inclusionAI/Ring-lite-rl-data huggingface dataset:
27 identical dice were glued together to form a $3 \times 3 \times 3$ cube in such a way that any two adjacent small dice have the same number of dots on the touching faces. How many dots are there on the surface of the large cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{189}
Given that \( A, B, C, D \) are four non-coplanar points on the surface of a sphere, and \( AB = BC = AD = 1 \), \( BD = AC = \sqrt{2} \), and \( BC \perp AD \), find the volume of the sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{2} \pi}
Suppose that $f(x)=x^2-10x+21$. Find all distinct real roots of $f(f(x)+7)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3,5,7}
Calculate the expected number of tosses to observe the first Head when rolling a fair six-sided die. If the die lands on 1 or 2, toss a fair coin (with probability 0.5 of showing Heads) until a Head appears; otherwise, if the die lands on 3 or more, toss a biased coin (with probability 0.25 of showing Heads) until a Head appears. Express the answer correct to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.33}
Calculate the effect changes in dimension of a rectangle will have on its area, if length is increased by 40% and its breadth is decreased by some percentage. The effect is 5%. What is the percentage decrease in breadth?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25\%}
Evaluate the limit: $$\lim_{n \to 0} \left[\frac{(2020+n)!}{2020!}\right]^{1/n}$$ and provide a numerical approximation for the limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2020.5}
Determine the domain of the function $f(\lg \frac {x^{2}+x}{2})$, given that the domain of the function $f(x)$ is $(0,1]$. Express your answer in interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-5,-2)\cup(1,4]}
在一个圆周上有 12 个红点. 求最小的正整数 n，使得存在以红点为顶点的 n 个三角形，满足以红点为端点的每条弦都是其中某个三角形的一条边.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Gary purchased a large beverage, but only drank m/n of it, where m and n are relatively prime positive integers. If he had purchased half as much and drunk twice as much, he would have wasted only 2/9 as much beverage. Find m+n.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
37
}
In triangle $ \triangle ABC $, the sides opposite angles A, B, C are respectively $ a, b, c $, with $ A = \frac{\pi}{4} $, $ \sin A + \sin(B - C) = 2\sqrt{2}\sin 2C $ and the area of $ \triangle ABC $ is 1. Find the length of side $ BC $.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}
Consider the system
$$
x + y = z + u,
$$
$$
2xy = zu.
$$

Find the greatest value of the real constant $ m $ such that
$ m \leq \frac{x}{y} $
for any positive integer solution $(x, y, z, u)$ of the system, with $ x \geq y $.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$m = 3 + 2\sqrt{2}$

}
Determine whether the random variable \( Y = aX - \lfloor aX \rfloor \), where \( X \sim U[0,1) \) and \( a \geq 2 \) is an integer, follows a uniform distribution \( U[0,1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Y \sim U[0,1)}
Vertices $A$ and $B$ of the prism $A B C A 1 B 1 C 1$ lie on the axis of the cylinder, while the other vertices lie on the lateral surface of the cylinder. Find the dihedral angle in this prism with edge $A B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
There are five courses at my school. 243 take algebra. 323 take language arts. 143 take scocial studies. 241 take biology. 300 take history. 213 take algebra and language arts. 264 take algebra and social studies. 144 take algebra and biology. 121 take algebra and history. 111 take language arts and social studies. 90 take language atrts and biology. 80 take language arts and history. 60 take social studies and biology. 70 take social studies and history. 60 take biology and history. 50 take algebra, language arts and social studies. 50 take algebra, language arts and biology. 50 take algebra, language arts and history. 50 take algebra, social studies and biology. 50 take algebra, social studies and history. 50 take algebra, biology and history. 50 take language arts, social studies and biology. 50 take language arts, social studies and history. 50 take language arts, biology and history. 50 take social studies, biology and history. 20 take algebra, language arts, social studies and biology. 15 take algebra, language arts, social studies and history. 15 take algebra, language arts, biology and history. 10 take algebra, social studies, biology and history. 10 take language arts, social studies, biology and history. 5 take all three. None take none.

