inclusionAI/Ring-lite-rl-data huggingface dataset:
10.4. On the side $A C$ of triangle $A B C$, a point $M$ is taken. It turns out that $A M=B M+M C$ and $\angle B M A=\angle M B C+\angle B A C$. Find $\angle B M A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Let the three-digit number \( n = abc \). If the digits \( a \), \( b \), and \( c \) can form an isosceles (including equilateral) triangle, how many such three-digit numbers exist? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{165}
Calculate the covariance function for the process \( V(t) = e^{t}W(e^{-2t}) \) for \( t \geq 0 \), where \( W(t) \) represents standard Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-|s - t|}}
Determine the smallest integer that can replace 13 in the statement: "Every simple planar graph with minimum degree \(\delta \geq 3\) has an adjacent pair with \(\deg(u) + \deg(v) \leq\) [integer]," such that the statement remains true for all planar graphs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
The parabola  $\mathcal P$  given by equation  $y=x^2$  is rotated some acute angle  $\theta$  clockwise about the origin such that it hits both the  $x$  and  $y$  axes at two distinct points.  Suppose the length of the segment  $\mathcal P$  cuts the  $x$ -axis is  $1$ .  What is the length of the segment  $\mathcal P$  cuts the  $y$ -axis?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the exact number of non-negative integer solutions to the following system of equations:
\[ x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 6 \]
\[ 2x_1 + x_2 + x_3 = 4 \]
\[ x_2 + 2x_4 + x_5 = 4 \]
\[ x_3 + x_5 + 2x_6 = 4 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Find a positive integer $n$ such that there exists a subfield of $\mathbb{Q}_n$, where $\mathbb{Q}_n = \mathbb{Q}(\omega)$ and $\omega$ is a primitive $n$th root of unity, that is not a cyclotomic extension of $\mathbb{Q}$. Provide the smallest such $n$. \(\text{Note: A cyclotomic extension is a field extension obtained by adjoining a root of unity to } \mathbb{Q}.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the rate at which the professor writes a letter when there are fewer than \( k \) requests in a continuous-time Markov chain model.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mu}
Given the piecewise function \( f(x) \) defined as:
\[
f(x) = \begin{cases} 
ax(x-1) + b &; x < 1 \\
x + 2 &; 1 \le x \le 3 \\
px^2 + qx + 2 &; x > 3 
\end{cases}
\]
where it is continuous for all \( x \) in \( \mathbb{R} \) except at \( x = 1 \), and \( |f(x)| \) is differentiable everywhere with \( f'(x) \) continuous at \( x = 3 \). If \( |a + p + b + q| = k \), find the value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the volume of the solid formed by the union of triangles generated by the points $A(t) = (t,t^3,t)$, $B(t) = (t,t,t)$, and $C(t) = (0,2t,t)$ for $t \in [0,1]$. Consider the triangle to be empty if the points are collinear.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{15}}
Given propositions $p$ and $q$:
- Proposition $p$: The equation $\frac{x^2}{2} + \frac{y^2}{m} = 1$ represents an ellipse with foci on the $y$-axis.
- Proposition $q$: The function $f(x) = \frac{4}{3}x^3 - 2mx^2 + (4m - 3)x - m$ is strictly increasing on $(-\infty, +\infty)$.
Find the range of values for $m$ if the proposition $(\neg p) \land q$ is true.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, 2]}
Danai is decorating her house for Halloween. She puts 12 plastic skulls all around the house. She has 4 broomsticks, 1 for each side of the front and back doors to the house. She puts up 12 spiderwebs around various areas of the house. Danai puts some pumpkins around the house. She also places a large cauldron on the dining room table. Danai has the budget left to buy 20 more decorations and has 10 left to put up. She will put up 83 decorations in all. What is the ratio of the number of pumpkins to the number of spiderwebs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
Given the function \( L = f \cdot y \) where \( f = X \cdot W + b \), with matrix dimensions \( X.shape=(1, m), W.shape=(m,10), b.shape=(1, 10), y.shape=(10, 1) \), find the derivative \( \frac{\partial L}{\partial W} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X^\top y^\top}
Find the smallest possible value of the expression\[\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor\]
in which $a,~ b,~ c$, and $d$ vary over the set of positive integers.

