inclusionAI/Ring-lite-rl-data huggingface dataset:
Given real numbers $b_0,b_1,\ldots, b_{2019}$ with $b_{2019}\neq 0$, let $z_1,z_2,\ldots, z_{2019}$ be the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019}b_kz^k.
\]
 Let $\mu = (|z_1|+ \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\ldots, z_{2019}$ to the origin.  Determine the largest constant $M$ such that $\mu\geq M$ for all choices of $b_0,b_1,\ldots, b_{2019}$ that satisfy
\[
1\leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[2019]{\frac{1}{2019}}}
Chords $ AB$ and $ CD$ of a circle intersect at a point $ E$ inside the circle. Let $ M$ be an interior point of the segment $ EB$. The tangent line at $ E$ to the circle through $ D$, $ E$, and $ M$ intersects the lines $ BC$ and $ AC$ at $ F$ and $ G$, respectively. If
\[ \frac {AM}{AB} = t,
\]
find $\frac {EG}{EF}$ in terms of $ t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[\frac{EG}{EF} = \frac{t}{1 - t}\]}
某建筑物的底部是周长约是75.36米的圆，它的占地面积大约是多少平方米？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{452.16}
Given that the largest real root of the equation \(x^{3} - 7x^{2} + 1 = 0\) is \(t\), what is the remainder when \(\left[t^{20100}\right]\) is divided by 7?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Jack and Jenny start hiking from the trailhead to the fire tower at the top of the mountain. Jenny, carrying a very heavy backpack, walks very slowly. Jack starts at a speed of 4 mph and, upon reaching halfway, where the trail becomes steep, his speed decreases to 2 mph. After reaching the fire tower, Jack immediately returns and descends the steep trail at a speed of 3 mph, meeting Jenny halfway down. What was Jenny's average speed (in mph) before meeting Jack? Express your answer as a fraction or decimal value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{12}{13}}
(2009•宝山区一模)某抛物线形拱桥的跨度为20米，拱高是4米，在建桥时，每隔4米需用一根柱支撑，其中最高支柱的高度是多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.84}
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}
Suppose matrices A and B satisfy the equation \( A^{-1} + B^{-1} = (A+B)^{-1} \). Given that \(|A| = -3\), determine the possible values of \(|B|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Evaluate \( \lim_{n\to\infty} n^2 \int_0^1 x^n f(x) \, dx \) given that \( f(x) = (x-1) e^{-\frac{1}{x+5}} \) and \( f(1) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-e^{-\frac{1}{6}}}
Let $\{W_t\}$ be a standard Brownian motion and define the process $\{X_t\}$ by $X_t= {W_t}^{2}-t ,\ \ t\geq0.$ Calculate $P(X_5>0\mid W_3=1)$. Provide your answer as a probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.202}
Determine the minimum dimension n such that the system of differential equations \( x' = Ax \) is satisfied, where \( A \) is an \( n \times n \) matrix and the first coordinate of the solution \( x(t) \) is given by \( x_1(t) = 5t^2\cos(3t) + 2\sin(3t) - e^{2t}\sin(3t) + 4t \). What is the minimum value of n?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a simple linear regression model $Y_i = \beta_0 + \beta_1x_i + \epsilon_i$ where $\epsilon_i \sim^{indep} N(0, \sigma^2)$ for $i = 1,...,n$, let $\hat{\beta_{0}}$ and $\hat{\beta_{1}}$ be the maximum likelihood estimators of $\beta_0$ and $\beta_1$. The $i$th residual is $\hat{\epsilon_{i}} = Y_i - \hat{Y_{i}}$, where $\hat{Y_i} = \hat{\beta_{0}} + \hat{\beta_{1}}x_i$. Derive the covariance $Cov(\hat{\epsilon_{i}},\hat{Y_i})$. \( \text{Express your answer in terms of known quantities.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $(a_n)_n\geq 0$ and $a_{m+n}+a_{m-n}=\frac{1}{2}(a_{2m}+a_{2n})$ for every $m\geq n\geq0.$ If $a_1=1,$ then find the value of $a_{2007}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2007^2}
