inclusionAI/Ring-lite-rl-data huggingface dataset:
Let \( A \) and \( B \) be the vertices of the major axis of ellipse \( \Gamma \). \( E \) and \( F \) are the foci of \( \Gamma \). Given that \( |AB| = 4 \) and \( |AF| = 2 + \sqrt{3} \). Point \( P \) lies on \( \Gamma \) and satisfies \( |PE| \cdot |PF| = 2 \). Find the area of \( \triangle PEF \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the maximum possible value of the inradius of a triangle whose vertices lie in the interior, or on the boundary, of a unit square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5}-1}{4}}
For $\pi \le \theta < 2\pi$ , let 
$$\begin{align*} P &= \frac12\cos\theta - \frac14\sin 2\theta - \frac18\cos 3\theta + \frac{1}{16}\sin 4\theta + \frac{1}{32} \cos 5\theta - \frac{1}{64} \sin 6\theta - \frac{1}{128} \cos 7\theta + \cdots \end{align*}$$
 and 
 $$\begin{align*} Q &= 1 - \frac12\sin\theta -\frac14\cos 2\theta + \frac18 \sin 3\theta + \frac{1}{16}\cos 4\theta - \frac{1}{32}\sin 5\theta - \frac{1}{64}\cos 6\theta +\frac{1}{128}\sin 7\theta + \cdots \end{align*}$$
 so that $\frac{P}{Q} = \frac{2\sqrt2}{7}$ . Then $\sin\theta = -\frac{m}{n}$ 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{36}
Solve the differential equation: $$y' + \frac{x^2-y^2}{2xy} = 0$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 + y^2 = Cx}
There are $n$ sticks which have distinct integer lengths. Suppose that it's possible to form a non-degenerate triangle from any $3$ distinct sticks among them. It's also known that there are sticks of lengths $5$ and $12$ among them. What's the largest possible value of $n$ under such conditions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Compute the value of the triple integral \( \int\int\int_{E}{|xyz|dxdydz} \) where \( E=\{(x,y,z)|\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\leq1\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{a^2 b^2 c^2}{6}}
Evaluate the integral \( I = \frac{1}{2\pi i}\oint_{|z+1|=2}\frac{\zeta'(z)+1}{\zeta(z)+z}dz-1 \), where \( \zeta \) is the Riemann Zeta function, and determine if \( I = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the largest $k$ such that for all competitive graphs with $2019$ vertices, if the difference between the in-degree and out-degree of any vertex is less than or equal to $k$, then this graph is strongly connected.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the formal name for an additive abelian group equipped with a biadditive multiplication that satisfies left and right distributivity over addition, without assuming associativity or the existence of a unit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{non-associative ring}
Evaluate the flux integral \(\iint_{S} \mathbf{F} \cdot d\mathbf{S}\) using the Divergence Theorem, where \(\mathbf{F}(x, y, z) = y\mathbf{j} - z\mathbf{k}\) and \(S\) is the surface composed of the paraboloid \(y = x^2 + z^2\) for \(0 \leq y \leq 1\) and the disk \(x^2 + z^2 \leq 1\) at \(y = 1\), both oriented outward. Determine the flux through the paraboloid alone.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi}
Let $\alpha$ be an element in an algebraic closure of $GF(64)$ such that $\alpha^4 = \alpha + 1$. Determine the value of $r \in \mathbb{N}$ for which the field extension $GF(64)[\alpha]$ is isomorphic to $GF(2^r)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given the function $f(x)=\sin x+\cos x$ and its derivative $f'(x)=3f(x)$, find the value of $\tan 2x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{4}{3}}
Given three sets $t_i$, $t_j$, and $t_k$, suppose that for any element $x$, if $x$ is in the symmetric difference of $t_i$ and $t_j$, then $x$ is not in the symmetric difference of $t_i$ and $t_k$. Similarly, if $x$ is in the symmetric difference of $t_i$ and $t_k$, then $x$ is not in the symmetric difference of $t_j$ and $t_k$. Determine if it is necessarily true that if $x$ is in the symmetric difference of $t_j$ and $t_k$, then $x$ is not in the symmetric difference of $t_i$ and $t_j$. Answer with 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
Let \( \mathbf{X} \in \mathbb{R}^n \) be a vector such that each of its components is subGaussian with the same parameter \( s \). Determine if the vector \( \mathbf{X} \) is subGaussian, and if so, find its subGaussian parameter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{s\sqrt{n}}
A school has 6 sections of chemistry in Class X having 50, 35, 45, 42, 38, and 48 students. The mean marks obtained in chemistry tests are 50, 60, 55, 45, 55, and 50 respectively for the 6 sections. However, the tests in each section carry different weights: the first and second section's average is out of 70, the third and fourth's averages are out of 80, and the last two sections' averages are out of 100. Determine the overall weighted average of marks per student.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52.15}
In a Hilbert space, determine whether the annihilation operator $c(f)$ is a bounded linear operator on the fermionic Fock space. If it is bounded, find its norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\|f\|}
Let the function $f(x) = \ln \left( \frac{\sum_{i=1}^{n-1}i^x + n^x a}{n} \right)$, where $a \in \mathbb{R}$, for any positive integer $n$ ($n \geq 2$). If the inequality $f(x) > (x-1)\ln n$ has solutions in the interval $[1, +\infty)$, then the range of values for the real number $a$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > \frac{1}{2}}
12 In an equilateral triangle $ABC$ with side length 1, take points $D$ and $E$ on sides $AB$ and $AC$ respectively, such that when the triangle is folded along segment $DE$, vertex $A$ lands exactly on side $BC$. The minimum length of $AD$ is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["2\\sqrt{3}-3"]}
Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are face cards (Jacks, Queens, or Kings) totaling to a numeric value of 20?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{221}}
Given any natural number \( n \) (where \( n > 3 \)), if the set \( S_{n} = \{3, 4, \cdots, n\} \) is divided into two groups arbitrarily, there will always be three numbers \( a, b, c \) (allowing \( a = b \)) in one of the groups such that \( ab = c \). Find the smallest value of \( n \) for which this property holds.

