inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the number of self-intersections in a regular \((2018 ; 25)\)-star, where \(2018\) and \(25\) are coprime, \(2018 \geq 5\), and \(25 < \frac{2018}{2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48432}
fractions are corrected by using each of the numbers $1,2,...,26$ once.Example:$\frac{12}{5},\frac{18}{26}.... $
What is the maximum number of fractions which are integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
12
}
Determine the number of conjugacy classes in a group of order $625$ if its center has order $25$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{145}
Two decks, each containing 36 cards, were placed on a table. The first deck was shuffled and placed on top of the second deck. For each card in the first deck, the number of cards between it and the same card in the second deck was counted. What is the sum of these 36 numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1260}
Let $a_1, a_2, \ldots, a_{2017}$ be the 2017 distinct complex numbers which satisfy $a_i^{2017} = a_i + 1$ for $i = 1, 2, \ldots, 2017$. Compute $$\sum_{i=1}^{2017} \frac{a_i}{a_i^2 + 1}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
Evaluate the integral \( \int \int \int_A \frac{dxdydz}{(x+y+1)^2} \) where \( A=\{(x,y,z):0<x<1, 0<x+y<1, 0<z(x+y+1)<1\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{8}}
Find a real number $x$ such that $x^5 - x - 1 = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.1673}
Determine the number of roots of the equation \(4\cos(e^x) = 2^x + 2^{-x}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the integral \( \int_0^\infty \frac{\sin x(1-\cos x)}{x^2} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
In a cube with edges of length 2 units, point Q is the intersection of the diagonals of one face. Calculate the length of QR. Express your answer as a radical.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{6}}
In a restaurant, clients arrive following a homogeneous Poisson process with an average rate of 10 clients per hour. Each client independently has a 20% probability of eating at the restaurant and an 80% probability of taking away. On average, how many clients should be expected before one eats at the restaurant?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $N \subset 2^X$. Let $M$ be the $\sigma$-algebra generated by $N$ and let $M'$ be the algebra generated by $N$. What is the relationship between $M$ and $M'$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{M' \subset M}
Find four consecutive odd numbers, none of which are divisible by 3, such that their sum is divisible by 5. What is the smallest possible value of this sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Given real numbers $x$ and $y$ that satisfy the equation $\frac{1}{x} + \frac{2}{y} = \sqrt{xy}$, find the minimum value of $xy$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Evaluate the integral \( \int_{-1}^{2} (x^3 - 2) \, d[x] \), where \([x]\) is the floor function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral \( \int_0^{\frac{\pi}{2}} \log\left(\frac{1+a\cos(x)}{1-a\cos(x)}\right)\frac{1}{\cos(x)}dx \) for \( |a| \le 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \arcsin a}
Dave walks to school and averages 85 steps per minute, with each step being 80 cm long. It now takes him 15 minutes to get to school. Jack, walking the same route to school, takes steps that are 72 cm long and averages 104 steps per minute. Find the time it takes Jack to reach school.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13.62}
In a game with 111,400 rounds numbered from 0 to 111,400, a player has won 3,000 rounds. For each winning round, the last three digits are noted (e.g., Round 89,345 becomes "345"). What is the expected number of three-digit endings, from "000" to "999", that do not appear among these 3,000 winning rounds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Calculate the variational derivative of the expression $-f\nabla^2f$ with respect to $f$. Express your answer as $\frac{\delta(-f\nabla^2f)}{\delta f}$. What is the result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\nabla^2 f}