(Here $\lfloor x\rfloor$ denotes the biggest integer which is smaller than or equal to $x$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
9
}
一副三色纸牌，共有 32 张，其中红黄蓝每种颜色的牌各 10 张，编号分别是 $1,2, \cdots, 10$ ；男有大小王牌各一张，编号均为 0 。从这副牌中任取若干张牌，然后按如下规则计算分值：每张编号为 $k$ 的牌计为 $2^{k}$ 分。若它们的分值之和为 2004 ，则称这些牌为一个"好牌组". 试求"好牌组"的个数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1006009}
How many distinct composition series does the group $D_{12}$ have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Pick out three numbers from $0, 1, \ldots, 9$, such that their sum is an even number and not less than $10$. We have ________ different ways to pick these numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
What is the lowest dimension of a faithful ordinary representation of the projective unitary group $\rm{PU}(d)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d^2 - 1}
Determine the number of integer values of \( y \) such that there are no more than 2021 integer values of \( x \) satisfying the inequality \( \log_2(x + y^2 + 1) - 3^{y^2 + y - 3x} < 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78}
Determine whether the limit \( \lim_{x \to +\infty } \int_{x^2}^{x^3} t \sinh^{-1}(t^4) \, dt \) converges to a finite value or diverges. If it converges, find the limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let $X$ be a smooth projective curve of genus $3$. Let $K$ be a canonical divisor with degree $4$. If $|K|$ induces a morphism into $\mathbb{P}^{2}$, determine the degree of this morphism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $H$ be a Hilbert space, and let $y$ and $z$ be fixed vectors in $H$. Define the operator $T: H \to H$ by $T(x) = \langle x, z \rangle y$ for all $x \in H$. Calculate the operator norm $\|T\|$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\|y\| \|z\|}
Find the smallest integer \( n \) such that there are no twin primes between \( n \) and \( 2n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Kelvin the Frog and $10$ of his relatives are at a party. Every pair of frogs is either [i]friendly[/i] or [i]unfriendly[/i]. When $3$ pairwise friendly frogs meet up, they will gossip about one another and end up in a [i]fight[/i] (but stay [i]friendly[/i] anyway). When $3$ pairwise unfriendly frogs meet up, they will also end up in a [i]fight[/i]. In all other cases, common ground is found and there is no fight. If all $\binom{11}{3}$ triples of frogs meet up exactly once, what is the minimum possible number of fights?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Given $k\in R$, if the equation $x^{2}+\left(k+2i\right)x+2+ki=0$ has a real root $a$, then $a=\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±\sqrt{2}}
Line $DE$ cuts through triangle $ABC$, with $DF$ parallel to $BE$. Given that $BD = DF = 10$ and $AD = BE = 25$, find $BC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Evaluate the integral \( \int_1^4 x^2 \,\text{d}\left[\frac{x}{3}\right] \) given that for \( n = 1 \), we have \( 1 \leq \frac{x}{3} < 2 \) and \( \left[\frac{x}{3}\right] = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the value of x. (47% of 1442 - some percentage of 1412) + 63 = x. If x is 3, what is the percentage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52.25\%}
Brian starts at the point $\left(1,0\right)$ in the plane. Every second, he performs one of two moves: he can move from $\left(a,b\right)$ to $\left(a-b,a+b\right)$ or from $\left(a,b\right)$ to $\left(2a-b,a+2b\right)$. How many different paths can he take to end up at $\left(28,-96\right)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
70
}
Divide the numbers 1, 2, 3, …, 49, 50 into 10 groups arbitrarily, each group containing 5 numbers. Take the median (the middle number) of each group. What is the maximum possible sum of these 10 medians?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{345}
Given that the perimeter of triangle ABC is 20, the area is 10√3, and angle A is 60°, find the length of side BC. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Triangles $ABC$ and $AFG$ have areas $3009$ and $9003,$ respectively, with $B=(0,0), C=(331,0), F=(800,450),$ and $G=(813,463).$ What is the sum of all possible $x$-coordinates of $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1400}