Given the function $F = e^a b(c(t))$, where $a \in \mathbb{R}$ and $b(c(t))$ is a function with $c(t)$ as its argument, determine the derivative of $F$ with respect to $\dot{c}(t)$. Is this derivative equal to zero?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Who is credited with the conjecture that all odd squares are solitary numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Dean Hickerson}
Calculate the expected number of days it will take for two independent seeds to sprout, given that the probability of a seed not sprouting on day \( n \) is \( \frac{1}{2^n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{3}}
4. Let $z=1-2 i$. Find $\frac{1}{z}+\frac{2}{z^{2}}+\frac{3}{z^{3}}+\cdots$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2i-1}{4}}
Let $A,B,C$ be angles of an obtuse triangle with 
$$
\begin{align*} \cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\ \cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9} \end{align*} 
$$
There are positive integers $p$, $q$, $r$, and $s$ for which $$\cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B = \frac{p-q\sqrt{r}}{s},$$ where $p+q$ and $s$ are relatively prime and $r$ is not divisible by the square of any prime. Find $p+q+r+s$.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{222
}
A trapezoid has area $2\, m^2$ and the sum of its diagonals is $4\,m$. Determine the height of this trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \sqrt{2} \, \text{m} }
A $4 \times 4 \times h$ rectangular prism can fit 8 small spheres with a radius of 1 and 1 large sphere with a radius of 2. Determine the minimum value of $h$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 + 2\sqrt{7}}
How many pairs of positive integers \((x, y)\) satisfy the equation \(x^2 - 10! = y^2\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
Let \( V \) be a finite-dimensional vector space, and let \( \mathcal{L}(V) \) denote the space of all endomorphisms of \( V \). For any \( \phi \in \mathcal{L}(\mathcal{L}(V)) \), there is a unique bilinear map \( \mathcal{L}(V) \otimes \mathcal{L}(V) \to \mathcal{L}(V) \) (denoted by \( \Phi(\phi) \)) satisfying \( \Phi(\phi) (a \otimes b) = \phi(ab) \) for any \( a, b \in \mathcal{L}(V) \). Determine the rank of the linear map \( \Phi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^4}
Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$75$}
Determine the additional variable \( \eta \) required to make the transformation \( \Phi : (x,t) \rightarrow (\xi,\eta) \) a diffeomorphism when using Boltzmann's transformation with \( \xi = \frac{x}{2\sqrt{t}} \) in the context of converting Fick's second law to an ordinary differential equation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t}
Determine whether the sequence \(\sqrt[n]{(n+1)!}\) converges or diverges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Consider a Poisson process $N_t$ and a continuous process of finite variation $X_t$. Is the quadratic covariation $\langle X, N \rangle_t$ equal to zero?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In trapezoid $ABCD$, sides $AB$ and $CD$ are parallel, and diagonal $BD$ and side $AD$ have equal length. If $m\angle DBC=110^\circ$ and $m\angle CBD =30^\circ$, then $m \angle ADB=$

