inclusionAI/Ring-lite-rl-data huggingface dataset:
The circles \( S_1 \) and \( S_2 \) with centers \( O_1 \) and \( O_2 \) respectively touch each other externally; a line touches the circles at distinct points \( A \) and \( B \) respectively. It is known that the intersection point of the diagonals of quadrilateral \( O_1 A B O_2 \) lies on one of the circles. Find the ratio of the radii of the circles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:2}
Find the limit of the ratio of consecutive coefficients in the power series representation of a function that has a simple pole at a non-zero complex number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{a}}
Determine the smallest possible degree of the polynomial $f(x)$ in the expression \( \frac{d^{100}}{d x^{100}} \frac{p(x)}{x^3-x} =\frac{f(x)}{g(x)} \), where $p(x)$ is a polynomial of degree less than 100 and not divisible by $x^3-x$, and $f(x)$ and $g(x)$ are polynomials.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Determine the values of \( n \) for which insertion sort, which runs in \( 8n^2 \) steps, is faster than merge sort, which runs in \( 64n\log_2n \) steps.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \leq n \leq 43}
Evaluate the limit: $$\lim_{n\to\infty}\frac{\int_{0}^1\left(x^2-x-2\right)^n dx}{\int_{0}^1\left(4x^2-2x-2\right)^n dx}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Compute the Euler characteristic of the $n$-sphere $S^n$ for $n \ge 2$ using the standard triangulation of the $(n+1)$-simplex.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + (-1)^n}
Determine the dimension of the vector space $V$ defined by $$ V = \{P \in \mathbb{R}_n [X] : P (X^3)-( X^{n+1} + 1) P (X) =0\} $$ for $n \in \mathbb {N}$. \( \mathbb{R}_n [X] \) denotes the space of real polynomials of degree at most \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A large university has a 13-day registration period for students. The number of elapsed days before a randomly selected student registers follows a continuous distribution with a density function that is symmetric about day 6.5 and proportional to \( \frac{1}{t+1} \) between days 0 and 6.5. A student registers at the 60th percentile of this distribution. Calculate the number of elapsed days in the registration period for this student.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.99}
Sonya the frog chooses a point uniformly at random lying within the square \( \left\lbrack  {0,6}\right\rbrack   \times  \left\lbrack  {0,6}\right\rbrack \) in the coordinate plane and hop
 that point. She then randomly chooses a distance uniformly at random from \( \left\lbrack  {0,1}\right\rbrack \) and a direction uniformly at random fron
 \{north, south east, west\}. All he choices are independent. She now hops the distance in the chosen direction. What is the
 probability that she lands outside the square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{1}{12} $
}
Write down $1, 2, 3, ... , 2015$ in a row on a whiteboard. Every minute, select a pair of adjacent numbers at random, erase them, and insert their sum where you selected the numbers. (For instance, selecting $3$ and $4$ from $1, 2, 3, 4, 5$ would result in $1, 2, 7, 5$.) Repeat this process until you have two numbers remaining. What is the probability that the smaller number is less than or equal to $2015$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\tfrac{63}{2014}$

}
Calculate the contour integral \( \int_{2-i\infty}^{2+i\infty} \frac{1}{t} \, \mathrm{d}t \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i\pi}
The circle with radius $4$ passing through $A$ and $B$ is tangent to the circle with radius $3$ passing through $A$ and $C$, where $|AB|=2$. If the line $BC$ is tangent to the second circle, what is $|BC|$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$ \sqrt{7} $}
Given a constant $C$ where $0 < C < \frac{\pi}{2}$, determine the values of $A_1, A_2, \ldots, A_n > 0$ that minimize the sum $\sum_{i=1}^n \tan(A_i)$ subject to the constraint $\sum_{i=1}^n A_i = C$. Provide the values of $A_1, A_2, \ldots, A_n$ that achieve this minimum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{C}{n}}
Find the remainder when \[\prod_{i=0}^{100}(1-i^2+i^4)\] is divided by $101$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine the values of \(\alpha\) for which the double integral \(\int_{0}^{1}\left(\int_{1}^{1+x} \frac{1}{(x-y)^{\alpha}} \ dy\right)dx\) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha < 2}
Evaluate the line integral \( \oint_C P \, dx + Q \, dy \) using Green's theorem, where \( P(x,y) = \frac{y}{x^2+y^2} \) and \( Q(x,y) = \frac{-x}{x^2+y^2} \), and \( C \) is a piecewise smooth Jordan curve contained in \( S = \{(x,y) | x^2 + y^2 > 0\} \) such that the origin \((0,0)\) lies interior to \( C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi}
In a family of 4, each member exercises at least one of the 7 deadly sins, with no sin being shared between two distinct members. Determine the number of ways to assign the sins to the family members such that no single member exercises both greed and gluttony.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6840}
Calculate the arc length of the curve defined by the parametric equations $x(t) = \frac{\cos(3t)}{t^3}$ and $y(t) = \frac{\sin(3t)}{t^3}$ for $t \geq \frac{1}{\sqrt{3}}$. Use the arc length formula $l = \int_{t_1}^{t_2} \sqrt{(x'(t))^2 + (y'(t))^2} \, dt$ to find the total arc length. Determine the appropriate limits of integration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let $a$ be a prime number, and let $b$ be a positive integer such that $a \equiv b \equiv 1 \pmod{4}$. Given the biconditionals:

$$(A \equiv 1 \pmod{4}) \iff (a \equiv b \pmod{24})$$

$$(A \equiv 3 \pmod{4}) \iff (a \equiv b + 4 \pmod{8})$$

If $a \equiv b \equiv 5 \pmod{12}$, determine the value of $A$ modulo 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given that there is at least one solution to the equation \(a^{2} + 2b^{2} = p^{11}q^{13}\), where \(a, b \in \mathbb{Z}\) and \(p, q\) are odd prime numbers, determine the total number of integer solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{336}
Consider a hat containing slips numbered from \(kr+1\) to \((k+1)r\), where \(k\) is a positive integer. What is the expected number of slips that need to be drawn from the hat until all numbers drawn are unique?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r H_r}
Evaluate the integral \( \int_{-\infty}^\infty (f(x+1)-f(x)) \, dx \) given that \( \lim_{x \rightarrow \infty} f(x) = 2 \) and \( \lim_{x \rightarrow -\infty} f(x) = 1 \). Assume that the function \( f(x) \) is Riemann-integrable over every bounded interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the values of \( s \) for which the integral \( \int_{\mathbb{R}^n}\left(\frac{1}{1+|x|^2}\right)^{s}dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{s > \dfrac{n}{2}}
Determine the minimum expected number of bits required to encode sequences of length 200 from the typical set $S$, where $S$ consists of all sequences $x$ with at most three 1s. Assume $x \in \{0,1\}^{200}$ and $P(0) = 0.99$. Use $\log_{2}|S|$ to encode such sequences.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20.35}
What is the standard notation for the space \(\mathcal{H}_2\), which contains stochastic processes that are adapted and have finite \(L^2\) norm?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathcal{H}^2}
A 10-sided die with numbers 1 to 10 is rolled $n$ times. Determine the smallest value of $n$ such that the probability of rolling each of the numbers 1 to 5 at least once is 50%.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Find all possible values of \(M\) such that the expression \(M^n + 12^{2n-1}\) is divisible by 133 for all positive integers \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
Point \( O \) is located inside an isosceles right triangle \( ABC \). The distances from \( O \) to the vertices \( A \), \( B \), and \( C \) are 6, 4, and 8, respectively. Find the area of triangle \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20 + 6\sqrt{7}}
Determine the value of \( \lim_{N \to \infty} \frac{f(N)}{N} \), where \( f(N) \) is the number of integers \( 1 \leq n \leq N \) such that the number of divisors of \( n \), denoted \( \tau(n) \), is greater than \( \ln{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
What is the result of the Fourier Transform of the conjugate of a function in the Fourier plane, i.e., FFT($T^*(fx,fy)$)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t^*(x, y)}
Al is at $(0, 0)$. He wants to get to $(4, 4)$, but there is a building in the shape of a square with vertices at $(1, 1)$, $(1, 2)$, $(2, 2)$, and $(2, 1)$. Al cannot walk inside the building. If Al is not restricted to staying on grid lines, what is the shortest distance he can walk to get to his destination?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt{5}+\sqrt{13}$}
Determine the kernel of the ring homomorphism \( \varphi : R[u,v] \to R[x,1/x] \) defined by \( \varphi(p(u,v)) = p(x,1/x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(uv - 1)}
Find the limit of the function \( v(x,t) \) as \( t \to \infty \), where \( v(x,t) \) is a bounded solution of the heat equation \( v_t = v_{xx} \) with the initial condition \( v(x,0) = \frac{e^{-x^2}-1}{e^{-x^2}+1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Find the minimal area of a rhombus circumscribed around the region bounded by the curves $y = \cos(x)$ and $y = -\cos(x)$ for $x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. The rhombus is formed by drawing a tangent line to the upper curve at a point $A = (x, \cos x)$, where $x \in \left(0, \frac{\pi}{2}\right)$, and reflecting the intersection points of this tangent line with the axes to form the vertices of the rhombus. What is the minimal area of such a rhombus?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
What is the graph diameter of the 1-skeleton of the omnitruncated $E_8$ polytope?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Consider a block matrix \( C \) defined as follows, where \( m, n \in \mathbb{N} \), \( J \) is an all-ones matrix, and \( A \) is an \( n \times n \) matrix with rank \( n \), such that \( A \neq J, I, J-I \):

\[ C = \begin{pmatrix} (J-I)_{m\times m} & J_{m\times n}\\ J^T_{n\times m} & A_{n\times n}\end{pmatrix} \]

