inclusionAI/Ring-lite-rl-data huggingface dataset:
Bob, a bomb-defuser, encounters a bomb with two wires: red and green. Initially, he decides to cut the green wire. However, every second from time \( t = 15 \) seconds down to \( t = 1 \) second, there is a \(\frac{1}{2t^2}\) chance that Bob, due to panic, will switch his wirecutters to the other wire. At \( t = 1 \) second, Bob will cut the wire his wirecutters are currently on without switching. What is the probability that Bob cuts the green wire?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{23}{30}}
If $A$ and $B$ are two positive definite matrices such that $A < B$, determine the relationship between $A^{-1}$ and $B^{-1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A^{-1} > B^{-1}}
A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105263157894736842}
A frog is positioned at the origin of the coordinate plane. From the point $(x, y)$, the frog can jump to any of the points $(x + 1, y), (x + 2, y), (x, y + 1)$, or $(x, y + 2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0, 0)$ and ends at $(4, 4)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{556}
Let $ n$ be a positive integer. Find the number of pairs $ P,Q$ of polynomials with real coefficients such that
\[ (P(X))^2+(Q(X))^2=X^{2n}+1\]
and $ \text{deg}P<\text{deg}{Q}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2^{n+1}$

}
Find the maximum value of the function \( f(x) = 12x^2\int_0^1 yf(y) \, dy + 20x\int_0^1 y^2 f(y) \, dy + 4x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{8}}
Let \( n \geq 3 \) be an odd integer. Define \( [\minus{}n,n] \) as the set of all integers from \( -n \) to \( n \), inclusive. Player \( A \) selects a positive integer \( k \), and player \( B \) chooses a subset \( S \) of \( k \) distinct elements from \( [\minus{}n,n] \). Player \( A \) wins if every number in \( [\minus{}n,n] \) can be expressed as the sum of exactly \( n \) distinct elements from \( S \). Otherwise, player \( B \) wins. Determine the smallest value of \( k \) such that player \( A \) can always win.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n}
Find the last two digits of the number \( A = (2016^{2015^{2014}} + 2014^{2015^{2016}} + 2017)^{2017} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77}
Suppose there are ten genes that determine tallness, each represented by two bits (either 0 or 1). Each gene can be 00, 01, 10, or 11, and a person's height is determined by the number of '11' genes they possess. If both parents have a height of 7, meaning they each have seven '11' genes, what is the expected height of their offspring, assuming the bits of the three remaining non-'11' genes are uniformly distributed? Inheritance is such that for each gene, one bit is randomly chosen from the male and the other from the female.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{22}{3}}
A club has increased its membership to 12 members and needs to elect a president, vice president, secretary, and treasurer. Additionally, they want to appoint two different advisory board members. Each member can hold only one position. In how many ways can these positions be filled?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{665,280}
In an equilateral triangle $ABC$, three lines are drawn perpendicular to each side, intersecting within the triangle. Blue points are placed such that any line connecting two blue points is parallel to one of the sides of the triangle. Each perpendicular line divides the parallel lines it intersects into three segments, with each segment containing at most one blue point. What is the maximum number of blue points that can be placed within the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let $m = 30030$ and let $M$ be the set of its positive divisors which have exactly $2$ prime factors. Determine the smallest positive integer $n$ with the following property: for any choice of $n$ numbers from $M$, there exist 3 numbers $a$, $b$, $c$ among them satisfying $abc = m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Determine whether there exists a field of characteristic 0 that is not algebraically closed and cannot be ordered to form an ordered field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Q}(i)}
Evaluate the surface integral \( \iint_{S} \vec{F} \cdot \text{d}S \) where \( \vec{F} = 3x\hat{i} + 2y\hat{j} - 5z\hat{k} \), and \( S \) is the portion of the surface \( y = x^2 + z^2 \) that lies in the region \( y \leq 1 \), oriented with the normal vector pointing in the positive \( y \)-direction. Use Gauss' theorem to find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Find the smallest constant \( k \) such that
 \( \frac{x}{\sqrt{x + y}} + \frac{y}{\sqrt{y + z}} + \frac{z}{\sqrt{z + x}} \leq  k\sqrt{x + y + z} \)
 for all positive \( x,y,z \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$k = \frac{5}{4}$}
Evaluate the limit \( \lim_{n\to \infty}\left(\dfrac{1}{\sqrt{n^2}}+\dfrac{1}{\sqrt{n^2+1}}+\cdots+\dfrac{1}{\sqrt{n^2+2n}}\right) \) using the Squeeze Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
## 265. Math Puzzle $6 / 87$

For a motorcycle, the optimal rotational speed of the engine shaft is $6000 \mathrm{U} / \mathrm{min}$.

What is the gear ratio of the rear axle to the engine shaft produced by the transmission if the optimal rotational speed results in a speed of $60 \mathrm{~km} / \mathrm{h}$?