How many people are in my school?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
472
}
For a positive sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, and there exists a positive number $t$, such that for any positive integer $n$, $$\sqrt {tS_{n}}= \frac {t+a_{n}}{2}$$ holds. If $$\lim_{n\rightarrow +\infty } \frac { \sqrt {S_{n}}}{a_{n}}<t$$, then the range of $t$ is \_\_\_\_\_\_\_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \sqrt[3]{\frac {1}{4}}, +\infty \right)}
In a probability space \((\Omega, \mathcal{A}, P)\), consider a sequence of identically independently distributed (i.i.d.) random variables \((X_n)_{n \in \mathbb{N}}\), which are not integrable. Define \(C_n = \{|X_n| \geq nK\}\) for each \(n \in \mathbb{N}\), where \(K > 0\). Determine whether it is true that \(P\left(\limsup_{n \to \infty} C_n \right) = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine which of the following sets is recursively enumerable: \( L_1 = \{ M \mid \text{Turing Machine } M \text{ terminates for at least 637 inputs} \} \) and \( L_2 = \{ M \mid \text{Turing Machine } M \text{ terminates for at most 636 inputs} \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{L_1}
Determine whether the function \( f(x,y) = \begin{cases} \frac{x^2y^{\frac{4}{3}}}{x^4+y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} \) is continuous at the point \((0,0)\). Specifically, evaluate whether \( \lim_{(x,y)\to(0,0)} f(x,y) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a trapezoid, the lengths of the parallel sides are \(a\) and \(b\), and the height is \(m\). What is the length of the line parallel to \(a\) and \(b\) that divides the trapezoid into two parts whose areas are in the ratio \(p: q\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{pa^2 + qb^2}{p + q}}}
Determine the radius of convergence of the power series expansion of the function $c(\zeta)$ around the origin, given its implicit equation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
If \( \frac{\partial u}{\partial v} = a \), is it true that \( \frac{\partial v}{\partial u} = \frac{1}{a} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{a}}
$\underline{\text { Folklore }}$

Among the actors of Karabas Barabas theater, a chess tournament was held. Each participant played exactly one game with each of the others. One solido was given for a win, half a solido for a draw, and nothing for a loss. It turned out that among any three participants, there would be a chess player who earned exactly 1.5 solido in games with the other two. What is the maximum number of actors that could have participated in such a tournament?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
There are $10000$ trees in a park, arranged in a square grid with $100$ rows and $100$ columns. Find the largest number of trees that can be cut down, so that sitting on any of the tree stumps one cannot see any other tree stump.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2500}
Find the minimum length of cable required to connect an antenna to the points A, B, C, and D. The antenna must be placed on the same axis as point A, and its position can be between A and B, or behind either A or B. Use the relationship:

\[ D_1 + D_2 = \sqrt{2^2 + X^2} + \sqrt{(12-x)^2+1^2} \]

where \(D_1\) and \(D_2\) are the distances from the antenna to points A and B, respectively. Determine the optimal position of the antenna to minimize the total cable length.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{17}}
For what value of $x$ does

$$\log_{\sqrt{2}}\sqrt{x}+\log_{2}{x}+\log_{4}{x^2}+\log_{8}{x^3}+\log_{16}{x^4}=40?$$