(Source: 1988 National Training Team Test, China)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243}
Say that a polynomial with real coefficients in two variables, $x, y$, is balanced if the average value of the polynomial on each circle centered at the origin is $0$. The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\mathbb{R}$. Find the dimension of $V$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2020050}
Let $n$ be a integer and let $n=d_1>d_2>\cdots>d_k=1$ its positive divisors.
Determine the three positive integers such that
\[d_1-d_2+d_3-...+(-1)^{k-1}d_k=n-4\].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10,12,25}
Let $A, B, C$ be unique collinear points$ AB = BC =\frac13$. Let $P$ be a point that lies on the circle centered at $B$ with radius $\frac13$ and the circle centered at $C$ with radius $\frac13$ . Find the measure of angle $\angle PAC$ in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
In the polar coordinate system, if a line passing through point $A(3,0)$ and perpendicular to the polar axis intersects the curve $\rho=4\cos \theta$ at points $A$ and $B$, then $|AB|=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}}
Given an ant crawling inside an equilateral triangle with side length $4$, calculate the probability that the distance from the ant to all three vertices of the triangle is more than $1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1- \dfrac { \sqrt {3}\pi}{24}}
Given an infinite sequence $\left\{a_{n}\right\}$ satisfies: $a_{n}$ is a positive integer, $a_{1}=1$, and for any positive integers $k$ and $n$,
$$
a_{n} \mid\left(a_{k}+a_{k+1}+\cdots+a_{k+n-1}\right) \text {. }
$$