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$100^\circ$}
Determine whether \( f(n!) \to +\infty \) as \( n \to +\infty \), where \( f(x) = x (1 + [x] - x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{+\infty}
Chris and Paul each rent a different room of a hotel from rooms $1 - 60$. However, the hotel manager mistakes them for one person and gives ”Chris Paul” a room with Chris’s and Paul’s room concatenated. For example, if Chris had $15$ and Paul had $9$, ”Chris Paul” has $159$. If there are $360$ rooms in the hotel, what is the probability that ”Chris Paul” has a valid room?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{153}{1180}}
Find the sum of all integers $n$ with $2 \le n \le 999$ and the following property: if $x$ and $y$ are randomly selected without replacement from the set $\left\{ 1,2,\dots,n \right\}$, then $x+y$ is even with probability $p$, where $p$ is the square of a rational number.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{600}
Determine the dimension of the vector space \(V = \mathbb{R}^+ \times \mathbb{R}^+ \times \mathbb{R}^+\), where \(\mathbb{R}^+\) is the set of all positive real numbers. The operations in \(V\) are defined as \((x_1, y_1, z_1) \oplus (x_2, y_2, z_2) = (x_1x_2, y_1y_2, z_1z_2)\) and \(\alpha(x, y, z) = (x^\alpha, y^\alpha, z^\alpha)\) for \(\alpha \in \mathbb{R}\). Provide a basis for \(V\) that includes the vector \((e, e, 1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A cylinder is filled with gas at atmospheric pressure (103.3 kPa). Assuming the gas is ideal, calculate the work done (in joules) during the isothermal compression of the gas by a piston that moves inward by 1.0 meter. The initial height of the gas column is 2.0 meters, and the radius of the cylinder is 0.4 meters. Use the equation of state \( \rho V = \text{const} \), where \( \rho \) is the pressure and \( V \) is the volume.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72000}
Evaluate the Riemann-Stieltjes integral \( \int_0^3 x^3 \, \text{d}\left[\frac{x}{2}\right] \) using the properties that \([x/2] = 0\) for \(0 \leq x < 2\) and \([x/2] = 1\) for \(2 \leq x < 4\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the largest positive integer $n$ such that the following is true:
There exists $n$ distinct positive integers $x_1,~x_2,\dots,x_n$ such that whatever the numbers $a_1,~a_2,\dots,a_n\in\left\{-1,0,1\right\}$ are, not all null, the number $n^3$ do not divide $\sum_{k=1}^n a_kx_k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let \( u_n \) be a sequence defined by \( u_0 = 1 \) and \( u_{n+1} = 1 + \frac{n}{u_n} \). Determine whether the sequence \( u_n \) diverges to \( +\infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{+\infty}
Evaluate the contour integral \( \int_\gamma \frac{\mathrm{Re} (z)}{z - \beta} dz \), where \( \beta \) is a complex number such that \( 0 < |\beta| < 4 \) and \( \gamma \) is the circle oriented in the negative sense, centered at the origin with radius 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ - \pi i \beta }
Let $\pi_1$ and $\pi_2$ be two distinct irreducible representations of a finite group $G$, both having the same degree. Determine whether the following sum equals the zero matrix:
\[\sum_{g \in G}  \pi_1(g)   \pi_2(g^{-1})^\top = 0.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
(fidgetboss_4000) In the diagram below, $ABC$ is an isosceles right triangle with a right angle at $B$ and with a hypotenuse of $40\sqrt2$ units. Find the greatest integer less than or equal to the value of the radius of the quarter circle inscribed inside $\triangle ABC$.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
28
}
In a plane covered with equilateral triangles, each vertex is shared by 6 triangles. Initially, one triangle is colored red. In each subsequent step, color all triangles that share at least one vertex with a red triangle and are not yet colored. How many triangles are colored in the 100th step?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1200}
Suppose that the series \(\sum_{j=1}^{\infty}a_j\) is convergent, with \(a_j\geq 0\). Determine whether the following limit is true: \[\lim_{n\to \infty}S_n=0,\qquad S_n:=\sum_{j=1}^{n-1} \frac{a_j}{n-j}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the value of $a$ such that:

$$\lim_{x\to\frac{\pi}{6}}\bigg(\sqrt3 \tan x\bigg)^\frac{(a+1)\tan (3x)}{\sin x} = e^\frac{-8\sqrt3}{3}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $ABC$ be a triangle with incenter $I$. Let $M_b$ and $M_a$ be the midpoints of $AC$ and $BC$, respectively. Let $B'$ be the point of intersection of lines $M_bI$ and $BC$, and let $A'$ be the point of intersection of lines $M_aI$ and $AC$.

If triangles $ABC$ and $A'B'C$ have the same area, what are the possible values of $\angle ACB$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
If $f(x)=e^{ax+1}-x(\ln x-2)$ exists a monotonically decreasing interval, then the range of positive number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, e^{-2})}
Calculate the integral \( \int_{-\infty}^{\infty} \frac{x \sin(\pi x)}{(x-3)(x-2)} \, dx \) using the Residue Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5\pi}
Determine whether the sequence $P_n$ is equal to the sequence $G_n$, where $G_0=1$ and $G_{n+1}=F_0G_n+F_1G_{n-1}+\cdots+F_nG_0$, with $F_n$ being the $n$th term of the Fibonacci sequence defined by $F_0=F_1=1$ and $F_{n+1}=F_n+F_{n-1}$. The sequence $P_n$ is defined by $P_0=P_1=1$, $P_2=2$, and $P_{n+1}=2P_n+P_{n-1}$ for $n>1$. Is $P_n=G_n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{P_n = G_n}
A question and answer forum has 200 members. The average number of answers posted by each member on the forum is three times as many as the number of questions asked. Each user posts an average of 3 questions per hour. How many questions and answers are posted on the forum by its users in a day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43,200}
Find the derivative with respect to \( a \) of the solution to the initial value problem at \( a = 0 \):

\[ x' = x^2 - x\cos t, \quad x(0) = a \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-\sin t}}
4. Find the number of all 5-digit numbers (in base 10) each of which contains the block 15 and is divisible by 15 . (For example, 31545,34155 are two such numbers.)

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{479}
Suppose $f$ is an analytic function on the unit disk $D = \{ z \in \mathbb{C} : |z| < 1 \}$ and continuous up to the closure $\bar{D}$. If $f$ vanishes on a non-null segment of the boundary, such as $\{ e^{it}, 0 \leq t \leq \pi/2 \}$, is $f$ necessarily zero everywhere on the boundary? Answer "yes" or "no".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
Let $f$ be a differentiable function defined over a convex set $X$. Suppose that for all $x, \overline{x} \in X$ with $x \neq \overline{x}$, the following strict inequality holds: $\langle \nabla f(\overline{x}), x - \overline{x}\rangle > 0$. Does this imply that $\overline{x}$ is a local minimum? Answer "yes" or "no."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
Find the equation of the tangent line to the curve $y = g(u(x), v(x))$ at $x = 1$, given that $(u'(1), v'(1)) = (3, 1)$ and the tangent plane to the surface $z = g(u, v)$ at the point $(1, 2)$ is $4 - z - 2u + 5v = 0$. Assume $g(u, v)$ and $g(u(x), v(x))$ are related functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = -x + 13}
Let \( a,b,c,d,e, \) and \( f \) be real numbers. Define the polynomials
 \( P\left( x\right)  = 2{x}^{4} - {26}{x}^{3} + a{x}^{2} + {bx} + c\; \) and \( \;Q\left( x\right)  = 5{x}^{4} - {80}{x}^{3} + d{x}^{2} + {ex} + f. \)
  Let \( S \) be the set of all complex numbers which are a root of either \( P \) or \( Q \) (or both).
 Given that \( S = \{ 1,2,3,4,5\} \) ,compute \( P\left( 6\right)  \cdot  Q\left( 6\right) \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2400
}
Given that \(a^2 + b^2 + c^2 = 1\) where \(a, b, c\) are real numbers, find the maximum value of the expression \((4a-3b)^2 + (5b-4c)^2 + (3c-5a)^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Determine the range of values of \( p \) for which the improper integral \( \int_{0}^{1} \frac{\ln x}{x^p(1-x)^{1-p}} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
Ricsi, Dénes, and Attila often play ping pong against each other, with two of them standing on one side. Dénes and Attila win against Ricsi three times as often as they lose; Dénes wins against Ricsi and Attila as often as he loses; finally, Attila wins against Ricsi and Dénes twice as often as he loses. Recently, they spent an entire afternoon playing, during which they played six matches, with two players on each side. What is the probability that Ricsi won at least once, either alone or as part of a pair?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15}{16}}
There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,$ $2,$ $3,$ $...,$ $n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that


($1$) for each $k,$ the mathematician who was seated in seat $k$ before the break is seated in seat $ka$ after the break (where seat $i + n$ is seat $i$);


($2$) for every pair of mathematicians, the number of mathematicians sitting between them after the break, counting in both the clockwise and the counterclockwise directions, is different from either of the number of mathematicians sitting between them before the break.