Dan is holding one end of a 26-inch long piece of light string that has a heavy bead on it with each hand (so that the string lies along two straight lines). If he starts with his hands together and keeps his hands at the same height, how far does he need to pull his hands apart so that the bead moves upward by 8 inches?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24 \ \text{inches}}
Evaluate the contour integral \(\oint_{|z| = 3} \frac{z^3e^{1/z}}{z^3+1}dz\) using the Residue Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
Three uncoordinated aliens launch a 3-day attack on 4 galaxies. Each day, each of the three aliens chooses a galaxy uniformly at random from the remaining galaxies and destroys it. They make their choice simultaneously and independently, so two aliens could destroy the same galaxy. If the probability that every galaxy is destroyed by the end of the attack can be expressed as \(\frac{m}{n}\) for relatively prime positive integers \(m,n\), what is \(m+n\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1151}
Determine the number of positive integers, each with exactly \(2^{2005}\) digits, where each digit is either 7 or 8, such that among any two chosen integers, at most half of their corresponding digits are the same.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2006}}
Let \( F: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) be a linear transformation such that \( F(0,0,1) = (2,-1,\alpha) \). The vectors \( (1,0,-1) \) and \( (0,1,0) \) are eigenvectors of \( F \) with associated eigenvalues \( 3 \) and \( -9 \) respectively. Determine the value of \( \alpha \) given that \( -2 \) is also an eigenvalue of \( F \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
Find the smallest integer \( n \) such that a square can be decomposed into \( n \) non-overlapping triangles, all of whose angles are acute.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A necklace consists of 100 beads of red, blue, and green colors. It is known that among any five consecutive beads, there is at least one blue bead, and among any seven consecutive beads, there is at least one red bead. What is the maximum number of green beads that can be in this necklace? (The beads in the necklace are arranged cyclically, meaning the last one is adjacent to the first one.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Calculate the value of $S_{10} + I_{10}$, where $S_n = \sum_{i=1}^n \frac{1}{i}$ and $I_n = \int_{1}^{n} \frac{x - \lfloor x \rfloor}{x^2} \, dx$. Given that $I_{10} = \ln(10) - \int_{1}^{10} \frac{\lfloor x \rfloor}{x^2} \, dx$, evaluate the integral $\int_{1}^{10} \frac{\lfloor x \rfloor}{x^2} \, dx$ to find $S_{10} + I_{10}$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \ln 10}
Let $A$ be a $6 \times 6$ matrix with characteristic polynomial $x(x+1)^2(x-1)^3$. Find the dimension of the centralizer $\gamma = \{B \in M_6(\mathbb{R}) : AB = BA\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Lucas and Monk play chess, and the first to win a match receives a book as a reward. Each match can result in a win, loss, or tie. The probability of there being $k$ matches until someone wins is $\frac{1}{2^{k+2}}$, where $k=1,2,...$. Lucas' chance of winning the book is twice that of Monk. What is the probability that Monk will win the book?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
Determine the distribution of the second greatest value among three independent random variables X1, X2, and X3, each uniformly distributed over the interval (0,1).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6x(1 - x)}
Let  $a$  be real numbers. Given a matrix  $A=\left(\begin{array}{cc} a & 1-a  1-a & a \end{array}\right).$  Let  $\left(\begin{array}{c} x_0  y_0 \end{array}\right)=\left(\begin{array}{c} 1  0 \end{array}\right),$  and the sequences  $\{x_n\},\ \{y_n\}$  are defined by the following equation.