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256
}
$(\text{b})$ Do there exist 21 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\le p \le 13$?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
一台压路机的滚筒长1.5米，直径是6米．如果它每分钟转100圈，那么这种压路机每小时可以压路面多少平方米．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{169560}
Determine whether the sequence \(\left(\dfrac{1}{n^3 \cos n}\right)\) converges or diverges. If it converges, find its limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the largest possible value of the modulus of the entire function $f(z)$ at $z = \ln(2)$, given that $f(z)$ is analytic for all $z \in \mathbb{C}$ and satisfies $|f(z)| \leq |e^z|$ along the contour $|z| = 1$. Can the Maximum Modulus Principle provide a better upper bound than $e$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider a random walk $X_t = \sum_{s=1}^t D_s$ with i.i.d. increments $D_t \in \mathbb{R}^n$, where $X$ is a martingale with $\mathbb{E}[D_t]=\vec{0} \in \mathbb{R}^n$. The support of $D_t$ is bounded, and $D_{t,i}$ has strictly positive variance for all $i \in \{1,\ldots,n\}$. Is it true that the probability that the random walk eventually pointwise falls below some threshold $k < 0$ equals $1$, i.e., $$ \mathbb{P}[ \exists t \text{ such that } X_{t,i} \leq k \text{ for all } i \in \{1,\ldots,n\} ] = 1 $$ for all $k<0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider a random permutation of the set $\{1, 2, . . . , 2015\}$. In other words, for each $1 \le i \le 2015$, $i$ is sent to the element $a_i$ where $a_i \in \{1, 2, . . . , 2015\}$ and if $i \neq j$, then $a_i \neq a_j$. What is the expected number of ordered pairs $(a_i, a_j )$ with $i - j > 155$ and $a_i - a_j > 266$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{651222}
Define a permutation $a_1a_2a_3a_4a_5a_6$ of the set $1, 2, 3, 4, 5, 6$ to be $\text{ factor-hating }$ if $\text{ gcd }(a_k, a_{k+1}) = 1$ for all $1 \leq k \leq 5$. Find the number of $\text{ factor-hating }$ permutations.



Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
72
}
Find a pair of coprime integers \(x\) and \(y\) such that \((p^y + 1) \bmod (p^x + 1) = 0\) for a given prime number \(p\). If such a pair exists, provide an example.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 3)}
Evaluate the limit of the expression obtained after substituting \( x = \frac{1}{t} \) in the original limit expression, given that \( \frac{1}{2} < \sigma < 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider the linear operator $T: l_2 \rightarrow l_2$ defined by \[T(\bar{x}) = (3x_1 + 2x_2, x_3 - 4x_4, 3x_5 + 2x_6, ...)\] where $\bar{x} = (x_1, x_2, x_3, ...,)$ is an element of the space of square-summable sequences $l_2$. Determine whether $T$ is bounded and find bounds for its operator norm $\|T\|$. Provide your answer as a single value or expression for the norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{17}}
Determine if the residue theorem can be used to evaluate the integral \( I = \int_0^{\infty} \frac{dx}{x^6+1} \) using a pie-shaped contour subtending an angle of \( \frac{\pi}{3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
7. In any $m$ consecutive natural numbers, if there must be a number whose sum of digits is a multiple of 6, then the smallest $m$ is


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine whether the function $f \equiv 1$ is the unique normalized solution to the equation \( f(x) = \frac{1}{x} \int_0^x f(t) \, dt \) for \( f \in L^1(0,1) \), where \( f \) is non-negative almost everywhere and satisfies \( \int_0^1 f(x) \, dx = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = 1}
Consider a stochastic process defined as \(x(t) = a \sin(2\pi f_0 t) + b \cos(2\pi f_0 t)\), where \(a\) and \(b\) are independent Gaussian random variables with zero mean and variance \(\sigma^2\). Determine if \(x(t)\) is a Gaussian random process and find its mean function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the improper integral \(\int_0^\infty \frac{\sin x(1-\cos x)}{x^2}dx\) and determine whether it converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
Given that the probability of winning each bet is \( p = \frac{1}{3} \), find the probability that the gambler first goes broke after 5 bets using the generating function derived in the original answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{16}{243}}
A PE class has 12 students, 6 girls and 6 boys. The coach has 4 jerseys in each of 3 colors to mark 3 teams for a soccer tournament. If the coach wants at least one girl and at least one boy on each team, how many ways can he give out the jerseys? (Jerseys of the same color are indistinguishable.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29700}
## Task Condition

Calculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4_{\text {and }}}$ its height dropped from vertex $A_{4 \text { to the face }} A_{1} A_{2} A_{3}$.

