inclusionAI/Ring-lite-rl-data huggingface dataset:
In the arithmetic sequence $\{a_{n}\}$, if $\frac{{a}_{9}}{{a}_{8}}＜-1$, and its sum of the first $n$ terms $S_{n}$ has a minimum value, determine the minimum value of $n$ for which $S_{n} \gt 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Find the smallest integer $n \geq 5$ for which there exists a set of $n$ distinct pairs $\left(x_{1}, y_{1}\right), \ldots,\left(x_{n}, y_{n}\right)$ of positive integers with $1 \leq x_{i}, y_{i} \leq 4$ for $i=1,2, \ldots, n$, such that for any indices $r, s \in\{1,2, \ldots, n\}$ (not necessarily distinct), there exists an index $t \in\{1,2, \ldots, n\}$ such that 4 divides $x_{r}+x_{s}-x_{t}$ and $y_{r}+y_{s}-y_{t}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the sum of $453_6$, $436_6$, and $42_6$ in base 6.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1415_6}
Three points are chosen uniformly at random on a circle. What is the probability that no two of these points form an obtuse triangle with the circle's center?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{16}}
Sally, Mark, Sarah, Steve, Jim, Laura, and Matthew are inheriting the money from their grandmother. She left them $874,532.13 to split equally between all of them. If the money is split equally, how much money will each person receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$124,933.16}
In the acute triangle $ABC$, the distance between the feet of the altitudes to sides $AB$ and $BC$ is $24$. Given that $\angle B = 37^\circ$, calculate the circumradius of triangle $ABC$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Determine the maximum order of persistence of excitation for the periodic input signal $u(t)$ given by:
\[
u(t) = \begin{cases} 0, & t = 0,3 \\ 1, & t = 1,4 \\ -1, & t = 2,5 \end{cases}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( n \) be a positive integer. Calculate
\[ 
P = \prod_{k=0}^{n-1} \sin \left(\theta + \frac{k \pi}{n} \right).
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sin(n \theta)}{2^{n-1}}}
How many ways are there to arrange the numbers $1$ through $8$ into a $2 \times 4$ grid such that:
1. The sum of the numbers in each of the two rows is a multiple of $6$.
2. The sum of the numbers in each of the four columns is a multiple of $3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
Determine the smallest positive integer $n$ for which $\sin\left(\frac{\pi}{n}\right)$ cannot be expressed in terms of radicals of integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given that the function \( f(x) = \sin(2x) - 8(a+1)\sin x + (4a^2 + 8a - 14)x \) is strictly increasing for all \( x \in \mathbb{R} \) and has no critical points for all \( x \in \mathbb{R} \), and the set of all values of the parameter \( a \) is \((-
fty, m - \sqrt{n}) \cup (\sqrt{n}, \infty)\), where \( m \) and \( n \) are prime numbers, find the value of \( m^2 + n^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Henry scored 50 points on his Geography test, 70 on his Math test, 66 on his English test, 84 on his Science test, and 75 on his French test. If his History score is a weighted average of these 5 scores, where Geography has a weight of 25%, Math and English have a weight of 20% each, Science has a weight of 15%, and French has a weight of 10%, what was his total score across all 6 subjects?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{404.8}
For any positive integer $n$, we define the integer $P(n)$ by :

$P(n)=n(n+1)(2n+1)(3n+1)...(16n+1)$.