Find the number of possible values of $n$ with $1 < n < 1000.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{332}
6. 22 For a given positive integer $k$, define $f_{1}(k)$ as the square of the sum of the digits of $k$, and let
$$f_{n+1}(k)=f_{1}\left(f_{n}(k)\right),$$

Find: $f_{1991}\left(2^{1990}\right)$.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
For the curve defined by \(x^3 + y^3 = c^3\), the tangent at the point \((a, b)\) passes through the point \((a_1, b_1)\). Given that \(\frac{a_1}{a} + \frac{b_1}{b} = -k\), find the value of \(k\). Assume that \((a_1, b_1)\) also lies on the curve.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Four siblings BRYAN, BARRY, SARAH and SHANA are having their names monogrammed on their towels. Different letters may cost different amounts to monogram. If it costs $21 to monogram BRYAN,$25 to monogram BARRY and $18 to monogram SARAH, how much does it cost to monogram SHANA?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
What is the largest integer that is a divisor of \[
(n+1)(n+3)(n+5)(n+7)(n+9)
\]for all positive even integers $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the values of $x$ for which the series $$\sum_{n=2}^{\infty} \frac{x^{n^2}}{n \log(n)}$$ converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 1)}
A rectangular box has integer side lengths in the ratio $1: 3: 4$. Which of the following could be the volume of the box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96
}
99.2. Consider 7-gons inscribed in a circle such that all sides of the 7-gon are of different length. Determine the maximal number of $120^{\circ}$ angles in this kind of a 7-gon.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the degree of the splitting field of the polynomial \( f = X^4 + 2X^2 + 2 \) over \( \mathbf{Q} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
One rainy afternoon you write the number  $1$  once, the number  $2$  twice, the number  $3$  three times, and so forth until you have written the number  $99$  ninety-nine times. What is the  $2005$  th digit that you write?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Consider the inequality
\[(a_1+a_2+\dots+a_n)^2\ge 4(a_1a_2+a_2a_3+\cdots+a_na_1).\]
 Find all $n\ge 3$ such that the inequality is true for positive reals.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$n\geq4$

}
A biased coin has a $ \dfrac{6 + 2\sqrt{3}}{12} $ chance of landing heads, and a $ \dfrac{6 - 2\sqrt{3}}{12} $ chance of landing tails. What is the probability that the number of times the coin lands heads after being flipped 100 times is a multiple of 4? The answer can be expressed as $ \dfrac{1}{4} + \dfrac{1 + a^b}{c \cdot d^e} $ where $ a, b, c, d, e $ are positive integers. Find the minimal possible value of $ a + b + c + d + e $.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
8、A unit square has its center coinciding with the center of a circle and is located inside the circle. If a point is randomly selected on the circle, and the probability that from this point one can see two complete sides of the square is $\mathbf{1 / 2}$, then the radius of the circle is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{1+\frac{\sqrt{2}}{2}}}
Use Parseval's identity to find the length \(|f|\) of the function \(f(x) = 2 \cos(14x) + 4 \cos(11x) + 2 \sin(27x) - \cos(19x) + 5 \cos(140x)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\sqrt{2\pi}}
Identify the space obtained as the $L^2$-closure of polynomials $p(z)$ in the $z$-variable, where $z \in \Omega \subset \mathbb{C}$ for a given open and bounded set $\Omega$. Does this space have a classical interpretation in complex analysis?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A^2(\Omega)}
Let  $ABC$  be a triangle with circumcenter  $O$  such that  $AC = 7$ . Suppose that the circumcircle of  $AOC$  is tangent to  $BC$  at  $C$  and intersects the line  $AB$  at  $A$  and  $F$ . Let  $FO$  intersect  $BC$  at  $E$ . Compute  $BE$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{2}}