\[ \left(\begin{array}{c} x_n  y_n \end{array}\right)=A \left(\begin{array}{c} x_{n-1}  y_{n-1} \end{array}\right),\ \ n=1,2,\ \cdots \]

then find the necessray and sufficient condition for  $a$  such that the sequence  $\{x_n\}$  converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 < a < 1}
已知复数数列 $\left\{z_{n}\right\}$ 满足 $z_{1}=1, z_{n+1}=\overline{z_{n}}+1+n \mathrm{i}$ ，其中 $n=1,2, 3\cdots$ ，其中 i 是虚数单位, $\overline{z_{n}}$ 表示 $z_{n}$ 的共轭复数, 则 $z_{2015}$ 的实部值为 $\qquad$。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015}
Given 2022 lines in the plane, such that no two are parallel and no three are concurrent. We denote \( E \) as the set of their intersection points. We want to assign a color to each point in \( E \) such that any two points on the same line, whose connecting segment does not contain any other point from \( E \), have different colors. What is the minimum number of colors required to achieve this coloring?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the size of the conjugacy class of the permutation $(123)(456)$ in the alternating group $A_7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{280}
将 $1,2,3,4,5,6,7,8$ 分别放在正方体的八个顶点上，使得每一个面上的任意三个数之和均不小于 10 。求每一个面上四个数之和的最小值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Compute the flux of the vector field \( \vec{F} \) through the surface \( S \), where \( \vec{F} : U \subseteq \mathbb{R}^3 \to \mathbb{R}^3 \) is given by:
\[
\vec{F} = \left( \frac{x}{(x^2 + y^2 + z^2)^{3/2}}, \frac{y}{(x^2 + y^2 + z^2)^{3/2}}, \frac{z}{(x^2 + y^2 + z^2)^{3/2}} \right)
\]
where \( U = \mathbb{R}^3 \setminus \{(0, 0, 0)\} \). The surface \( S \) is a sphere of radius 12345 centered at the origin, with a small section chopped off from the top and replaced by a flat disk, ensuring \( S \) remains closed. Assume \( S \) is oriented with an outward-pointing normal. Calculate the flux of \( \vec{F} \) through \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
A regular  $2008$ -gon is located in the Cartesian plane such that  $(x_1,y_1)=(p,0)$  and  $(x_{1005},y_{1005})=(p+2,0)$ , where   $p$  is prime and the vertices, \[(x_1,y_1),(x_2,y_2),(x_3,y_3),\cdots,(x_{2008},y_{2008}),\]
are arranged in counterclockwise order. Let \begin{align*}S&=(x_1+y_1i)(x_3+y_3i)(x_5+y_5i)\cdots(x_{2007}+y_{2007}i),T&=(y_2+x_2i)(y_4+x_4i)(y_6+x_6i)\cdots(y_{2008}+x_{2008}i).\end{align*} Find the minimum possible value of  $|S-T|$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the constant-coefficient linear differential equation of the smallest possible order that has \( y(x) = x^3 \) as a solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y'''' = 0}
Given a power function $y = f(x)$ whose graph passes through the point $(2, \frac{\sqrt{2}}{2})$, if $f(m) = 2$, then $m = \_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Calculate the Euler characteristic of a topological space formed by joining two projective planes, each with two punctures, along their punctures.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Elbert and Yaiza each draw $10$ cards from a $20$-card deck with cards numbered $1,2,3,\dots,20$. Then, starting with the player with the card numbered $1$, the players take turns placing down the lowest-numbered card from their hand that is greater than every card previously placed. When a player cannot place a card, they lose and the game ends.

Given that Yaiza lost and $5$ cards were placed in total, compute the number of ways the cards could have been initially distributed. (The order of cards in a player’s hand does not matter.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{324}
In the plane, two points $A$ and $B$ are given, with the distance between them being $d$. Construct a square such that points $A$ and $B$ lie on its boundary, in a way that minimizes the sum of the distances from point $A$ to the vertices of the square. What is this minimum sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1 + \sqrt{2})d}
Find the closed-form expression for the limit:

$$\lim_{n\to \infty}\left(\frac{1}{ \Gamma\left(\displaystyle \frac{1}{1}\right)}+ \frac{1}{ \Gamma\left( \displaystyle \frac{1}{2}\right)}+ \cdots + \frac{ 1}{ \Gamma \left( \displaystyle \frac{1}{ n}\right) }- \log\left( \Gamma\left(\displaystyle\frac{1}{n}\right)\right)\right),$$

