inclusionAI/Ring-lite-rl-data huggingface dataset:
Given the asymptote of a hyperbola as the line $5x - 4y + 5 = 0$ and a tangent to the hyperbola as the line $4x + 5y - 7 = 0$, find the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{41}}{4}}
Find the permutation $X$ such that $\tau \circ X = \sigma$, given the permutations $\tau = (1,3,5)(2,4)$ and $\sigma = (1,3)(2)(4,5)$. Express $X$ in cycle notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 4, 3, 5)}
Given a Poisson process with rate \(\lambda = 3\), where three customers arrive in the first hour, calculate the probability that the third customer arrives between the 30th and 60th minute, conditional on the total of three arrivals. Express this probability as \(P(X_3 \in [0.5, 1] \mid N(1) = 3)\), knowing that \(X_3 \sim \text{Gamma}(3, 3)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{8}}
Find the volume of the solid $D$ bounded below by the sphere $x^2 + y^2 + (z - 1)^2 = 1$ and above by the cone $z = \sqrt{x^2 + y^2}$ using cylindrical coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
Determine the largest power of three that divides the largest product of several natural numbers whose sum is 2011.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^{669}}
Compute the expectation \( M_T = \mathbb{E}\left[W_T\int_0^T t \, dW_t \right] \) where \( 0 < t < T \) and \( W = (W_t)_{t \geq 0} \) is a standard Brownian motion starting at 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{T^2}{2}}
Compute \( 5^{11469} \mod 1911 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1448}
Find the smallest symmetric group $S_n$ into which the group $G = S_4 \times \mathbb{Z}/5\mathbb{Z}$ can be embedded, given that there exists an element $(\sigma, i) \in G$ with order 15, where $\sigma$ is a four-cycle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A standard deck of 52 cards is distributed equally among 4 players, with each player receiving 13 cards. What is the probability that one of the players receives 13 cards, each with a different rank? Express your answer as a fraction or a decimal rounded to four decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0004}
Calculate the probability of drawing at least one repeated ball when drawing 10 times with replacement from a set of 100 balls numbered from 0 to 99.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.3718}
Let $A \in \mathbb{R}^{n \times n}$ be a symmetric positive semi-definite matrix such that $\operatorname{Tr}(A) \le n$. Determine a lower bound for the quantity $$\operatorname{Tr} \bigl( \bigl( (A+I )^{-1} \bigr)^2\bigr),$$ where $I$ is the identity matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n}{4}}
Find the least value of \(b\) such that at least one permutation of the numbers consisting of one \(1\), two \(2\)'s, three \(3\)'s, ..., nine \(9\)'s can be expressed in the form \(a^b\), where \(a\) and \(b\) are natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the integral: \[ \int_{-1}^{1} \LegP[n, x] \frac{d}{dx} (\LegP[n+1,x]) \, dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find a number $a$, with $a < 2000$, that is not divisible by 7 and satisfies the congruence $a \equiv 7^{3003} \pmod{m}$, given that $\phi(m) = 1000$. Use Euler's Theorem, which implies $7^{1000} \equiv 1 \pmod{m}$, to solve this without a calculator.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1454}
Given that the coefficient of the $x^{\frac{3}{2}}$ term in the expansion of $\left(\sqrt{x} - \frac{a}{x}\right)^6$ is 30, find the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5}
Calculate the number of permutations of the letters ABCDEFGHI such that none of the sequences ABC, DEF, or GHI appear consecutively in the permutation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{348114}
Determine the value of $r$ for which the sequence $a_n = \dfrac{4^n}{n^3} + (r^2)^n$ and the series $S = \sum_{n=0}^\infty \frac{4^n}{n^3 + (r^2)^n}$ converge.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|r| > 2}
To each pair of nonzero real numbers $a$ and $b$, a real number $a*b$ is assigned such that:

1. $a*(b*c) = (a*b)*c$ for all $a, b, c$.
2. $a*a = 1$ for all $a$.

