H = \langle a \rangle$ in $GF(2^9)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{73}