where the sequence is known as the Gammaharmonic series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\gamma}
Students in a class participate in a tree-planting activity. There are three different types of saplings, labeled A, B, and C, in the nursery. They are to be planted in a row of 5 pits, with the condition that the same type of sapling cannot be adjacent, and only type A saplings can be planted in the first and the fifth pits. Calculate the total number of planting methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \( n > 0 \) be an integer. We have a two-pan balance and \( n \) weights, with masses \( 2^{0}, 2^{1}, \ldots, 2^{n-1} \). We want to place these \( n \) weights onto the balance one by one in such a way that the right pan is never heavier than the left pan. At each step, we choose one of the weights that has not yet been placed on the balance and place it either in the left pan or the right pan, continuing until all the weights have been placed. Determine the number of ways to do this.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n - 1)}
By mistakenly subtracting 20 from a number gives 52, when 4 should have divided by a number. How much do you get when you calculate it correctly?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Calculate the fundamental group of $S^3 - \gamma$, where $\gamma$ is a circle in $\mathbb{R}^3$ and $S^3 = \mathbb{R}^3 \cup \{\infty\}$ is the one-point compactification of $\mathbb{R}^3$. Provide your answer as a group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Mark on the line with red all points of the form $81x + 100y$, where $x, y$ are natural numbers, and paint the remaining integer points blue. Find a point on the line such that any symmetrical integer points relative to it are painted in different colors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4140.5}
Line \( AB \) is the common perpendicular segment of skew lines \( a \) and \( b \). Point \( A \) is on line \( a \) and point \( B \) is on line \( b \). Given that \( AB = 2 \) and the angle between the skew lines \( a \) and \( b \) is \( 30^\circ \), and \( AP = 4 \) on line \( a \), find the distance from point \( P \) to line \( b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Find the minimum page size $s \in \Bbb N$ such that the inequality \( \frac{i + l}{1 + \lfloor \frac{i}{s} \rfloor} \leq s \) holds, given that $i > l$ and $\{i, l\} \in \Bbb N$. What is the minimum value of $s$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the expression \( I(a) = \lim_{\Omega \to \mathbb{R}^3} \, \int_\Omega \nabla \left( \frac{1}{|x-a|} - \frac{1}{|x|} \right) dV \), where \(x\) and \(a\) are vectors in \(\mathbb{R}^3\). Determine the value of \(I(a)\) or explain why the evaluation fails.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4\pi}{3} \mathbf{a}}
Evaluate \( \mathop{\lim }\limits_{{x \rightarrow  {1}^{ - }}}\mathop{\prod }\limits_{{n = 0}}^{\infty }{\left( \frac{1 + {x}^{n + 1}}{1 + {x}^{n}}\right) }^{{x}^{n}} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{e}}
Evaluate the contour integral \( \frac{1}{2 \pi i} \int_{|z| = 1} \frac{6z^{98}}{23z^{99}-2z^{81}+z^4-7} \, dz \) using the residue theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{6}{23}}
Compute the residue of the function \( \frac{(\sin(z))^2}{(\sinh(z))^5} \) at \( z = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{7}{6}}
Given the polynomial \( f(x) = \frac{x^3}{\alpha} + \frac{x^2}{\beta} + \frac{x}{\gamma} + \frac{1}{\delta} \) where \( \alpha, \beta > 0 \), and the condition \( |f(x)| \leq 1 \) for \( |x| \leq 1 \), determine the maximum value of \( \frac{1}{\alpha} + \frac{1}{\beta} + \left| \frac{1}{\gamma} \right| + \left| \frac{1}{\delta} \right| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the largest value of \( k \) such that the \( k \)-th absolute moment \( E[|X|^k] \) is finite for a random variable \( X \) with the cumulative distribution function \( F_X(x) = \frac{1}{1+e^{-x}} \) for \( x \geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given a convex quadrilateral $ABCD$, points $A_1$, $B_1$, $C_1$, and $D_1$ are located on the extensions of sides $AB$, $BC$, $CD$, and $DA$, respectively, such that $|BB_1|=|AB|$, $|CC_1|=|BC|$, $|DD_1|=|DC|$, and $|AA_1|=|DA|$. Express the area of quadrilateral $A_1B_1C_1D_1$ in terms of the area of quadrilateral $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given a positive integer \( n \), let \( G \) be a simple, undirected graph with \( 2n + 1 \) vertices such that there is no cycle of length greater than 3. Find the maximum number of edges in \( G \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3n}