Determine the rank of the matrix \( C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m + n - 1}
Determine the number of group homomorphisms from \(\Bbb Z/2\Bbb Z \times \Bbb Z/2\Bbb Z\) to \(S_3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Solve Burgers' equation \(u_t + uu_x = 0\) with the initial condition \(u(x,-1) = x^2\). Find the value of \(u(1,2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7 - \sqrt{13}}{18}}
Express the integral \( I_{n} = \int_{\mathbb{R}^{n}} \exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x \) in terms of \( I_1 \), where \( \left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots x_{n}^{2}\right)^{\frac{1}{2}} \) is the Euclidean norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I_n = (I_1)^n}
Evaluate the limit: $$\lim_{x\to0}\frac{\Phi^{-1}(1-x)}{\Phi^{-1}(1-x/n)}$$ where $\Phi^{-1}$ denotes the inverse cumulative distribution function (CDF) of the standard normal distribution, and $n>0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find a generator for the cyclic group of units in \( \mathbb{Z}/(\mathbb{2017Z}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the number of non-negative integer solutions to the equation \(a+b+c+d+e=79\) given the constraints \(a \geq 7\), \(b \leq 34\), and \(3 \leq c \leq 41\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{968239}
Evaluate the integral: \[ \int_{-1}^1 x \frac{\sqrt{1-x}}{\sqrt{1+x}} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{2}}
Evaluate the limit: $$\lim_{q\to 1^-}\frac{\pi K'} {K} \cdot\frac{K} {2\pi^2}\cdot(E-k'^2K)$$ where $K, K', E$ are elliptic integrals and $k, k'$ are moduli corresponding to the nome $q$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Evaluate the double integral \( \iint_R (x+y) \, dA \) where \( R \) is the trapezoidal region with vertices at \((0,0), (5,0), (5/2, 5/2), (5/2, -5/2)\). Use the transformation \( x = 2u + 3v \) and \( y = 2u - 3v \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{125}{8}}
Given $2n$ people, each having exactly $k$ friends, find the smallest positive integer $k$ such that it is always possible to divide the $2n$ people into two groups of $n$ people each, where every person in each group has at least one friend in their own group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In triangle $ABC, AB=\sqrt{30}, AC=\sqrt{6},$ and $BC=\sqrt{15}.$  There is a point $D$ for which $\overline{AD}$ bisects $\overline{BC}$ and $\angle ADB$ is a right angle.  The ratio \[ \frac{\text{Area}(\triangle ADB)}{\text{Area}(\triangle ABC)} \] can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers.  Find $m+n.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Determine the function \( f(x) \) such that the differential equation \( xy'' + 2y' - xy = f(x) \) corresponds to the solution \( xy = Ae^x + Be^{-x} + x^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 - x^2}
Find the limit of the expression as \( n \) approaches infinity: $$\lim_{n\rightarrow\infty} \frac{(n+1)^{2n^2+2n+1}}{(n+2)^{n^2+2n+1}n^{n^2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{e}}
Find the least possible sum of the distances from a point E inside a rectangle ABCD to its vertices, given that these distances are all distinct integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
What is the minimum number of states required for a non-deterministic finite automaton (NFA) to recognize the complement of the set of binary strings of length $i$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i + 2}
Consider a population that, in each generation, has the following probabilities: it dies out with probability 0.2, stays constant with probability 0.5, and doubles with probability 0.3. What is the expected long-term behavior of this population?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the Lipschitz constant of the radial retraction function for a norm that does not originate from an inner product.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the probability that when 36 people are randomly split into 12 groups of 3, and then split again into 12 groups of 3, at least one group will have the same members as in the first grouping.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{12}{595}}
A triangle is composed of circular cells arranged in 5784 rows: the first row has one cell, the second has two cells, and so on. The cells are divided into pairs of adjacent cells, so that each cell belongs to exactly one pair. A pair of adjacent cells is called diagonal if the two cells in it aren't in the same row. What is the minimum possible number of diagonal pairs in this division?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2892}
Given four non-coplanar points \(A, B, C, D\) in space where the distances between any two points are distinct, consider a plane \(\alpha\) such that the distances from three of the points to \(\alpha\) are equal, and the distance from the fourth point to \(\alpha\) is twice the distance of one of the three points. Determine the number of such planes \(\alpha\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the number of continuous functions \( f(x) \) defined on \([0,1]\) that satisfy the following conditions: \( I_1 = \int_0^1 f(x) \, dx = 1 \), \( I_2 = \int_0^1 x f(x) \, dx = a \), and \( I_3 = \int_0^1 x^2 f(x) \, dx = a^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
A closed orientable surface $R$ has a genus of 5. If there is a degree 2 covering map from $R$ to another surface $S$, what is the genus of the surface $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the number of different subgroups of order 9 that are isomorphic to $C_3 \times C_3$ in the symmetric group $S_6$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Annie has $24$ letter tiles in a bag; $8$ C’s, $8$ M’s, and $8$ W’s. She blindly draws tiles from the bag until she has enough to spell “CMWMC.” What is the maximum number of tiles she may have to draw?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
18
}
Determine the time complexity of the recurrence relation: \[ T(n) = n + \sum_{k=1}^{n-1} [T(n-k) + T(k)] \] with the base case \( T(1) = 1 \). Find the order of \( T(n) \) without considering a term for \( T(0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(3^n)}
Calculate the surface area of the portion of the cylinder \(x^2 + y^2 = 2ay\) that is contained within the sphere \(x^2 + y^2 + z^2 = 4a^2\), given that \(a > 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16a^2}
In a triangular prism formed by the planes $P_1: 2x + y + z = 2$, $P_2: x + y - z = 3$, and $P_3: x + 2z = 2$, the normal vectors of the planes are co-planar. Determine the area of the base of the prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9\sqrt{14}}{28}}
Determine the eigenvalues of the operator \( T: \ell^2 \to \ell^2 \) defined by \( T(x_1, x_2, x_3, \ldots) = \left(0, x_1, \frac{x_2}{2}, \frac{x_3}{3}, \ldots\right) \), where \( \ell^2 \) is the space of square-summable sequences.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Let \( m = 2007^{2008} \). Determine the number of natural numbers \( n \) such that \( n < m \) and \( n(2n + 1)(5n + 2) \) is divisible by \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the surface integral \( \oint_{\varGamma} \dfrac{\partial \ln r}{\partial \vec{n}} \mathrm{d}s \) where \( \varGamma: x^2+y^2+z^2=R^2 \) and \( r=\sqrt{x^2+y^2+z^2} \). Here, \( \vec{n} \) is the outward unit normal vector on the surface \( \varGamma \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi R}
Evaluate the limit of the integral as \( k \) approaches infinity:

\[ \lim_{k\rightarrow\infty}\int_{-\infty}^\infty\frac{1}{1+x^2+\cdots+x^{2k}}\ \text{dx} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
A planar, 3-regular graph has a proper edge coloring with 3 colors. Every alternating path of red and yellow edges with a length of 6 forms a cycle, and every alternating path of yellow and blue edges with a length of 6 forms a cycle. However, every alternating path of red and blue edges with a length of 4 forms a cycle. How many vertices does this graph have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Square \(CASH\) and regular pentagon \(MONEY\) are both inscribed in a circle. Given that they do not share a vertex, determine the number of intersection points between these two polygons.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Consider the space \(\mathbb{R}^3\). Let \(L\) be the set of all lines and \(P\) be the set of all planes in \(\mathbb{R}^3\). Define topologies \(\tau_1\) and \(\tau_2\) on \(\mathbb{R}^3\) generated by the subbases \(L\) and \(P\), respectively. Determine the relationship between \(\tau_1\) and \(\tau_2\). Are these topologies equivalent, or is one finer than the other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\tau_1 = \tau_2}
Find the specific solution to the differential equation \( y' = \frac{y(y + 2x - 1)}{x + y} \) given the general solution \( y^2 + 2xy - c e^{2x} = 0 \) and the initial condition \( y(0) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y^2 + 2xy = 0}
$ \mathrm{p}19.$How many ways are there to put the six letters in “$MMATHS$” in a two-by-three grid such that the two “$M$”s do not occupy adjacent squares and such that the letter “$A$” is not directly above the letter “$T$” in the grid? (Squares are said to be adjacent if they share a side.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{176}
Evaluate the integral \( \int_{0}^{1} \frac{\ln x}{x^2 + 1} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-G}
Find all subgroups of the multiplicative group of non-zero real numbers, \(\mathbb{R}^*\), that have an index of 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}^+}
A ball with mass 0.15 kg is thrown upward with an initial velocity of 20 m/s from the roof of a building 30 m high. The force due to air resistance is given by \(|v|/30\), directed opposite to the velocity, where \(v\) is the velocity in m/s. Determine the time at which the ball hits the ground.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.13}
Given that \( x, y, \) and \( z \) are real numbers satisfying \( x^2 + z^2 = 1 \) and \( y^2 + 2y(x + z) = 6 \), find the maximum value of \( y(z - x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the number of elements in the group $\left(\mathbb{Z}_{/(2^k\mathbb{Z})}\right)^*$ that have an order of $1$, $2$, or $4$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}