Given a positive integer $m$, find the maximum value of $a_{2 m}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{2^m - 1}.}
How many letters in the word \text{UNCOPYRIGHTABLE} have at least one line of symmetry?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In a trapezoid \(ABCD\) with bases \(AD=12\) and \(BC=8\), circles constructed on the sides \(AB\), \(BC\), and \(CD\) as diameters intersect at one point. The length of diagonal \(AC\) is 12. Find the length of \(BD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\overline{DM}$ has integer length, and the lengths of $\overline{MA}, \overline{MC},$ and $\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MABCD?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24\sqrt{5}}
Given \( a_{1}, a_{2}, \cdots, a_{2019} \in [0, 2] \). Find the range of values of \( \sum_{1 \leqslant j \leqslant k \leqslant 2019} \left| a_{j} - a_{k} \right| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 2038180]}
How many seconds will a train 100 meters long take to cross a bridge 150 meters long if the speed of the train is 53.7 kmph and the train's speed decreases by 3% due to wind resistance? Keep in mind, the train's speed may vary along the travel.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17.28}
Given a circle $C:x^{2}+y^{2}-2ax-2by+a^{2}+b^{2}-1=0$ $(a < 0)$ whose center is on the line $\sqrt{3}x-y+\sqrt{3}=0$, and the maximum distance from a point on circle $C$ to the line $\sqrt{3}x+y=0$ is $1+\sqrt{3}$, find the value of $a^{2}+b^{2}$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
$n$ 个人参加由 12 轮比赛组成的训练赛, 每一轮比赛后, 每个参赛者都根据他在该轮比赛中的名次 $k$ 获得一个分数 $a_k\left(a_k \in \mathbf{N}_{+}, k=1,2, \cdots, n, a_1>a_2>\cdots>a_n\right)$. 所有比赛结束后, 再根据每人在 12 轮比赛中所得分数之和排列总名次. 试求最小正整数 $n$, 对于这个 $n$, 可恰当选取 $a_1, a_2, \cdots, a_n$. 使得在倒数第二轮结束后, 无论排名如何, 都至少有 2 人具有最终获得冠军的可能性.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Determine the Krull dimension of the ring \( A = \mathbb{Q}[\pi,\sqrt{11},x,y]/I \), where \( I = \langle x^2-y^3+xy+3 \rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( H \) be the collection of all absolutely continuous functions \( f: [0,1] \to \mathbb{C} \) such that \( f(0) = 0 \) and \( f' \in L^2(0,1) \). Define the inner product \( \langle f, g \rangle = \int_0^1 f'(t) \overline{g'(t)} \, dt \) for \( f \) and \( g \) in \( H \). Fix \( t \), \( 0 < t \leq 1 \), and define \( L: H \to \mathbb{C} \) by \( L(h) = h(t) \). Find the operator norm \( \|L\| \) and a function \( h_0 \) such that \( L(h) = \langle h, h_0 \rangle \) for every \( h \) in \( H \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{t}}
Define a $\textit{tasty residue}$ of $n$ to be an integer $1<a<n$ such that there exists an integer $m>1$ satisfying \[a^m\equiv a\pmod n.\] Find the number of tasty residues of $2016$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{831}
The group $G$ is generated by two elements $\sigma$ and $\tau$, with orders $5$ and $4$ respectively, and it satisfies $\tau\sigma\tau^{-1}=\sigma^2$. Given that $G$ is of order $20$ and every element $g$ in $G$ can be uniquely expressed as $g=\sigma^k\tau^m$, where $0\le k<5$ and $0\le m<4$, determine the number of subgroups of $G$. \( \text{Provide your answer as a single integer.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Let $V$ be an $n$-dimensional $\mathbb{R}$-vector space, and let $\langle \cdot , \cdot \rangle : V \times V \to \mathbb{R}$ be a scalar product on $V$. Let $F: V \to V$ be a self-adjoint linear map such that $v, F(v), F^2(v), \ldots, F^{n-1}(v)$ is an orthogonal basis for $V$. Determine whether $\det F < 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\det F < 0}
Evaluate \[\frac{3}{\log_8{5000^4}} + \frac{2}{\log_9{5000^4}},\] giving your answer as a fraction in lowest terms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
【Question 11】8 different non-zero natural numbers are arranged in ascending order. The average of the first 3 numbers is 9, the average of the 8 numbers is 19, and the average of the last 3 numbers is 29. What is the maximum difference between the second largest and the second smallest number? $\qquad$ _.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
In the alternating group $A_5$, let $x$ be an element of order $2$ and $y$ be an element of order $3$ such that the subgroup generated by $x$ and $y$ is $A_5$. Determine if the order of the element $xy$ must be $5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $z_1, z_2, \ldots, z_{20}$ be the twenty (complex) roots of the equation:

$z^{20} - 4z^{19} + 9z^{18} - 16z^{17} + \cdots + 441 = 0$

Calculate the value of:

$\cot{\left(\sum\limits_{k = 1}^{20} \cot^{ - 1}{z_k}\right)}$


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{241}{220}}
A large tank is initially filled with 100 gallons of pure water. Brine containing 3 pounds of salt per gallon is pumped into the tank at a rate of 4 gallons per minute, while the well-mixed solution is pumped out at a rate of 5 gallons per minute. Determine the amount of salt in the tank after 30 minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{159.58}
A sequence of \( n \) positive integers is considered full if for each \( k > 1 \), \( k \) occurs only if \( k-1 \) occurs before the last occurrence of \( k \). How many full sequences are there for each \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
Calculate the limit: $\lim_{n \rightarrow \infty} \sqrt [n] {nk \choose n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{k^k}{(k-1)^{k-1}}}
Let $B_t$ be a Brownian motion. Calculate the probability $P(3B_2 > 4)$. Assume $B_2 \sim N(0, 2)$. Use the property that if $X \sim N(\mu, \sigma^2)$, then $kX \sim N(k\mu, k^2\sigma^2)$ for a constant $k$. Determine the correct variance for $3B_2$ and find $P(3B_2 > 4)$. Provide the final probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.173}
A square pyramid with an initial side length of 100 cm and height of 100 cm is melting at a constant rate. The melting occurs such that all ice less than y cm from the surface disappears after y hours, including the bottom surface. Calculate the rate of change of the volume of the pyramid when the remaining height is 10 cm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-200}
In an infinite-dimensional Hilbert space $H$, consider two distinct unit vectors $x$ and $y$. Determine whether there exists a unitary operator $U$ such that $y = Ux$. If such an operator exists, find the number of unique unitary operators for the given pair of $x$ and $y$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let $G = \langle X_1, X_2 \rangle$ be a group. Determine whether, if $X_1$ or $X_2$ is a normal subgroup of $G$, it follows that $G = X_1X_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{G = X_1 X_2}
Calculate the limit \( \lim_{a\to 0}\int_{[a,1]^2}f(x,y)d\mathscr{L}^2 \), where the function \( f:[0,1]^2\rightarrow\mathbb{R} \) is given by \( f(x,y)=\frac{x^2-y^2}{(x^2+y^2)^2} \) for \( (x,y)\neq(0,0) \) and \( f(x,y)=0 \) otherwise. You have computed that \( \int_{0}^1\int_{0}^1f(x,y)dxdy=-\frac{\pi}{4} \) and \( \int_{0}^1\int_{0}^1f(x,y)dydx=\frac{\pi}{4} \). Determine the limit as \( a \to 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the surface integral \(\iint_{S} \text{curl}G\cdot dS\) where \(S\) is the surface formed by the union of a hemisphere and a cone, given that \(G(x,y,z)=(xy+z, yz+x, zx+y)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
若关于$z$的方程$z ^ {2 } + c \vert z \vert ^ { 2} = 1 + 2i$有复数解，求实数$c$的取值范围.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1,$\frac { \sqrt { 5 } } { 2 } ] $}
Determine the number of zeros of the function \( f(z) = 10 \sin(z^{2019}) - z \) within the unit disk \( D = \{ z \in \mathbb{C} : |z| < 1 \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019}
Given the expression \( y = w^T X + b \), where \( w \) and \( X \) are both \([13, 1]\) matrices, find the derivative \( \frac{dy}{dw} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X}
Determine the values of \( \alpha \in \mathbb{R} \) for which the function \( f(x,y) = \frac{\exp(-x)\sin(x)}{(1+xy)^{\alpha}} \) is integrable over the domain \((0, +\infty) \times (0, +\infty)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > 1}
Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.

Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n+p?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
Two particles are placed at opposite vertices of the same face of a cube. If they are on the same edge at some time \( t \), they remain in the same position at time \( t+1 \). If not, one of the particles is chosen with equal probability, and at time \( t+1 \), it moves to one of the three neighboring vertices with probability \( \frac{1}{3} \). What is the expected value of the minimum time they are on the same edge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the 100th derivative of the function \( y = \ln(2x - x^2) \) evaluated at \( x = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2 \cdot 99!}
Let \( f(x) \) and \( g(x) \) be a function defined on \( \mathbb{R} \), where \( f(x) \) is an odd function and \( g(x) \) is an even function. Given that \( f(x) + g(x) = 2^{x} \), for \( x \in [1,2] \), the inequality \( a f(x) + g(2x) \geq 0 \) holds. Determine the range of the real number \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[-\frac{17}{6}, +\infty \right]}
Three identical cones with vertex \( A \) and a vertex angle of \( \frac{\pi}{3} \) are externally tangent to each other. Each of these cones is internally tangent to a fourth cone with vertex at point \( A \). Find the vertex angle of the fourth cone. (The vertex angle of a cone is the angle between its generatrices in an axial section.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3} + 2 \arcsin \frac{1}{\sqrt{3}}}
Use Stokes' Theorem to evaluate the line integral \( \int_C \mathbf{F} \cdot \mathrm{d}\mathbf{r} \) for the vector field \( \mathbf{F}(x,y,z) = \langle 5y+\sin x, z^2+\cos y, x^3\rangle \), where \( C \) is the curve given by \( \mathbf{r}(t)=\langle \sin t, \cos t, 2\sin t\cos t\rangle, t\in[0,2\pi] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\pi}
$P$ is a point inside $\triangle ABC$, line $AC$ and $BP$ intersect at $Q$, line $AB$ and $CP$ intersect at $R$. It is known that $AR = RB = CP$ and $OQ = PQ$. Find the measure of $\angle BRC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
Determine the integer value of \( n \) such that the following limit is a finite non-zero number:
\[ \lim_{x\to 0} \frac{\cos^2 x - \cos x - e^x \cos x + e^x - \frac{x^3}{2}}{x^n} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
[ Application of trigonometric formulas (geometry).]

On the extension of side $A D$ of rectangle $A B C D$ beyond point $D$, a point $E$ is taken such that $D E=0.5 A D, \angle B E C=$ $30^{\circ}$.

Find the ratio of the sides of rectangle $A B C D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}:2}
Point $A$ lies on the line $y=\frac{8}{15} x-6$, and point $B$ on the parabola $y=x^{2}$. What is the minimum length of the segment $AB$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1334}{255}}
Given the circumcenter $O$ of $\triangle ABC$, and $2 \overrightarrow{OA} + 3 \overrightarrow{OB} + 4 \overrightarrow{OC} = \mathbf{0}$, find the value of $\cos \angle BAC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Determine the number of homomorphisms from a finite extension of \(\mathbb{Q}\), denoted as \(K\), to \(\mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r_1}
Given an arithmetic-geometric sequence {a_n}, the sum of its first n terms is represented as s_n. If s_6 : s_3 = 1 : 2, find the value of s_9 : s_3. Express your answer as a ratio in the form a/b.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/4}
The coefficients $a,b,c$ of a polynomial $f:\mathbb{R}\to\mathbb{R}, f(x)=x^3+ax^2+bx+c$ are mutually distinct integers and different from zero. Furthermore, $f(a)=a^3$ and $f(b)=b^3.$ Determine $a,b$ and $c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
a = -2, b = 4, c = 16
}
Determine if the operator $S: \ell_2 \to \ell_2$ defined by $(S(x))_n = \frac{x_{n+1}}{n+1}$ has any eigenvalues.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
$E$ is the midpoint of side $BC$ of parallelogram $ABCD$. Line $AE$ intersects the diagonal $BD$ at point $G$. If the area of triangle $\triangle BEG$ is 1, find the area of parallelogram $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}