For which values of \(a\) does the linear transformation \(L: \mathbb{R}^3 \rightarrow \mathbb{R}^3\) satisfy \(v_1, v_2 \in \ker(L)\) and \(L(v_3) = (1, 1, 1)\), where \(v_1 = (1, a, a^2)\), \(v_2 = (a^2, 1, a)\), and \(v_3 = (a, a^2, 1)\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \neq 1}
For an \( n \times  n \) matrix \( A \) , let \( {X}_{i} \) be the set of entries in row \( i \) , and \( {Y}_{j} \) the set of entries in column \( j \) , \( 1 \leq  i,j \leq  n \) . We say that \( A \) i
 golden if \( {X}_{1},\ldots ,{X}_{n},{Y}_{1},\ldots ,{Y}_{n} \) are distinct sets. Find the least integer \( n \) such that there exists a \( {2004} \times  {2004} \) golden
 matrix with entries in the set \( \{ 1,2,\ldots ,n\} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$n=13$}
Calculate the flux of the vector field \( F(x,y,z) = (2x+y-z, x-2y+z, x+y-2z) \) through the surface defined by the equation \( |x+y-z| + |x-2y+z| + |x+y-2z| = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{64}{9}}
Calculate the integral \( I = \int_0^1 |f(x) - f^{-1}(x)| \, dx \), where \( f(x) = x e^{x^2 - 1} \) and \( f: [0, 1] \to [0, 1] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{e}}
Consider a cross-shaped grid of size 2017, where each cell initially contains a 1. A transformation \( \pi \) is defined as selecting any 3x3 sub-cross of the grid and multiplying all the cells inside by -1. Determine the minimum number of \( \pi \) transformations required to change every cell in the grid to -1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
Calculate the exterior derivative \(d\omega(X,Y)\), where \(\omega = 2x\,dx \wedge dy + y^2\,dx \wedge dz\) is a differential form, and \(X = x^2y\frac{\partial}{\partial y} + x\frac{\partial}{\partial z}\), \(Y = x\frac{\partial}{\partial y}\) are vector fields.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral \( \int\limits_{t,t'}{\frac{dt\,dt'}{\|x(t)-y(t')\|}} \) where \( x(t) \) and \( y(t') \) are curves traversing the boundary of \([0,1]^2\) in \( \mathbb{R}^2 \) counterclockwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
A crazy physicist has discovered a new particle called an omon. He has a machine that takes two omons of mass $a$ and $b$ and entangles them. This process destroys the omon with mass $a$, preserves the one with mass $b$, and creates a new omon whose mass is $\frac{1}{2}(a+b)$. The physicist can then repeat the process with the two resulting omons, choosing which omon to destroy at every step. The physicist initially has two omons whose masses are distinct positive integers less than $1000$. What is the maximum possible number of times he can use his machine without producing an omon whose mass is not an integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let $S_n$ and $T_n$ be the respective sums of the first $n$ terms of two arithmetic series. If $S_n:T_n=(7n+1):(4n+27)$ for all $n$, the ratio of the eleventh term of the first series to the eleventh term of the second series is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4/3}
Find the number of permutations \( a_1, a_2, \ldots, a_{10} \) of the numbers \( 1, 2, \ldots, 10 \), such that \( a_{i+1} \) is not less than \( a_i - 1 \) for \( i = 1, 2, \ldots, 9 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{512}
Evaluate the double summation and provide a closed formula for it: $$\sum_{i=1}^b \sum_{j=i-b}^{i-1} j$$ where the evaluation considers different cases based on the relationship between \(i\) and \(b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
On a $4 \times 4 \times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing $A$, $B$, and $C$ is closest to
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.1}
Determine the internal angle that guarantees two isosceles triangles are similar. Express your answer as a single angle value in degrees, including the degree symbol (°).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60°}