$A_{1}(1 ; 2 ; 0)$

$A_{2}(3 ; 0 ;-3)$

$A_{3}(5 ; 2 ; 6)$

$A_{4}(8 ; 4 ;-9)$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7\frac{2}{7}}
Evaluate the limit: $$\lim_{n\to\infty}n\cdot \sum_{m=1}^{\infty}\left(1-\frac{1}{m}\right)^n\cdot \frac{1}{m^2}.$$ Is this limit equal to 1 as $n$ approaches infinity?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Triangle $\vartriangle ABC$ is inscribed in a circle $\omega$ of radius $12$ so that $\angle B = 68^o$ and $\angle C = 64^o$ . The perpendicular from $A$ to $BC$ intersects $\omega$ at $D$, and the angle bisector of $\angle B$ intersects $\omega$ at $E$. What is the value of $DE^2$?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{432}
NT 3. Find the largest integer $k(k \geq 2)$, for which there exists an integer $n(n \geq k)$ such that from any collection of $n$ consecutive positive integers one can always choose $k$ numbers, which verify the following conditions:

1. each chosen number is not divisible by 6 , by 7 and by 8 ;
2. the positive difference of any two different chosen numbers is not divisible by at least one of the numbers 6,7 or 8 .

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
If the radius of convergence of \(\sum_{n=1}^\infty a_n z^n\) is \(R_1\) and the radius of convergence of \(\sum_{n=1}^\infty b_n z^n\) is \(R_2\), where \(R_1 < R_2\), is the radius of convergence of \(\sum_{n=1}^\infty (a_n - b_n) z^n\) equal to \(R_1\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{R_1}
Let $\langle x_n\rangle$ be a positive sequence of real numbers such that $\displaystyle\limsup_{n\to\infty}x_n<\infty$. Let $\langle a_n\rangle$ be a sequence of real numbers such that $\displaystyle\sum_1^{\infty}\frac1{1+a_n}=\infty$. Determine whether $\displaystyle\sum_1^{\infty}\frac1{1+x_na_n}=\infty$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the minimum number of colors necessary to color the integers from $1$ to $2007$ such that if distinct integers $a$, $b$, and $c$ are the same color, then $a \nmid b$ or $b \nmid c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 6 }
12. B6 (FIN 3) Four distinct circles \( C, C_{1}, C_{2}, C_{3} \) and a line \( L \) are given in the plane such that \( C \) and \( L \) are disjoint and each of the circles \( C_{1}, C_{2}, C_{3} \) touches the other two, as well as \( C \) and \( L \). Assuming the radius of \( C \) to be 1, determine the distance between its center and \( L \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Consider the system $x + y = z + u,$ $2xy = zu.$ Find the greatest value of the real constant $m$ such that $m \leq x/y$ for any positive integer solution $(x,y,z,u)$ of the system, with $x \geq y$.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ m = 3 + 2\sqrt{2} \]}
Is it possible to draw some number of diagonals in a convex hexagon so that every diagonal crosses EXACTLY three others in the interior of the hexagon? (Diagonals that touch at one of the corners of the hexagon DO NOT count as crossing.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
An integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition.For example, 4 can be partitioned in five distinct ways:
$4$
$3 + 1$
$2 + 2$
$2 + 1 + 1$
$1 + 1 + 1 + 1$