The diameter of the rear wheel is $53 \mathrm{~cm}$ (circumference 1.67 m).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:10}
Let \( n \geq 3 \) be an integer. \( n \) balls are arranged in a circle and labeled with numbers \( 1, 2, \ldots, n \) in order. There are 10 colors, and each ball is colored with one of them such that two adjacent balls have different colors. The sum of the labeled numbers of balls of each color is the same for all colors. Find the minimum possible value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Determine the number of non-equivalent formulas that can be constructed using the propositions \( p_1, p_2, \ldots, p_n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2^n}}
Aerith and Bob take turns picking a nonnegative integer, each time changing exactly one digit from the other's last number. The first person to pick a number that (s)he picked before loses. If Aerith goes first, and both play optimally, who wins?

(Note: There are no leading zeroes, except in the number 0 itself. For instance, if one person picks 2020, the other could respond by picking 0020 = 20, however the reverse does not hold.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Bob}}
On a game show, Merble will be presented with a series of 2013 marbles, each of which is either red or blue. Each time he sees a marble, he can either keep it or pass, but cannot return to a previous marble; he receives 3 points for keeping a red marble, loses 2 points for keeping a blue marble, and gains 0 points for passing. All distributions of colors are equally likely, and Merble can only see the color of his current marble. If his goal is to end with exactly one point and he plays optimally, what is the probability that he fails?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2^{2012}}}
For a vector field $F(x,y)=\left(-\frac{y}{x^2 + y^2} , \frac{x}{x^2+y^2}\right)$ in $\mathbb{R}^2$, find the value of the line integral $\int_{\vert x \vert + \vert y \vert = 5 } -\frac{y}{x^2 + y^2} dx + \frac{x}{x^2 + y^2}dy $.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Evaluate the sum \( \sum_{k=0}^{2013} 2^k a_k \) for the sequence \( a_0, a_1, a_2, \ldots, a_n \) defined by \( a_0 = 2013 \) and \( a_n = -\frac{2013}{n} \sum_{k=0}^{n-1} a_k \) for \( n \geq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2013}
Let $r,s,t$ be the roots of $x^3+6x^2+7x+8$. Find
$$(r^2+s+t)(s^2+t+r)(t^2+r+s).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1100
}
If \(M\) is a linear operator on \(\mathbb{R}^3\) with unique and real eigenvalues \(\lambda_1 < \lambda_2 < \lambda_3\), and there exists a vector \(x \in \mathbb{R}^3 \setminus \{0\}\) such that \(\lim_{n \to \infty} ||M^n x|| = 0\), what are the possible values of \(\lambda_1\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1)}
Find the inverse of the element \(1010\) in the field \(GF(2^4)\) with multiplication modulo \(x^4 + x^3 + 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1011}
In a parallelogram \( ABCD \), \( \angle B = 111^\circ \) and \( BC = BD \). Point \( H \) is on segment \( BC \) such that \( \angle BHD = 90^\circ \). If \( M \) is the midpoint of side \( AB \), find the measure of angle \( AMH \) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132}
Calculate the sum of the series \(\sum_{k=0}^{n}(-1)^{k-1}k\binom{n}{k}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the interval of convergence of the series \(\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{4^n \ln(n)}.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-4, 4]}
Among a group of 120 people, some pairs are friends. A [i]weak quartet[/i] is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4769280}
Evaluate the integral:

$$\int_0^{2\pi}\frac{\cos^2\left(\frac{N+1}{2}x\right)}{2\pi(N+1)\cos^2(x/2)} dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider the interval $[0,1]$. At each stage, remove the closed middle third interval from each remaining segment. For example, from $[0,1]$, remove $[rac{1}{3}, rac{2}{3}]$ to get $[0, rac{1}{3}) \cup (\frac{2}{3}, 1]$. Continue this process indefinitely. How many elements are left in the intersection of all these sets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
Given $\sqrt{a-5}+\left(b-3\right)^{2}=\sqrt{c-4}+\sqrt{4-c}$, then the simplified result of $\frac{\sqrt{c}}{\sqrt{a}-\sqrt{b}}$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}+\sqrt{3}}
A player starts with $X and plays a betting game where they have a 49.8% chance to double their bet and a 50.2% chance to lose it. If the player's total wealth increases to $X + Y, what is the expected total amount of money they must wager to return to $X? For example, if a player starts with $100 and their wealth increases to $150, calculate the expected total amount they must wager to return to $100.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250Y}
Let the letters $F$,$L$,$Y$,$B$,$U$,$G$ represent distinct digits. Suppose $\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}$ is the greatest number that satisfies the equation \[8\cdot\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}=\underline{B}~\underline{U}~\underline{G}~\underline{B}~\underline{U}~\underline{G}.\] What is the value of $\underline{F}~\underline{L}~\underline{Y}+\underline{B}~\underline{U}~\underline{G}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1107$}
Determine the number of ordered quadruples \((a, b, c, d)\) of distinct positive integers such that \(\binom{\binom{a}{b}}{\binom{c}{d}} = 21\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
A criminal must undergo a series of 10 lie detector tests to secure his release from jail. The probabilities that each detector correctly identifies a lie are as follows: 0, \( \frac{1}{2} \), \( \frac{2}{3} \), \( \frac{3}{4} \), \( \frac{4}{5} \), \( \frac{5}{6} \), \( \frac{6}{7} \), \( \frac{7}{8} \), \( \frac{8}{9} \), \( \frac{9}{10} \). If a detector fails, the test restarts from the first detector. The criminal is released only if all tests are successfully passed in a row. Given that the criminal always lies and each test takes 1 second, determine the expected time (in seconds) for the criminal to be released from jail.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9864100}
Evaluate \( \sum_{d\mid N}\Lambda(d) \) for a given positive integer \( N \), where \( \Lambda(n) = \log p \) if \( n = p^r \) for some prime \( p \) and integer \( r \), and \( \Lambda(n) = 0 \) otherwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log N}
If $\sin \theta= \frac {3}{5}$ and $\frac {5\pi}{2} < \theta < 3\pi$, then $\sin \frac {\theta}{2}=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["-\\frac {3 \\sqrt {10}}{10}"]}
Given that \( \mathbb{E}[X] = 10 \), find a lower bound for the expression \( 0.8\mathbb{E}[X^2] + 0.2\mathbb{E}[X^4] \) in terms of \( \mathbb{E}[X] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2080}
Determine the number of elements of order 7 in a given group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the value of the Riemann zeta function at zero, \( \zeta(0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Consider the set $\mathbf{Z}_n = \{0, 1, \ldots, n - 1\}$ with modular addition. A subset $S_n$ of $\mathbf{Z}_n$ is such that the intersection of $(S_n + k)$ and $S_n$ is nonempty for all $k \in \mathbf{Z}_n$. Let $f(n)$ denote the minimum number of elements in such a subset. Determine the limit $$\lim_{n \to \infty} \frac{\ln f(n)}{\ln n}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Let \( X \) be a set with 56 elements. Determine the smallest positive integer \( n \) such that for any 15 subsets of \( X \), if the union of any 7 of these subsets contains at least \( n \) elements, then there must exist 3 of these 15 subsets whose intersection is non-empty.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Determine the greatest common divisor (GCD) of all possible values of the product $a \cdot b \cdot c \cdot d \cdot e \cdot f$, where $a, b, c, d, e, f$ are natural numbers satisfying the equation $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Compute the integral \( \int_{\mathbb{R}} \left(\int_{\mathbb{R}} f(x,y)\,dx \right)dy \) where the function \( f: \mathbb{R}^2 \rightarrow \mathbb{R} \) is defined as follows:

\[
f(x,y) = \begin{cases} 
1, & \text{for } x \ge 0, \ x \le y < x+1 \\
-1, & \text{for } x \ge 0, \ x+1 \le y < x+2 \\
0, & \text{elsewhere}
\end{cases}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
How many known classifications and types of indifferent fixed-points are there in complex dynamics?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
已知△ABC中，$AC=2\sqrt{3}$，D是BC边上的一点，且△ABD为等边三角形，则△ACD面积S的最大值为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt{3}$.}
Let $P$ be the transition matrix of a random walk on an $n$-cycle, where $n$ is odd. Determine the smallest integer $t$ such that $P^t(x, y) > 0$ for all states $x$ and $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n-1}
Given a random variable $X$ that follows a Student's $t$ distribution with $k$ degrees of freedom, determine the distribution of $Y = X^2$. The probability density function of $X$ is given by:
$$f_X(x;k) = \frac{\Gamma\left(\frac{k+1}{2}\right)}{\sqrt{k\pi} \ \Gamma \left(\frac{k}{2}\right)}{\left(1+\frac{x^2}{k}\right)}^{-\frac{k+1}{2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F(1, k)}
Let \( \omega = ze^{xy} dx \wedge dy \) be a 2-form on \( \mathbb{R}^3 \). Consider the map \( \varphi: \mathbb{R}^2_{\ne 0} \to \mathbb{R}^3 \) defined by \( \varphi(u,v) = \left( \frac{u}{v}, \frac{v}{u}, uv \right) \). Compute the pullback \( \varphi^*\omega \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A 24-ton mixture of crushed rock is needed in a construction job; it will cost $800. The mixture is composed of rock costing a certain amount per ton and $40 per ton. 8 tons of each should be added. How much does the cheaper rock cost per ton?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$30}
Calculate the limit: \[ \lim_{x\to\infty}\frac{\gamma(x,-c\sqrt{x}+x)}{\Gamma(x,c\sqrt{x}+x)} \] where \( \gamma(a,b) \) and \( \Gamma(a,b) \) are the lower and upper incomplete gamma functions, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the values of \( n \) for which the improper integral converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n > 1}
已知 $T=\{1,2, \cdots, 8\}$ 。对于 $A \subseteq T, A \neq \varnothing$ ，定义 $S(A)$ 为 $A$ 中所有元素之和。问： $T$ 有多少个非空子集 $A$ ，使得 $S(A)$ 是 3 的倍数，但不是 5 的倍数？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Determine the disk of convergence for the power series \( \sum_{n=1}^\infty n!\,(z-i)^{n!} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|z - i| < 1}
Find the value of the harmonic function $u(x, y)$ at the point $(1, 0)$ using Poisson's representation formula, given that the boundary condition is $u(x, y) = 2+3y^2$ on the circle $x^2 + y^2 = 9$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In the figure, $AB = 200$ is a diameter of the circle. Points $A$ and $B$ are assigned the numbers $1$ and $1$. The two semi-circles are bisected at points $C$ and $D$, which are assigned the numbers $2$ and $2$. Each quarter circle is then bisected and assigned the number $3$. This process continues, with each arc being bisected and the new point being assigned the sum of the numbers at the endpoints of the arc. This process is repeated until the sum of all numbers on the circle is at least equal to the product of $4$ times the area of the circle and $rac{1}{3}$ of the radius. Find the least number of points on the circle. (Take $\pi$ to be $3.14$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32768}
Choose a random subset of $\{2^1,2^2,⋯,2^{10}\}$ by selecting each of the 10 elements independently with probability $\frac{1}{2}$. Find the expected value of the smallest element in the subset.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a convex quadrilateral \(ABCD\), the midpoints of the sides \(AB, BC, CD,\) and \(DA\) are marked as \(M, N, K,\) and \(L\) respectively. Given that \(|AC| = |BD| = 2a\) and \(|MK| + |NL| = 2b\), find the area of quadrilateral \(MNKL\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b^2 - a^2}
Evaluate the limit: $$\lim_{x\rightarrow\infty}\Delta^x_{k=1}\sin \frac{k\pi}{2x}$$ Does this limit exist, and if so, what is its value?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the value of \( \alpha \) that makes the metric \( g = (1-x_1^2-x_2^2-\cdots-x_n^2)^{-\alpha}(dx_1\otimes dx_1+dx_2\otimes dx_2+\cdots+dx_n\otimes dx_n) \) geodesically complete on the set \( \{\vec{x}\in \mathbb{R}^n : x_1^2+x_2^2+\cdots+x_n^2<1\} \) in \( \mathbb{R}^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a parallelogram ABCD, let K and L be the midpoints of sides BC and CD, respectively. Given that ∠BAD = π/3 and the points A, B, K, and L are concyclic, determine the measure of ∠ADB.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Evaluate the limit \( \lim_{n\to\infty}\frac{\displaystyle\sum_{n=1}^{N}a_{n}}{\ln(n)} \), given that \( a_1 + a_2 + a_3 + \ldots + a_n = \int_{1}^{e}\ln^{k}(x)dx \) for \( n \ge 2 \) where \( a_n = \int_{0}^{1}x^{n}e^{x}dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Determine the order of the Galois group of the polynomial \( f(x) = x^2 + 2ix + 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine which function is asymptotically greater as $n$ approaches infinity: $f(n) = 3^{2^n}$ or $g(n) = n! \times n^3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(n)}
Find the constant term in the antiderivative of the function \( f(x) = \log(\log x) + \log^{-2}x \) given that the antiderivative passes through the point \((e, 1998-e)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1998}
Evaluate the integral of the 2-form \( x_2 \, dx_2 \wedge dx_3 + x_1 x_3 \, dx_1 \wedge dx_3 \) over the image set \( Y \) of the mapping \( \alpha: (0,1)^2 \to \mathbb{R}^3 \) given by \( \alpha(u,v) = (u,v,u^2+v^2+1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
In a right-angled triangle $\triangle ABC$ with $\angle B = 90^\circ$ and $\angle C = 15^\circ$, given that $|AC| = 7$, a point $D$ is chosen on $AC$. Perpendicular lines $DE$ and $DF$ are drawn to $AB$ and $AC$, respectively. Calculate the probability that $DE \cdot DF > 3$. Provide your answer as a decimal rounded to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.14}
Determine the values of \( p \) for which the series \( \sum_{n=N}^\infty \frac{1}{n(\log(n))(\log\log(n))\dotsm(\log_{j}(n))(\log_{j+1}(n))^p} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p > 1}
Calculate the expected value \( E(\ln(X)) \) where \( X \sim \text{Beta}(1/y, 1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-y}