Find the greatest common divisor of the integers $P(1)$, $P(2)$, $P(3),...,P(2016)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $\min (x, y)$ denote the smaller of the numbers $x$ and $y$. For what $\lambda$'s does the equation  
$$\int_{0}^{1} \min (x, y) f(y) d y=\lambda f(x)$$  
have continuous solutions which do not vanish identically in ( 0,1$)$? The largest $\lambda$ can be expressed in the form of $\frac{m}{n\pi^{2}}$. Find the value of $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let \((X, d)\) be a metric space and \(Y \subseteq X\). Suppose there exists an \(x \in X\) and \(r > 0\) such that \(Y \subseteq B(x, r)\). Determine the smallest \(r''\) such that \(conv(Y) \subseteq B(x, r'')\), where \(conv(Y)\) denotes the convex hull of \(Y\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2r}
Let \(ABCD\) be an isosceles trapezoid with parallel bases \(\overline{AB}\) and \(\overline{CD}\), where \(AB > CD\). Line segments from a point inside \(ABCD\) to the vertices divide the trapezoid into four triangles with areas \(2\), \(3\), \(4\), and \(5\), starting with the triangle with base \(\overline{CD}\) and moving clockwise. The original answer for the ratio \(\frac{AB}{CD}\) is in the form \(k + \sqrt{m}\). Please calculate the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \( p(x) \) be a polynomial of degree strictly less than 100. If \( \frac{d^{100}}{d x^{100}} \frac{p(x)}{x^2+1} =\frac{f(x)}{g(x)} \) for some polynomials \( f(x) \) and \( g(x) \), what is the smallest possible degree of \( f(x) \) when \( p(x) \) does not have \( x^2+1 \) as a factor?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Find the greatest number which leaves the same remainder when it divides 25, 57, and a number. The greatest number is 16. What is the third number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{73}
Evaluate the following limit without using L'Hopital's rule:
$$\lim_{x\to\infty}\left[\frac{2}{\pi}\left((x+1)\cos^{-1}\left(\frac{1}{x}\right)\right)-x\right]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi -2}{\pi }\)}
Among the eight vertices of the cube ABCD-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub>D<sub>1</sub>, a plane passes through four of them, and the remaining four vertices are all at equal distances from the plane. Calculate the number of such planes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The function $y=(m^2-m-1)x^{m^2-2m-1}$ is a power function, and it is a decreasing function on $x\in(0, +\infty)$. Then, the real number $m=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
If the inequality \(x^{2}-\log _{m} x<0\) always holds in the interval \(\left(0, \frac{1}{2}\right)\), find the range of values for the real number \(m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[ \frac{1}{16}, 1 \right)}
A 12% stock yielding 10% is quoted at, with a 3% tax rate on dividend income and 2% inflation rate. Calculate the adjusted yield taking into account tax and inflation implications.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.55\%}
Find the inclination angle of the line $\sqrt {2}x+ \sqrt {6}y+1=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\pi}{6}}
Find the number of solutions to
\[\sec \theta + \csc \theta = \sqrt{15}\]where $0 \le \theta \le 2 \pi.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In the 19th century, Danish mathematician Kierkegaard made outstanding contributions to mathematical analysis, especially in the concavity and convexity of functions and inequalities. It is defined that the derivative of a function $f(x)$ on the interval $(a, b)$ is denoted as $f'(x)$, and the derivative of $f'(x)$ on the interval $(a, b)$ is denoted as $f''(x)$. If $f''(x) < 0$ holds for all $x$ in $(a, b)$, then the function $f(x)$ on $(a, b)$ is called a "strictly convex function." If the function $f(x) = e^x - x\ln x - \frac{m}{2}x^2$ is a "strictly convex function" on $(1, 4)$, then the range of $m$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[e^4 - \frac{1}{4}, +\infty)}
Calculate the expected difference between the second and third highest numbers when selecting 5 numbers (with repetition allowed) from the set \( \{1, 2, \ldots, 100\} \) and arranging them in descending order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{50}{3}}