Solve the equation $x*36 = 216$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7776}
What relationship exists between $A$ and $p$ if

$$
A=\frac{x^{2}-3 y^{2}}{3 x^{2}+y^{2}}
$$

and

$$
\frac{p x y}{x^{2}-(2+p) x y+2 p y^{2}}-\frac{y}{x-2 y}=\frac{1}{2} ?
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9p^{2}-3}{27p^{2}+1}}
Given that $a+b+c+d=3$ and $a^2+b^2+c^2+d^2=45$, find the value of the expression:
\[ \frac{a^5}{(a-b)(a-c)(a-d)} + \frac{b^5}{(b-a)(b-c)(b-d)} + \frac{c^5}{(c-a)(c-b)(c-d)} + \frac{d^5}{(d-a)(d-b)(d-c)} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Determine the number of solutions in positive integers to the equation \(n_1 + n_2 + n_3 + n_4 + n_5 = 20\) under the condition that \(n_1 < n_2 < n_3 < n_4 < n_5\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Solve the recurrence relation: \( T(n) = T\left(\frac{n}{4}\right) + T\left(\frac{n}{2}\right) + n^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n^2)}
Determine the best bound constant \( C \) such that \( \sup_{R^3/D}|u| \leq C \cdot \sup_{\partial D}|g| \) for the Helmholtz equation \( \Delta u + k^2 u = 0 \) in the exterior region \( \mathbb{R}^3 \setminus D \), where \( D \) is an open and bounded region in three-dimensional space. Assume Dirichlet boundary condition \( u = g \) on \( \partial D \) and the usual radiation condition at infinity. Consider \( \partial D \) to be smooth and \( g \) continuous on \( \partial D \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the minimum number of people required to attend 20 meetings, each with 8 attendees, such that no pair of attendees appears more than once across all meetings.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Consider a method for randomly generating a classical Hilbert propositional calculus formula $F$ with $n$ variables. The formula $F$ is generated as follows:
- $F = x_i$ for each $i \leq n$ with probability $\frac{1}{n+2}$.
- $F = \neg F_1$, where $F_1$ is generated independently using the same method.
- $F = F_1 \to F_2$, where $F_1$ and $F_2$ are generated independently using the same method.

Given that the generation process terminates with probability 1, what is the probability that the formula $F$ is a tautology?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find all functions \(f(x)\) such that \(D^*f = Df\), where \(D^*f(x) = \lim_{h \to 0} \frac{f^2(x + h) - f^2(x)}{h}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = C}
Evaluate the triple integral \( \iiint_{G}(x+z) \,d x d y d z \), where the domain \( G \) is bounded by the planes \( x+y=1, x-y=1, x+z=1, z=0, x=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{12}}
Evaluate the integral \( \int_{-\infty}^{\infty}\frac{\sin^3{x}}{x^3}dx \) using contour integration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{4}}
Find the closed-form expression for the determinant of the matrix \( \mathbf{G} \), where
\[\mathbf{G} = a\mathbf{I} + b\boldsymbol{ee}^T\]
in which \(a\) and \(b\) are constants, \(\boldsymbol{e}\) is a column vector with all elements being \(1\), and \((\cdot)^T\) denotes the transpose operation. \(\mathbf{G}\) is a \(u\times u\) matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^{u-1}(a + bu)}
Given the function \( (u,v) = f(x,y) = (x + y, x^2 - y^2) \), compute the second partial derivative \( \frac{\partial^2 x}{\partial v^2} \) at the point \((u,v) = (2,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the largest possible size of an $n \times n$ matrix $A$ such that $A^3 = 0$ and $A$ has a Jordan form consisting of exactly 4 blocks.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the number of points of continuity of the function \( f(x) \) defined as \( f(x) = \tan x \) for rational numbers and \( f(x) = x^2 + 1 \) for irrational numbers within the interval \([0, 6\pi]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the values of the real number $a$ for which the expression \( a(x_1^2+x_2^2+x_3^2)+2(x_1x_2+x_1x_3+x_2x_3) \) can be expressed as \( \alpha x^2+\beta y^2+\gamma z^2 \), where \( \alpha, \beta, \gamma \) are real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Let $A$ be the number of 12-digit words that can be formed from the alphabet $\{0,1,2,3,4,5,6\}$ if each pair of neighboring digits must differ by exactly $1$. Find the remainder when $A$ is divided by $2008$. \( \text{Provide your answer in the form } \boxed{\text{remainder}}. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1392}
