inclusionAI/Ring-lite-rl-data huggingface dataset:
In acute triangle $ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and they satisfy $a\cos B = b(1 + \cos A)$. The area of triangle $ABC$ is $S=2$. Find the range of $(c + a - b)(c + b - a)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(8\sqrt{2} - 8, 8)}
Consider the group \( G = \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} : a, b, c \in \mathbb{R} \right\} \) and its action \( \pi \) on \( \mathbb{R}^2 \) defined by \( \pi_g(x) = gx \) for all \( x \in \mathbb{R}^2 \). Determine the number of distinct orbits of this action on \( \mathbb{R}^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the limit of the function \( f(x,y) = -\dfrac{y^4+x^3y-xy^3-x^4}{\left(y^2+x^2\right)^2} \) as \((x,y)\) approaches \((0,0)\) along the path \(x=y\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
5. Given six cards, on which the digits $1,2,4,5,8$ and a comma are written. From them, all possible numbers are formed (each card must be used exactly once, the comma cannot stand at the beginning or at the end of the number). What is the arithmetic mean of all such numbers?

(M. V. Karluкова)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1234.4321}
Kozhevnikov P.A

On each of two parallel lines $a$ and $b$, 50 points were marked.

What is the maximum possible number of acute-angled triangles with vertices at these points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41650}
In triangle \( ABC \) with sides \( AB = 8 \), \( AC = 4 \), \( BC = 6 \), the angle bisector \( AK \) is drawn, and on the side \( AC \) a point \( M \) is marked such that \( AM : CM = 3 : 1 \). Point \( N \) is the intersection point of \( AK \) and \( BM \). Find \( AN \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{18\sqrt{6}}{11}}
In how many ways can one fill a \(4 \times 4\) grid with a 0 or 1 in each square such that the sum of the entries in each row, column, and long diagonal is even?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
King Qi and Tian Ji are competing in a horse race. Tian Ji's top horse is better than King Qi's middle horse, worse than King Qi's top horse, Tian Ji's middle horse is better than King Qi's bottom horse, worse than King Qi's middle horse, and Tian Ji's bottom horse is worse than King Qi's bottom horse. Now, randomly select one horse from each side for a race, then the probability of Tian Ji's horse winning is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
11. In triangle $ABC$, the altitude $BH$ is drawn, point $O$ is the center of the circumscribed circle around it, and the length of its radius is $R$. Find the measure in radians of the largest of the angles $ACB$ and $BAC$, given that $R=(2 / 3) \cdot B H=(4 / 3) \cdot O H$. If necessary, round the found value to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.34}
Determine the values of $x$ for which the series \( \sum \frac{x^n}{x^{2n}-1} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -1) \cup (-1, 1) \cup (1, \infty)}
Find the ratio of the area of $\triangle BCY$ to the area of $\triangle ABY$ in the diagram if $CY$ bisects $\angle BCA$. [asy]
import markers;

real t=34/(34+28);
pair A=(-17.18,0);
pair B=(13.82,0);
pair C=(0,30);
pair Y=t*B+(1-t)*A;

draw(C--A--B--C--Y);

label("$A$",A,SW);
label("$B$",B,E);
label("$C$",C,N);
label("$Y$",Y,NE);

label("$32$",.5*(B+A),S);
label("$34$",.5*(B+C),NE);
label("$28$",.5*(A+C),NW);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{14}}
Determine the set of real numbers \( s > 0 \) for which the double integral \(\iint_{(a,b)^2}\frac{1}{|x-y|^s}\,dx\,dy\) converges, where \((a,b)\) is an arbitrary finite open interval of \(\mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1)}
How many natural numbers between 2 and 60 have no primitive roots and require at least four squares to be expressed as a sum of squares?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In the repeating decimal 0.2017, if the sum of all digits from the $m$-th digit to the $n$-th digit after the decimal point is 2017, find the value of $n$ when $m$ takes the minimum value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{808}
Find the coefficient of \(x^{12}y^{24}\) in the expansion of \((x^3 + 2xy^2 + y + 3)^{18}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{111890128896}
Let $A_{1}, \ldots, A_{2022}$ be the vertices of a regular 2022-gon in the plane. Alice and Bob play a game. Alice secretly chooses a line and colors all points in the plane on one side of the line blue, and all points on the other side of the line red. Points on the line are colored blue, so every point in the plane is either red or blue. (Bob cannot see the colors of the points.)