In the class, there are 30 students: excellent students, C-grade students, and D-grade students. Excellent students always answer questions correctly, D-grade students always make mistakes, and C-grade students answer the questions given to them strictly in turn, alternating between correct and incorrect answers. All students were asked three questions: "Are you an excellent student?", "Are you a C-grade student?", and "Are you a D-grade student?". 19 students answered "Yes" to the first question, 12 to the second, and 9 to the third. How many C-grade students are there in this class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given a right prism $ABC-A'B'C'$ with volume $V$, points $P$ and $Q$ are on edges $AA'$ and $OC'$ respectively, and $AP = C'Q$. Calculate the volume of the tetrahedron $B-APQC$ in terms of the original volume $V$. Express your answer as a fraction of $V$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3} V}
Steve writes a letter every 3rd day to his friend. He spends 20 minutes writing the letters. It takes 10 minutes to write 1 page. He also writes a long letter at the end of the month which he puts more thought into. It takes a certain ratio of time per page compared to regular letters, but he spends 80 minutes writing. He writes 24 pages a month. What is the ratio of time spent per page for the long letter compared to a regular letter?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
Find the angle between the tangents to the curve \(3x^2+4xy+5y^2-4=0\) at the points where the lines \(y=(-3/2)x\) and \(y=(-2/5)x\) intersect the curve.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
From a sphere of what is the smallest radius can a regular quadrilateral pyramid with a base edge of 14 and an apothem of 12 be cut?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7\sqrt{2}}
Evaluate the limit: $$ \lim_{x\to\infty}\frac{\ln(1+e^{ax})}{\ln(1+e^{bx})} $$ as \( x \) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{a}{b}}
Solve the equation \(2^x = -2x + 11\) for \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.557}
Let $N$ denote the number of permutations of the $18$-character string $AAAAABBBBBBCCCCCC$ such that:
1. None of the first five letters is an $A$.
2. None of the next six letters is a $B$.
3. None of the last seven letters is a $C$.
Find the remainder when $N$ is divided by $500$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
Let \( T \) be the set of ordered triples \((x, y, z)\), where \( x, y, z \) are integers with \( 0 \leq x, y, z \leq 9 \). Players \( A \) and \( B \) play the following guessing game: Player \( A \) chooses a triple \((x, y, z)\) in \( T \), and Player \( B \) has to discover \( A \)'s triple in as few moves as possible. A move consists of the following: \( B \) gives \( A \) a triple \((a, b, c)\) in \( T \), and \( A \) replies by giving \( B \) the number \( |x+y-a+b|+|y+z-b+c|+|z+x-c+a| \). Find the minimum number of moves that \( B \) needs to be sure of determining \( A \)'s triple.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
在梯形 $A B C D$ 中, $A D / / B C, M$ 在边 $C D$ 上, 有 $\angle A B M=\angle C B D=\angle B C D$, 则 $\frac{A M}{B M}$ 的取值范围是一个区间，该区间的有限端点形如\frac{m}{n}，请给出m+n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let \( X_1, X_2, \ldots, X_n \) be a random sample from a distribution with the following probability mass function (pmf): \( P(X = k_1) = \frac{1 - \theta}{2} \), \( P(X = k_2) = \frac{1}{2} \), and \( P(X = k_3) = \frac{\theta}{2} \), where \( 0 < \theta < 1 \). Find a non-trivial sufficient statistic for \( \theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(N_1, N_3)}
Given that the random vector $(X, Y, Z)$ follows a multivariate normal distribution with expected values $E[X] = 0$, $E[Y] = 1$, $E[Z] = 1$, and covariance matrix:
\[
\begin{bmatrix}
1 & 1 & 0 \\
1 & 2 & 1 \\
0 & 1 & 2
\end{bmatrix}
\]
calculate the variance $\operatorname{Var}(X(Y+Z))$. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let $g: W_m \to W_m$ be an endomorphism such that $g^2 = 0$ and the rank of $g$ is $k$. Calculate the dimension of the intersection of the kernel and image of $g$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k}