Determine the values of $m$ for which the equation \( e^{\vert{x}\vert}=m(x+1) \) has exactly two distinct real solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -e^2) \cup (1, \infty)}
Evaluate the integral \( \int_{-\pi/2}^{\pi/2} \frac{\log(1+b\sin x)}{\sin x}\,\mathrm{d}x \) given that \(|b|<1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \arcsin b}
Use the Divergence Theorem to evaluate the surface integral \( \iint\limits_S F\cdot dS \) where \( F(x,y,z)=xi+y^2j+(z+y)k \) and \( S \) is the boundary of the region enclosed by the cylinder \( x^2+y^2=4 \) and the planes \( z=x \) and \( z=8 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64\pi}
Sum the series\[\sum_{m=1}^\infty\sum_{n=1}^\infty\dfrac{m^2n}{3^m(n3^m+m3^n)}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ S = \frac{9}{32} \]}
Given the system of equations:
$$
\begin{align*}
xw^3 + yz^2 + z^3 &= -1 \\
zw^3 - xz^3 + y^2w &= 1
\end{align*}
$$
Find the value of $\frac{\partial z}{\partial x}$ at the point $(x, y, z, w) = (1, -1, -1, 1)$ without eliminating the function $w$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{5}{4}}
Compute the sum of the irrational solutions of the equation
$$\frac{x^2+16x+54}{x^2+11x+35}=\frac{x^2+13x+35}{x^2+14x+54}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$-\dfrac{27}{2}$}
A grasshopper sits on a square of a 5x4 rectangular board. Each minute, it jumps at random to a neighboring square, with 2, 3, or 4 possible moves depending on its position (2 at corners, 3 at edges, 4 at the center), all equally likely. The grasshopper always remains on the board. What is the long-run average probability that the grasshopper will be on a corner square of the board after many jumps?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{31}}
Determine the maximum number of fleas that can be placed on a $10 \times 10$ board such that each flea jumps to an adjacent cell every minute, maintaining its direction as long as possible, and no two fleas ever occupy the same cell at the same time over the course of an hour.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Determine the number of arrangements of the 10 letters ABCDEFGHIJ such that none of the patterns ABE, BED, or HID appear in the arrangement.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3513600}
Calculate the surface integral \( \int \int_S (2z^2 - x^2 - y^2) \, dS \), where \( S \) is the surface defined by the intersection of the cone \( z = \sqrt{x^2 + y^2} \) and the cylinder \( (x-1)^2 + y^2 = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\sqrt{2}}{2} \pi}
Determine the number of Hahn-Banach extensions of a non-zero continuous linear functional $f:(c_{00},\|\cdot\|_1) \to \mathbb{C}$ to the space $(\ell^1,\|\cdot\|_1)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the expected duration of a randomly selected long-distance telephone call, given that the duration $Y$ has the following properties: $P(Y=3)=0.2$, $P(Y=6)=0.1$, and otherwise $Y$ has a continuous density function given by \( f(y) = \frac{1}{4}ye^{-y/2} \) for \( y > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the conditions on \( \alpha > 0 \) such that the function \[ f(\tau) = \frac{(x(t) - x(\tau))^2}{(t - \tau)^{\alpha + 1}} \] is integrable near \( \tau = t \), where \( x(t) \) is a continuous and differentiable function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 < \alpha < 2}