In each round, Bob chooses a point in the plane (not necessarily among $A_{1}, \ldots$, $A_{2022}$ ) and Alice responds truthfully with the color of that point. What is the smallest number $Q$ for which Bob has a strategy to always determine the colors of points $A_{1}, \ldots, A_{2022}$ in $Q$ rounds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Mr. Rose, Mr. Stein, and Mr. Schwartz start at the same point around a circular track and run clockwise. Mr. Stein completes each lap in  $6$  minutes, Mr. Rose in  $10$  minutes, and Mr. Schwartz in  $18$  minutes. How many minutes after the start of the race are the runners at identical points around the track (that is, they are aligned and are on the same side of the track) for the first time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Consider a random walker starting at $S_0 = 2$ on the integers. The walker moves according to the following rules:

- If the walker is on the $n$th prime number $p_n$, she moves to either $p_n + 1$ or $p_{n+1}$ with equal probability.
- If the walker is on a composite number $x$, she moves to one of the prime factors of $x$, each with probability $1/\omega(x)$, where $\omega(n)$ denotes the number of distinct prime factors of $n$.

Determine whether the probability $\mathbb{P}(\sup_{n\ge 0} S_n = \infty)$ is 1 or less than 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
(15) The number of planes equidistant from the four vertices of a spatial quadrilateral is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Ant Amelia starts on the number line at \(0\) and crawls in the following manner. For \(n=1,2,3,\) Amelia chooses a time duration \(t_n\) and an increment \(x_n\) independently and uniformly at random from the interval \((0,1).\) During the \(n\)th step of the process, Amelia moves \(x_n\) units in the positive direction, using up \(t_n\) minutes. If the total elapsed time has exceeded \(1\) minute during the \(n\)th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most \(3\) steps in all. Find the probability that Amelia’s position when she stops will be greater than \(1\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the value of the third derivative \( y''' \) at \( x = 1 \) for the function defined by the equation \( x^2 + xy + y^3 = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Evaluate the integral in a closed form: \[ \int_{25\pi/4}^{53\pi/4} \frac{1}{(1+2^{\sin x})(1+2^{\cos x})}\,dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7\pi}{4}}
Let \( x \) be a non-zero real number such that 

\[ \sqrt[5]{x^{3}+20 x}=\sqrt[3]{x^{5}-20 x} \]. 