Find the area enclosed by the ellipse given by the equation \(5x^2 + 6xy + 2y^2 + 7x + 6y + 6 = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Calculate the expected value of the random variable \( S = \sum_{i=1}^{N} \frac{X_i}{2^i} \), where \( X_1, X_2, \ldots \) are i.i.d. random variables uniformly distributed on \([0,1]\), and \( N \) is the smallest positive integer \( k \) such that \( X_k < X_{k+1} \). If no such \( k \) exists, then \( N = \infty \). Given that \( \Pr(N = n) = \frac{n}{(n + 1)!} \), find \( \mathbb{E}[S] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{e} - 3}
Use the power series expansion of \( \frac{z}{e^z-1} \) to find the value of \( B_5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
已知函数 $y=\left(a \cos ^{2} x-3\right) \sin x$ 的最小值为 -3 ，则实数 $a$ 的取值范围是。请提供该区间的两个端点之和的整数部分。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
146. Find the particular solution of the equation $y^{\prime \prime}-2 y^{\prime}+y=0$, if $y(0)=4, y^{\prime}(0)=2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{x}(4-2x)}
把边长为$1$的正方形$ABCD$沿对角线$AC$折起，当以$A,B,C,D$四点为顶点的三棱锥体积最大时，此三棱锥的外接球的表面积的大小等于__________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2\pi $}
What is the maximum value that the area of the projection of a regular tetrahedron with an edge length of 1 can take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
An $8 \times 8 \times 8$ cube is painted red on $3$ faces and blue on $3$ faces such that no corner is surrounded by three faces of the same color. The cube is then cut into $512$ unit cubes. How many of these cubes contain both red and blue paint on at least one of their faces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
56
}
In a classroom, there are ten students. Aline writes ten consecutive integers on the board. Each student chooses one of the ten integers written on the board, ensuring that no two students choose the same integer. Each student then calculates the sum of the nine integers selected by the other nine students. Each student whose result is a perfect square receives a gift.

What is the maximum number of students who will receive a gift?

A perfect square is an integer of the form \( n^{2} \), where \( n \) is a natural number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $T=[-1,0)\cup [1,2]$ and $g:T\to R$ be a strictly decreasing function such that $g(T)$ is connected. How many discontinuities does $g$ have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of lines that are both normal and tangent to the curve \(y^2 = x^3\), intersecting the curve at two distinct points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The equation $ax^3 + bx^2 + cx + d = 0$ has three distinct solutions. How many distinct solutions does the following equation have:

$4(ax^3 + bx^2 + cx + d)(3ax + b) = (3ax^2 + 2bx + c)^2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Compute the area of the region on the cylinder $x^2+y^2=R^2$ bounded by the planes $z=mx$ and $z=nx$, where $z \ge 0$, $m > n > 0$. Express your answer in terms of $R$, $m$, and $n$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2R^2(m - n)}
Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. Given that $AB=25$ and $PQ = QR = 2.5$, calculate the perimeter of $\triangle APR$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Maria gave a third of her money to her friend Isha, half of what she gave to Florence. Florence received three times as much money as Maria's cousin Rene. Maria gave her three friends $1650. How much money did Rene receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$300}
The sides of a regular polygon of $n$ sides, $n>4$, are extended to form a star. The number of degrees at each point of the star is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{(n-4)180}{n}$}
A sealed bottle, which contains water, has been constructed by attaching a cylinder of radius \(1 \mathrm{~cm}\) to a cylinder of radius \(3 \mathrm{~cm}\). When the bottle is right side up, the height of the water inside is \(20 \mathrm{~cm}\). When the bottle is upside down, the height of the liquid is \(28 \mathrm{~cm}\). What is the total height, in cm, of the bottle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Compute the integral \( \iint_D (3x^3 + 3x^2y + x + y) \, dA \) over the region \( D \) bounded by the curves \( y = x^3 - 4 \), \( y = x^3 + 4 \), \( x + y = 6 \), and \( x + y = 0 \), using the change of variables \( u = x^3 \) and \( v = x + y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