Jason's shelves can hold either 45 regular-sized books or 30 large-sized books each. If he has 240 regular-sized books and 75 large-sized books, how many shelves will Jason need to store all of his books?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Rectangle $ABCD$ has side lengths $AB = 6\sqrt{3}$ and $BC = 8\sqrt{3}$. The probability that a randomly chosen point inside the rectangle is closer to the diagonal $\overline{AC}$ than to the outside of the rectangle is $\frac{m}{n}$, 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{17}
Walking at 5/6th of its usual speed, a cab is 15 minutes late on a journey. If the cab encounters three traffic signals on the way, where the waiting times at each signal are 2 minutes, 4 minutes, and 5 minutes respectively, and the cab's speed varies by 10% due to road conditions, find the usual time it takes to cover the journey without considering traffic signals and varying speeds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Find the minimum positive period of the function f(x) = sin(x) * (4cos^2(x) - 1). Express your answer in radians as a fraction of pi, if applicable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\pi}{3}}
Find with all integers  $n$  when   $|n^3 - 4n^2 + 3n - 35|$   and  $|n^2 + 4n + 8|$   are prime numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n = 5}
A non-equilateral triangle  $\triangle ABC$   of perimeter  $12$  is inscribed in circle  $\omega$  .Points  $P$  and  $Q$  are arc midpoints of arcs  $ABC$  and  $ACB$  , respectively. Tangent to  $\omega$  at  $A$  intersects line  $PQ$  at  $R$ .
It turns out that the midpoint of segment  $AR$  lies on line  $BC$  . Find the length of the segment  $BC$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the minimum value of \(|z-i| + |z+i| + |z-7i| + |z+3i|\) for \(z \in \mathbb{C}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the immediate basin of attraction for the fixed point $1$ using Newton's method for the function $f(x) = 2 - \frac{1}{x}$. Provide your answer as a set of points or interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \infty)}
Determine the smallest integer \( n \) such that there exists a set \( \{g_1, g_2, \ldots, g_n\} \subset SO_4(\mathbb{R}) \) for which the subgroup generated by this set is dense in \( SO_4(\mathbb{R}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Shayan is playing a game by himself. He picks [b]relatively prime[\/b] integers $a$ and $b$ such that $1\u003Ca\u003Cb\u003C2020$. He wins if every integer $m\\geq\\frac{ab}{2}$ can be expressed in the form $ax+by$ for nonnegative integers $x$ and $y$. He hasn\u0027t been winning often, so he decides to write down all winning pairs $(a,b)$, from $(a_1,b_1)$ to $(a_n,b_n)$. What is $b_1+b_2+\\ldots+b_n$?\n\n[i]2020 CCA Math Bonanza Tiebreaker Round #2[\/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
The answer is \( 1020108 \).}
For any positive integer $n$, let $f(n)$ be the maximum number of groups formed by a total of $n$ people such that the following holds: every group consists of an even number of members, and every two groups share an odd number of members. Compute $\sum_{n=1}^{2022} f(n) \mod 1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{242}
A rower, passing under a bridge, lost his hat. 15 minutes later he noticed the loss and caught the hat one kilometer from the bridge. What is the speed of the river current?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \text{ km/h}}