Find the product of all possible values of \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5}
Given the function $y = x - 5$, let $x = 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5$, we can obtain 10 points on the graph of the function. Randomly select two points $P(a, b)$ and $Q(m, n)$ from these 10 points. What is the probability that $P$ and $Q$ lie on the same inverse proportion function graph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{45}}
Mr. Rose is making Platonic solids! If there are five different types of Platonic solids, and each Platonic solid can be one of three colors, how many different colored Platonic solids can Mr. Rose make?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Points $X$ and $Y$ lie outside regular pentagon $ABCDE$ such that $ABX$ and $DEY$ are equilateral triangles. Find the degree measure of $\angle XCY$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Evaluate the integral \( \int g_\epsilon(x)f(x)\,\mathrm{d}^3x \) where \( g_\epsilon(x) = \epsilon^{-3}g(\epsilon^{-1}x) \), \( g \in L^1(\mathbb{R}^3) \cap L^2(\mathbb{R}^3) \) with \( \int g(x)\,\mathrm{d}^3x = 1 \), and \( f(x) \) is a smooth function with compact support.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(0)}
Consider the permutation of $1,2,...,n$, which we denote as $\{a_1,a_2,...,a_n\}$. Let $f(n)$ be the number of these permutations satisfying the following conditions:
(1)$a_1=1$
(2)$|a_i-a_{i-1}|\le2, i=1,2,...,n-1$
what is the residue when we divide $f(2015)$ by $4$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Vanya thought of a seven-digit number and then subtracted the sum of all its digits except one. He got 9875352. What number would Vanya get if he subtracted the sum of all the digits except the second one from the left?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9875357}
Find the integral basis for the ring of algebraic integers in the quadratic field $\mathbb{Q}[\sqrt{D}]$ when $D \equiv 2 \pmod{4}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\{1, \sqrt{D}\right\}}
How many arrangements of the letters in "DISAPPEARANCES" satisfy the following conditions: (i) there are at least two letters between each 'A', (ii) the arrangement ends with an 'A', and (iii) the consonants are not in alphabetical order?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{179607780}
Find the mass percentage of C in CO
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42.88\%}
Given that \( a_1, a_2, a_3 \) form an arithmetic sequence with \( a_1 + a_2 + a_3 = 15 \), and \( b_1, b_2, b_3 \) form a geometric sequence with \( b_1 b_2 b_3 = 27 \), if \( a_1 + b_1, a_2 + b_2, a_3 + b_3 \) are positive integers and form a geometric sequence, find the maximum value of \( a_3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A function \( f\left( x\right) \) satisfies \( \left\{  \begin{array}{l} f\left( x\right)  =  - {f}^{\prime \prime }\left( x\right)  - \left( {{4x} - 2}\right) {f}^{\prime }\left( x\right) \\  f\left( 0\right)  = a,f\left( 1\right)  = b \end{array}\right. \)
 Evaluate \( {\int }_{0}^{1}f\left( x\right) \left( {{x}^{2} - x}\right) {dx} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$- \frac{b+a}{11}$}
A convex polyhedron has more faces than vertices. What is the minimum number of triangular faces among its faces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Bernado has an infinite amount of red, blue, orange, pink, yellow, purple, and black blocks. He puts them in the 2 by 2019 grid such that adjacent blocks are of different colors. What is the hundreds digit of the number of ways he can put the blocks in?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1
}
Calculate the value of \( \sqrt{M - 2} + 1 \), where \( M = \lim\limits_{x\to 0 } \left[\dfrac{1 - f(x)^3}{5 \tan^2x}\right] \), \( f(x) = \cos(2x) \cos(4x) \cos(6x) \cos(8x) \cos(10x) \), and \( M \) is finite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the kernel of the homomorphism $D : F[x] \to F[x]$ defined by $D(f(x)) = f'(x)$, where the characteristic of the field $F$ is a prime number $p$. What is the set of polynomials that form this kernel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F[x^p]}
Take the x-axis as horizontal and the y-axis as vertical. A gun at the origin can fire at any angle into the first quadrant (x, y ≥ 0) with a fixed muzzle velocity \( v \). Assuming the only force on the pellet after firing is gravity (acceleration \( g \)), which points in the first quadrant can the gun hit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y \leq \frac{v^2}{2 g} - \frac{g x^2}{2 v^2}}
Given that \( P_n(0) = -\frac{5}{16} \), find the value of \( \int_{-1}^{1} P_n^2(x) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{13}}
The integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that the first player removes three cards with consecutive integers. Finally, the second player removes four cards with consecutive integers.
What is th smallest value of $n$ for which the second player can ensure that he competes both his moves?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
The graph of the function y = sin(2x + π/3) is translated in such a way that the resulting graph is symmetric about the point (-π/12, 0). What is the translation type and amount? Express your answer in the format'move to the [left/right] by [amount]'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{move to the right by π/12}
Five girls ran a race. Fiona started first, followed by Gertrude, then Hannah, then India, and lastly Janice. Whenever a girl overtook another girl, she was awarded a point. Given that India finished first, followed by Gertrude, Fiona, Janice, and lastly Hannah, calculate the minimum total number of points that could have been awarded.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Point $P$ is a moving point on the parabola $y^{2}=4x$. The minimum value of the sum of the distances from point $P$ to point $A(0,-1)$ and from point $P$ to the line $x=-1$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Let $a > 1$ be a positive integer. The sequence of natural numbers $\{a_n\}_{n \geq 1}$ is defined such that $a_1 = a$ and for all $n \geq 1$, $a_{n+1}$ is the largest prime factor of $a_n^2 - 1$. Determine the smallest possible value of $a$ such that the numbers $a_1, a_2, \ldots, a_7$ are all distinct.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
For positive reals $p$ and $q$, define the remainder when $p$ is divided by $q$ as the smallest nonnegative real $r$ such that $\frac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_{1}$ and $r_{2}$ be the remainder when $a \sqrt{2}+b \sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ respectively. Find the number of pairs $(a, b)$ such that $a, b \leq 20$ and $r_{1}+r_{2}=\sqrt{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
What fraction of the volume of a parallelepiped is the volume of a tetrahedron whose vertices are the centroids of the tetrahedra cut off by the planes of a tetrahedron inscribed in the parallelepiped?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{24}}
Yesterday, Ben took his family fishing. They caught a total of 15 fish of different sizes and species:

- Ben caught 4 fish: two Bluefish that were 5 inches and 9 inches long, and two Yellowtail that were 9 inches long
- Judy caught 1 fish: a Red Snapper that was 11 inches long
- Billy caught 3 fish: a Bluefish that was 6 inches long, and two Yellowtail that were 6 inches and 10 inches long
- Jim caught 2 fish: a Red Snapper that was 4 inches long, and a Bluefish that was 8 inches long
- Susie caught 5 fish: a Red Snapper that was 3 inches long, two Yellowtail that were 7 inches and 12 inches long, and two Bluefish that were 12 inches long

Each fish species has its own unique minimum size limit to be kept:
- Bluefish: 7 inches
- Yellowtail: 6 inches
- Red Snapper: 8 inches

If each fish can give them 2 filets, how many fish filets will Ben and his family have from the fish that meet the size limits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Two ants sit at the vertex of the parabola $y = x^2$. One starts walking northeast (i.e., upward along the line $y = x$) and the other starts walking northwest (i.e., upward along the line $y = -x$). Each time they reach the parabola again, they swap directions and continue walking. Both ants walk at the same speed. When the ants meet for the eleventh time (including the time at the origin), their paths will enclose 10 squares. What is the total area of these squares?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{770}
How many elements of order $p$ are there in $GL_{2}(\Bbb Z_{p})$, where $p$ is a prime?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2-1}
In a grid of numbers and letters, starting from the letter A in the middle of the grid, calculate the total number of different paths that can spell out "AMC8". Each path must move to an adjacent (up, down, left, or right) letter or number only once (no diagonal moves allowed). Provide your answer as a single integer representing the total number of paths.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
用 $6$ 根等长的细铁棒焊接成一个正四面体形框架，铁棒的粗细和焊接误差不计。设此框架能容纳得下的最大球的半径为 $R_{1}$,能包容此框架的最小球的半径为 $R_{2}$ 。则 $\frac{R_{1}}{R_{2}}$等于。原始答案为 $\frac{\sqrt{m}}{n}$ 格式，请给出 $m + n$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
For each pair of real numbers \((x, y)\) with \(0 \leq x \leq y \leq 1\), consider the set
\[ A = \{ x y, x y - x - y + 1, x + y - 2 x y \}. \]