Given the equations \( \frac{1}{\omega+a}+\frac{1}{\omega+b}+\frac{1}{\omega+c} = 2\omega^2 \) and \( \frac{1}{\omega^2+a}+\frac{1}{\omega^2+b}+\frac{1}{\omega^2+c} = 2\omega \), where \( \omega \) is a complex cube root of unity, find the value of \( \frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The Lucas sequence is defined by these conditions: \( {L}_{0} = 2,{L}_{1} = 1 \) ,and \( {L}_{n + 2} = {L}_{n + 1} + {L}_{n} \) for all \( n \geq  0 \) . Determine
 the remainder when \( {L}_{2019}^{2} + {L}_{2020}^{2} \) is divided by \( {L}_{2023} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is 11/4. To the nearest whole percent, what percent of its games did the team lose?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27%

}
Determine the largest possible size of a subset $M$ of the set $\{1, 2, \ldots, 2006\}$ such that for any three elements $x, y, z$ in $M$ with $x < y < z$, the sum $x + y$ does not divide $z$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1004}
Consider the function $g : \mathbb{R}^3 \to \mathbb{R}$ defined by $g(x,y,z) = x^3 - 2y^2 + z^4$. Determine the values of $w$ for which $g^{-1}(w)$ is a submanifold of $\mathbb{R}^3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{w \neq 0}
Compute the value of the integral: \[ \int_{-\infty}^\infty \Gamma(1+ix)\Gamma(1-ix)\,dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Given the rational function \( Q(x) \) such that \( x \cdot Q(x+2018) = (x-2018)Q(x) \) for all \( x \neq 2018 \) and \( x \neq 0 \), and \( Q(1) = 1 \), find the value of \( Q(2017) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
Evaluate the integral \( \int_0^1{\frac{(2x^3-3x^2)f'(x)}{f(x)}}\,dx \) given that the function \( f(x) \) is differentiable on \([0, 1]\) and satisfies the conditions \( f(1) = 1 \) and \( f(x)f(1-x) = e^{x^2 - x} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{10}}
Determine the number of distinct groups of order 2015.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the coefficient of \(x^{20}\) in the expansion of \((x^2 + x^4 + x^6 + \dots)^3 (x + x^3 + x^6 + \dots)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Let \((r_k)_{k \in \mathbb{N}}\) be an enumeration of \(\mathbb{Q}\). Define \(U_{1/p} = \bigcup_{k \in \mathbb{N}} \left(r_k - \frac{1/p}{2^k}, r_k + \frac{1/p}{2^k}\right) \subset \mathbb{R}\). Determine the set \(\bigcap_{p \in \mathbb{N}^*} U_{1/p}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Q}}
Given that the area of triangle $\vartriangle ABC$ with vertices $(a,a^2),(b,b^2),(c,c^2)$ is $\frac{1}{4}$ square units, and the area of triangle $\vartriangle PQR$ with vertices $(p,p^2),(q,q^2),(r,r^2)$ is $3$ square units, determine the value of the determinant:

$$ \begin{vmatrix} (1+ap)^2 & (1+bp)^2 & (1+cp)^2 \\ (1+aq)^2 & (1+bq)^2 & (1+cq)^2 \\ (1+ar)^2 & (1+br)^2 & (1+cr)^2 \\ \end{vmatrix} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the smallest integer n such that the inequality \( \prod_{k=1}^n \frac{\arctan\left( \frac{3k+1}{\sqrt{8k^3+8k^2+2k}}\right)}{\arccos \left( \frac{1}{\sqrt{2k+1}} \right)}>1000 \) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the number of imaginary roots of the equation \( \frac{A_1^2}{x-a_1} + \frac{A_2^2}{x-a_2} + \frac{A_3^2}{x-a_3} + \cdots + \frac{A_n^2}{x-a_n} = k \), where \(a_1, a_2, \ldots, a_n, A_1, A_2, \ldots, A_n,\) and \(k\) are real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the values of \( r \) for which the integral \( \int_{1}^{\infty} \frac{\cos(x)}{x^r} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \infty)}
Determine the number of curly brackets in the expression \( \wp^5(\varnothing) \), where \( \varnothing \) is the empty set and counts as \( \{ \} \) (1 curly bracket).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3211265}
Aerith records a time duration in both days:hours:minutes:seconds format and in seconds. Bob subtracts the number of seconds from the concatenated days:hours:minutes:seconds format, ignoring punctuation. What is the largest number that always divides the result of this subtraction?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Evaluate the integral \( \displaystyle \int^{1}_{-1}\frac{x^3}{\sqrt{1-x^2}}\ln\left(\frac{1+x}{1-x}\right)dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5\pi}{3}}