The number of partitions of n is given by the partition function $p(n)$. So $p(4) = 5$.
Determine all the positive integers so that $p(n) + p(n + 4) = p(n + 2) + p(n + 3)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
n = 1, 3, 5
}
Let $F$ be a free group with a basis $X$, and let $X' = \{xF' : x \in X\}$, where $F'$ is the commutator subgroup of $F$. Determine if $|X| = |X'|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|X| = |X'|}
Is the unit circle the only infinite compact subgroup of the multiplicative group of non-zero complex numbers? Specifically, if $G$ is an infinite subgroup of $(\mathbb{C} \setminus \{0\}, \cdot)$ that is compact as a subset of $\mathbb{C}$, is it true that $G = S^1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S^1}
Verify whether the Fourier coefficient \(B_n\) is zero for all \(n\) when \(f(x) = 0.01\sin{3x}\) using the formula \(B_n=\frac{2}{\pi} \int_0^{\pi} f(x) \sin{(nx)} \, dx\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{B_3 = 0.01}
Let $I$ be the set of points $(x,y)$ in the Cartesian plane such that$$x>\left(\frac{y^4}{9}+2015\right)^{1/4}$$Let $f(r)$ denote the area of the intersection of $I$ and the disk $x^2+y^2\le r^2$ of radius $r>0$ centered at the origin $(0,0)$. Determine the minimum possible real number $L$ such that $f(r)<Lr^2$ for all $r>0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
${\dfrac{\pi}{3}}$}
Is the area of the fundamental parallelogram of the lattice generated by \((n, n\alpha \mod 1)\) always 1, even if \(\alpha = \sqrt{3}\) or another number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let the series \(\sum_{k=0}^\infty a_k x^k\) have a radius of convergence 1. Assume further that \(\sum_{k=0}^\infty a_k = \infty\). Determine whether \(\lim_{x\to 1^-} \sum_{k=0}^\infty a_k x^k = \infty\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the largest positive integer $n$ such that the following statement holds:
If $a_1,a_2,a_3,a_4,a_5,a_6$ are six distinct positive integers less than or equal to $n$, then there exist $3$ distinct positive integers ,from these six, say $a,b,c$ s.t. $ab>c,bc>a,ca>b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ n = 107 }
Identify and define the left adjoint to the functor \( \delta: \text{Set} \to \text{Cat} \) that maps a set to a discrete category consisting of its elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi_0}
Each day Maria must work $8$ hours. This does not include the $45$ minutes she takes for lunch. If she begins working at $\text{7:25 A.M.}$ and takes her lunch break at noon, then her working day will end at

 

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4:10 P.M.}
Let  $H$  be a rectangle with angle between two diagonal  $\leq 45^{0}$ . Rotation  $H$  around the its center with angle  $0^{0}\leq x\leq 360^{0}$  we have rectangle  $H_{x}$ . Find  $x$  such that  $[H\cap H_{x}]$  minimum, where  $[S]$  is area of  $S$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ x = 45^\circ }
In a school, $60\%$ of the students are boys and $40\%$ are girls. Given that $40\%$ of the boys like math and $50\%$ of the people who like math are girls, what percentage of girls like math?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$60\%$}
安迪和鲍勃去食堂买零食。他们花了相同的金额。安迪买了一罐价值1美元的苏打水和两个价值每个2美元的汉堡。鲍勃点了两个价值3美元的三明治和一罐果汁。鲍勃的果汁花了多少钱？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine whether the limit \( \lim_{n \to \infty} a_{n} \) exists for the sequence \( a_{1}, a_{2}, \ldots, a_{n}, \ldots \) defined by the recursion \( a_{n+1} = (2 a_{n})^{1 / a_{n}} \), where \( a_{1} \) is any positive number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For any nonnegative integer $n$, let $S(n)$ be the sum of the digits of $n$. Let $K$ be the number of nonnegative integers $n \le 10^{10}$ that satisfy the equation
 \[
        S(n) = (S(S(n)))^2.
 \]
Find the remainder when $K$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{632}
Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$119$
}
Evaluate the convergence and compute the value of the integral: $$\int_{0}^{\infty} \frac{\ln(x)}{x^2+x+1} \, dx$$ without using advanced techniques such as the calculus of residues.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Compute the differential \(d(x_1x_2)\) using the product rule for the stochastic differential equations \(dx_1 = a_1 dt + b_1 dw_1(t)\) and \(dx_2 = a_2 dt + b_2 dw_2(t)\), where \(w_1\) and \(w_2\) are independent Wiener processes. What is the term \((dx_1)(dx_2)\) when \(w_1\) and \(w_2\) are independent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the signature of the quadratic form given by:
\[ f(x,y,z) = 3x^2 + 4y^2 + z^2 - 9(5xy + 3xz + yz) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 1)}