Let the maximum value of the elements in set \(A\) be \(M(x, y)\). Find the minimum value of \(M(x, y)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4/9}
An ellipse satisfies the following optical property: a ray of light emitted from one focus of the ellipse, after being reflected by the ellipse, passes through the other focus. Now, consider a horizontally placed elliptical billiard table satisfying the equation: $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$, where points $A$ and $B$ are its two foci. When a stationary ball is placed at point $A$ and starts moving in a straight line, after being reflected by the elliptical wall (not at the endpoints of the major axis) and returning to point $A$, what is the shortest distance the ball travels? Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Evaluate the integral $$\int_0^{\pi/2} \sin^{10}x\cdot \cos(12x)\,\text{d}x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The graph of the function $y=g(x)$ is shown below. For all $x > 5$, it is true that $g(x) > 0.5$. If $g(x) = \frac{x^2}{Dx^2 + Ex + F}$, where $D, E,$ and $F$ are integers, then find $D+E+F$. Assume the function has vertical asymptotes at $x = -3$ and $x = 4$ and a horizontal asymptote below 1 but above 0.5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-24}
Let $c_1, c_2, \ldots, c_{6030}$ be 6030 real numbers. Suppose that for any 6030 real numbers $a_1, a_2, \ldots, a_{6030}$, there exist 6030 real numbers $\{b_1, b_2, \ldots, b_{6030}\}$ such that:
\[a_n = \sum_{k=1}^{n} b_{\gcd(k,n)}\]
and
\[b_n = \sum_{d\mid n} c_d a_{n/d}\]
for $n=1, 2, \ldots, 6030$. Find $c_{6030}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{528}
在 $\triangle A B C$ 中, $\angle B A C=30^{\circ}, \angle A B C=70^{\circ}, M$ 为形内一点, $\angle M A B=\angle M C A=20^{\circ}$, 求 $\angle M B A$ 的度数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$30^{\circ}$.}
Find all integer pairs \((x, y)\) such that \(2x^3 - 7y^3 = 3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
The cubic polynomial $p(x)$ satisfies $p(2) = 1,$ $p(7) = 19,$ $p(15) = 11,$ and $p(20) = 29.$  Find
\[p(1) + p(2) + p(3) + \dots + p(21).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{315}
Barry has three sisters. The average age of the three sisters is 27. The average age of Barry and his three sisters is 28. What is Barry's age?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
The diagonal of an isosceles trapezoid bisects its obtuse angle. The shorter base of the trapezoid is 3 cm, and the perimeter is 42 cm. Find the area of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Differentiate the expression $\theta^TX^TX\theta$ with respect to the vector $\theta$. Provide the resulting expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 X^T X \theta}
Given the sequence \( 1, (1+2), (1+2+2^{2}), \cdots, (1+2+2^{2}+\cdots+2^{n}) \), find the sum of the first \( n \) terms of this sequence in terms of \( n \). Express your answer as a formula in terms of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n+1} - n - 2}
How many polynomials of degree 3 are contained in the ideal \( I = (x^5 + x^2 + x + 1, x^7 + x^5 + x^3 + x) \) in \( \mathbb{F}_2[x] \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Solve the equation$$arc \sin (\sin x) + arc \cos (\cos x)=0$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
x = 0
}
A number of market women sold their various products at a certain price per pound (different in every case), and each received the same amount - 2s. 2 1/4d. What is the greatest number of women there could have been? The price per pound in every case must be such as could be paid in current money.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
An urn contains b blue balls and r red balls. You repeatedly and independently remove balls from the urn (without returning them) until the first blue ball is drawn. All balls currently in the urn have an equal probability of being selected each time a ball is removed. Define the random variable X as the number of balls that are drawn (number of red balls that are removed plus the first blue ball that is removed). Find E[X], the expected value of random variable X.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + r/(b+1)}
Determine the values of $z \in \mathbb{C}$ for which the series \( \sum_{n=1}^{\infty} \frac{1}{(n+2)^4}(z-2i)^{n^3} \) converges using the root test.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|z - 2i| \leq 1}