The graph of the function $y=\sin 2x - \sqrt{3} \cos 2x$ is translated to the left by $\frac{\pi}{6}$ units, resulting in the graph of the function $y=g(x)$. Find the value of $g(\frac{5\pi}{6})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\sqrt{3}}
Determine the smallest integer $N$ such that among the $\binom{N}{3}$ circles formed by every triple of $N$ points in general position in 3-D space, there exists at least one pair of linked circles. Note that circles sharing one of the $N$ points do not count as linked.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given that \(T_1\) is a lower \(\alpha \times 100\%\) confidence limit and \(T_2\) is an upper \(\alpha \times 100\%\) confidence limit for the parameter \(\theta\), such that \(P(T_1 < T_2) = 1\), determine the \((2\alpha - 1) \times 100\%\) confidence interval for \(\theta\). Assume \(T_1\) and \(T_2\) are not independent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[T_1, T_2]}
由曲线 $x^{2}=4 y, x^{2}=-4 y, x=4, x=-4$ 围成的图形绕 $y$ 轴旋转一周所得旋转体的体积为 $V_{1}$ ；满足 $x^{2}+y^{2} \leqslant 16, x^{2}+(y-2)^{2} \geqslant 4, x^{2}+(y+2)^{2} \geqslant 4$ 的点组成的图形绕 $y$ 轴旋转一周所得旋转体的体积为 $V_{2}$ ，求 $V_{1}$ 和 $V_{2}$ 的比值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For each integer $m\ge 2$, find the value of $f(2^m)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$$2^{m-1}$$}
Bob chooses a $4$-digit binary string uniformly at random and examines an infinite sequence of uniformly and independently random binary bits. If $N$ is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of $N$. For example, if Bob's string is $0000$ and the stream of bits begins $101000001\ldots$, then $N = 7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Evaluate the contour integral \( \int_\gamma \frac{z}{z^2-i} \, dz \), where \( \gamma \) is a counterclockwise unit square in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi i}
Find the values of \( \lambda \) where the functions \( f(\lambda) = e^{\frac{\ln(\frac{-\lambda\theta^k}{3\lambda-2T})}{k}} \) and \( g(\lambda) = \frac{\ln(\frac{-\lambda}{3\lambda-2T})}{k}+\theta \) intersect, given that \( \theta, k, \) and \( T \) are positive real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{T}{2}}
Determine the minimum number of spheres required to completely shield a point source of light from all directions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
How can we express the sum $\sum_{n=1}^\infty \frac{\Lambda(n)}{n^3 \ln^2 n}$ in terms of the prime zeta function?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sum_{k \geq 1} \frac{1}{k^2} P(3k)}
Given that $a$, $b$, and $c$ are the roots of the cubic equation $x^3 - 4x + 1 = 0$, find the value of the expression \[ \frac{a^2bc}{a^3+1} + \frac{ab^2c}{b^3+1} + \frac{abc^2}{c^3+1}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{3}{4}}
What is the minimum number of vertices required in a bipartite graph with a minimum degree of $d$ to ensure the existence of a cycle of length at least $2d+1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2d + 2}
Find the third derivative \( y'''(x) \) of the function implicitly defined by the equation \( x^2 + xy + y^3 = 1 \) at the point where \( x = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
The angles \( \alpha ,\beta ,\gamma \) of a triangle are in arithmetic progression. If \( \sin {20\alpha },\sin {20\beta } \) , and \( \sin {20\gamma } \) are in arithmetic progression,
 how many different values can \( \alpha \) take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4
}
In the trapezoid \(ABCD\) with bases \(AD\) and \(BC\), the diagonals \(AC\) and \(BD\) intersect at point \(E\). A circle is circumscribed around triangle \(ECB\), and the tangent to this circle at point \(E\) intersects line \(AD\) at point \(F\) such that points \(A, D,\) and \(F\) lie consecutively on this line. Given that \(AF = a\) and \(AD = b\), find \(EF\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{a(b - a)}}
Determine the set of all possible values of \(\varepsilon\) such that the set \(\mathcal{O}\), defined by the condition \(\exists \varepsilon > 0 : \forall x \in \mathcal{O} \; (x-\varepsilon,x + \varepsilon ) \subseteq \mathcal{O}\), is equal to the set of all real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \infty)}