From a container filled to the brim with $100\%$ juice, fifth-grader Masha drank 1 liter of juice in a day and in the evening refilled the container with 1 liter of water. The next day, after thoroughly mixing the contents, she drank 1 liter of the mixture and in the evening refilled the container with 1 liter of water. On the third day, after mixing the contents again, she drank 1 liter of the mixture and in the evening refilled the container with 1 liter of water. The next morning, her parents discovered that the volume of water in the container was 1.5 liters more than the volume of the remaining juice. How many liters of juice did Masha ultimately drink? If the answer is ambiguous, provide the sum of all possible values of the desired quantity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.75}
Calculate the sum of the series: $$\sum_{n=1}^\infty \frac{\ln^{5} (2n^{7}+13) + 10 \sin n}{n \ln^6 (n^{7/8} + 2 \sqrt n - 1) \ln (\ln (n + (-1)^{n}))}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\sin^2(\pi Rx)}{R(\pi x)^2}dx \) for \( R > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A particle of mass $m$ is trapped in an infinitely deep one-dimensional potential well between $x = 0$ and $x = a$. If the particle is initially described by the wave function $$ w(x,t=0) = \frac{1}{\sqrt{2}} \sin \left(\frac{3\pi x}{a} \right) \cos\left(\frac{5\pi x}{a} \right), $$ what is the energy of the particle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{17 \pi^2 \hbar^2}{ma^2}\)}
Given a set of $n$ elements, $\mathbb{A} = \{1, ..., n\}$, where $k$ elements are fixed. Consider a random permutation $\pi : \mathbb{A} \rightarrow \mathbb{A}$, which can be represented by the product of cycles. What is the probability that all $k$ fixed elements belong to a single cycle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{k}\)}
Determine the values of $a \in \mathbb{R}$ for which the integral \( \int_1^\infty x^a e^{-x^3 \sin^2 x} \, dx \) is finite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, \frac{1}{2})}
For a real number \( x \), let \( [x] \) be \( x \) rounded to the nearest integer and \( \langle x \rangle \) be \( x \) rounded to the nearest tenth. Real numbers \( a \) and \( b \) satisfy \( \langle a \rangle + [b] = 98.6 \) and \( [a] + \langle b \rangle = 99.3 \). Compute the minimum possible value of \( [10(a+b)] \).

(Here, any number equally between two integers or tenths of integers, respectively, is rounded up. For example, \( [-4.5] = -4 \) and \( \langle 4.35 \rangle = 4.4 \).)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{988}
Determine the winding number of the circle defined by \( r(t) = 1 + 2e^{2it} \) for \( t \in [0, 2\pi] \) around the point \( z = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral \( \int_0^\pi \cos^4\theta \sin^3\theta \,d\theta \) using de Moivre's theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{35}}
The lifespan of incandescent bulbs produced by a bulb factory is denoted as $X$, where it is known that $X \sim N(1000, 30^2)$. To ensure that the probability of the bulbs' average lifespan being 1000 hours is 99.7%, what should be the minimum lifespan of the bulbs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{910}
Say that a 4-digit positive integer is \textit{mixed} if it has 4 distinct digits, its leftmost digit is neither the biggest nor the smallest of the 4 digits, and its rightmost digit is not the smallest of the 4 digits. For example, 2013 is mixed. How many 4-digit positive integers are mixed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1680}
Given an affine subspace $H \subset \mathbb{R^3}$ defined by the equation $x_1 + 2x_2 - x_3 = 2$, and a line $L$ in $\mathbb{R^3}$ with the parametric equation $(0,1,1) + t(2,0,1)$, find the equation of the plane $f(H_1)$, where $H_1$ is another plane defined by the equation $2x_1 - x_2 + x_3 = 1$. The symmetry $f$ reflects $L$ about $H$. What is the equation of the plane $f(H_1)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7x_1 - x_2 + 2x_3 = 5}
Determine the number of 4x4 matrices over the field of two elements that are conjugate to a diagonal matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{802}
Evaluate the limit: $$\lim_{n\to\infty}\prod_{k=1}^\infty \left(1-\frac{n}{\left(\frac{n+\sqrt{n^2+4}}{2}\right)^k+\frac{n+\sqrt{n^2+4}}{2}}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