Evaluate the integral \( \int_{-\infty}^\infty e^{-(x-ti)^2} \, dx \), where \( t \in \mathbb{R} \) and \( i = \sqrt{-1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\pi}}
Find the Fourier transform of the function \( \delta(x)f(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(0)}
Evaluate the contour integral \( \int_{|z|=3} \frac{z^{17}}{(z^2+2)^3(z^3+3)^4} \, dz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
Given a real-valued function $f(x)$ that satisfies the functional equation \[ f(x-y) = f(x) \cdot f(y) - f(a-x) \cdot f(a+y) \] where $a$ is a constant, and knowing that $f(0) = 1$ and $f(2a) = -1$, find the value of $f(2a+x)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-f(x)}
Compute the integral \( \int_A f \) where \( A = \{(x,y) : 0 \le x + y \le 1, 0 \le x + 3y \le 3\} \subseteq \mathbb{R}^2 \) and \( f(x,y) = \sqrt{x+y} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of linear conditions that must be added to the set \( \mathcal{T} \) to ensure that \(|M_{\gamma_{1}}(Q, \mathcal{T'})| < |M_{\gamma_{2}}(Q, \mathcal{T'})|\) holds for any choice of linear conditions, where \( \mathcal{T'} \) is a subset of \( \mathcal{T} \) defined by these linear conditions. Assume \( 0 < \gamma_1 < \gamma_1 + 1 \leq \gamma_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\gamma_2 - \gamma_1}
Find the minimum number of integer lattice points contained in a centrally symmetric convex subset of \(\mathbb{R}^N\) with a volume that is not an integer multiple of \(2^N\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the order of the pole in the expansion of \( \frac{z - \sin z}{z^3} \) around \( z = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit: $$\lim_{n\to \infty} \frac{1}{n} \cdot \sum_{i=1}^n \frac{1}{1+(\frac{i}{n})^i}$$ using the concept of definite integrals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the order of the differential equation for the family of parabolas with a fixed directrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the angle in the axial section of a cone if a sphere with its center at the vertex of the cone, touching its base, divides the volume of the cone in half.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \arccos\left(\frac{1 + \sqrt{17}}{8}\right)}
Find a prime number \( p \) such that there exists a sequence of integers \( S_n \) where \( (S_n)^2 \) converges to 21 under the \( p \)-adic norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the dimension of the vector space \( \mathbb{R}(x) \) over the field \( \mathbb{R}(x^3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let \((X_1, X_2, \ldots)\) be an independent sequence of random variables, where each \(X_n\) is uniformly distributed on \([0, n]\). Calculate the probability \(P(\{w: X_n(w) \rightarrow \infty \text{ as } n \rightarrow \infty\})\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find a Gaussian integer \( \alpha \in \mathbb{Z}[i] \) such that the ideal \((3+5i,1+3i)\) in \( \mathbb{Z}[i] \) is equal to the principal ideal \((\alpha)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - i}
Suppose $u$ and $v$ are differentiable functions of $r$ and $s$, such that $ru^2+sv^3-uv=2$ and $uv^2-ru+sv=3$. Determine the value of $\frac{\partial u}{\partial r}$ at the point $(r,s,u,v)=(1,2,1,2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{29}{69}}
Evaluate the improper integral \( \int_0^\infty \frac{x\sin x}{1+x^2} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2e}}
For real numbers \(x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty)\), let \(S_k = \sum_{i = 1}^{333} x_i^k\) for each \(k\). Given that \(S_2 = 777\), find the minimum possible value of \(S_3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{999}
What is the closed-form expression for the constant $c_n$, defined as follows?
$$ c_n := n! \sum_{i+j+k+l=n} \binom{2i}{i}\binom{2j}{j}\binom{2k}{k}\binom{2l}{l} $$
Here, $i, j, k, l$ are nonnegative integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4^n(n+1)!}