In $\triangle ABC$, the sides $a$, $b$, and $c$ correspond to the internal angles $A$, $B$, and $C$ respectively. Given that $b=2$, $c=2\sqrt{2}$, and $C=\frac{\pi }{4}$, find the area of $\triangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}+1}
Given that there are 6 male doctors and 3 female nurses who need to be divided into three medical teams, where each team consists of two male doctors and 1 female nurse, find the number of different arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{540}
Find the maximum value of \( a \) such that \( \mathbb{E}[\exp(Z^a)] < +\infty \), where \( Z \sim \mathcal{N}(0,1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of $2 \times 2$ matrices $A$ with entries in the finite field $\mathbb{Z}/3\mathbb{Z}$ such that $A^{50} = I$, where $I$ is the identity matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Determine the dimension of the tangent space at a point \((X_0, Y_0)\) on the set of pairs \((X, Y)\) of real \(n \times k\) matrices \((k \le n)\) defined by the condition \(X^T Y = I\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2nk - k^2}
Determine the values of \( p \in [1, \infty) \) for which the function \( f: \mathbb{R}^d \to \mathbb{R} \) defined by \( f(x) = \begin{cases} \frac{1}{|x|(1+|x|)^d}, & x \neq 0 \\ 0, & x = 0 \end{cases} \) belongs to \( \mathbb{L}^p(\mathbb{R}^d) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, d)}
6. (5 points). Point $\boldsymbol{D}$ lies on the extension of side $\boldsymbol{A} \boldsymbol{C}$ of triangle $\boldsymbol{A B C}$, the area of which is equal to $\boldsymbol{S}$; point $\boldsymbol{A}$ is located between $\boldsymbol{D}$ and $\boldsymbol{C}$. Let $\boldsymbol{O}$ be the point of intersection of the medians of triangle $\boldsymbol{A B C}$. It is known that the area of triangle $\boldsymbol{D O C}$ is $\boldsymbol{S}_{l}$. Express the area of triangle $\boldsymbol{D O B}$ in terms of $\boldsymbol{S}$ and $\boldsymbol{S}_{\boldsymbol{l}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2S_{1}-\frac{S}{3}}
Determine the first digit of \((13)^{139}\) without directly computing the entire number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
For which values of $p \in [1, \infty]$ is the function $f(x) = \begin{cases} \frac{1}{|x|(1+|x|)^d}, & x \neq 0, \\ 0, & x = 0 \end{cases}$ in $L^p(\mathbb{R}^d)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, d)}
A lane runs perpendicular to a road that is 64 feet wide. If a pole 125 feet long can be carried horizontally from the road into the lane, what is the minimum width of the lane?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Determine the degree of the field extension $F(x)$ over $F(x^n)$, where $F$ is a field and $x$ is transcendental over $F$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Find the residue of $$\frac{e^z}{z^3\sin(z)}$$ at $z=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
Given $\frac{1}{x}+\frac{1}{y}=2$, then the value of the fraction $\frac{2x+5xy+2y}{x-3xy+y}$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-9}
Given a rectangular prism with dimensions 4 x 4 x 3, find the shortest distance from vertex D to the plane ABC, where A, B, C, and D are adjacent vertices. Express your answer as a decimal value rounded to one decimal place.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.1}
The extreme value of the function $f\left(x\right)=\frac{1}{3}x^{3}+3x^{2}+5x+2$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{3}}
Two boys start moving from the same point $A$ on a circular track but in opposite directions. Their speeds are $5$ ft. per second and $9$ ft. per second. If they start at the same time and finish when they first meet at the point $A$ again, find the number of times they meet, excluding the start and finish.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Calculate the moment of inertia with respect to the $z$-axis for the surface $S$, which is the boundary of the upper half of the solid cone defined by $x^2 + y^2 < z^2 < 2$ in the $xyz$-space. Assume the surface has a constant density of $3$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\pi(1 + \sqrt{2})}