inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the degree of the image of a curve $C \subset \mathbb{P}^n$ of degree $a$ under the Veronese map $\nu_{n,d}: \mathbb{P}^n \longrightarrow \mathbb{P}^{\binom{d+n}{n}-1}$, where the map is defined by $[...,x_j,...] \longrightarrow [...,M_k,...]$ with $M_k$ being monomials of degree $d$ in $\{...,x_j,...\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a d}
Calculate the limit $$\lim_{n \to \infty} \frac{f^{n}(0)}{2^{n}n!}$$ given that $\Omega = D(0,2) \setminus \left\{\frac{1}{2}\right\}$, $f$ is a holomorphic function in $\Omega$, and $\frac{1}{2}$ is a simple pole with residue $1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Find the sum of all prime numbers $p$ such that $\binom{20242024p}{p}\equiv 2024\pmod{p}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
41
}
Let $T$ be the sum of all the real coefficients of the expansion of $(1 + ix)^{2018}$. What is $\log_2(T)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1009}
计算下列回路积分．

$\oint _ { l } \frac { d z } { ( z ^ { 2 } + 1 ) ( z - 1 ) ^ { 2 } } ( l$的方程是$x ^ { 2 } + y ^ { 2 } - 2 x - 2 y = 0 )$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$-\pi i/2$}
Given a hyperbola $\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$ where $a>0$ and $b>0$, the right focus is $F(4,0)$. If the line that passes through point F and has an inclination of $60^\circ$ intersects with only one point on the right branch of the hyperbola, determine the range of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 < a \leq 2}
Mrs. Wilson, the math teacher, has 30 logic puzzles, 18 visual puzzles, and 12 word puzzles that she wants to group into sets for students who finish their tests early. Mrs. Wilson wants each set to be identical, containing the same combination of logic puzzles, visual puzzles, and word puzzles, with no puzzles left over. Additionally, each set must have a minimum of 5 puzzles and the ratio of logic puzzles to visual puzzles should be 2:1. What is the greatest number of sets she can create?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
For a complex number constant $c$, and real number constants $p$ and $q$, there exist three distinct complex values of $x$ that satisfy the equation:
\[
x^3 + cx + p(1 + qi) = 0
\]
Suppose $c$, $p$, and $q$ were chosen such that all three complex roots $x$ satisfy the inequality:
\[
\frac{5}{6} \leq \frac{\mathrm{Im}(x)}{\mathrm{Re}(x)} \leq \frac{6}{5}
\]
where $\mathrm{Im}(x)$ and $\mathrm{Re}(x)$ are the imaginary and real parts of $x$, respectively. The largest possible value of $|q|$ can be expressed as a common fraction \( \frac{m}{n} \), where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let \( S \) be the set of all 3-digit numbers with all digits in the set \( \{1,2,3,4,5,6,7\} \) (so in particular, all three digits are nonzero). For how many elements \( \overline{abc} \) of \( S \) is it true that at least one of the (not necessarily distinct) "digit cycles" \( \overline{abc}, \overline{bca}, \overline{cab} \) is divisible by 7? (Here, \( \overline{abc} \) denotes the number whose base 10 digits are \( a, b \), and \( c \) in that order.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{127}
已知$f:\mathbf{Z}\rightarrow\mathbf{Z}$为偶函数，且$f(1)=1,f(2017)\neq1$。若对任意的$x、y\in\mathbf{Z}$均满足
$$2f(x+y)-f(x)-f(y)\leqslant|f(x)-f(y)|$$，求$f(2018)$的所有可能值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
“斯诺克(Snooker)”是台球比赛的一种，意思是“阻碍、障碍”，所以斯诺克台球有时也被称为障碍台球，是四大“绅士运动”之一，随着生活水平的提高，“斯诺克”也成为人们喜欢的运动之一．$现甲、乙两人进行比赛，比赛采用5局3胜制，各局比赛双方轮流开球(例如：若第一局甲开球，则第二局乙开球，第三局甲开球\ldots \ldots )，没有平局.已知在甲的“开球局”，甲获得该局比赛胜利的概率为 \frac{1}{3} ，在乙的“开球局”，甲获得该局比赛胜利的概率为 \frac{1}{2} ，并且通过“猜硬币”，甲获得了第一局比赛的开球权．$
求甲以3:1赢得比赛的概率；
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{5}{36}$}
The equation $AB \times CD = EFGH$, where each letter $A$, $B$, $C$, $D$, $E$, $F$, $G$, and $H$ represents a different digit, and the values of $A$, $C$, and $E$ are all nonzero, has many solutions. For example, $46 \times 85 = 3910$. Find the smallest value of the four-digit number $EFGH$ for which there is a solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1058}
Determine the big O notation for the function \(f(n, k)\) defined as follows, given that \(n < k\):

\[f(n, k)=\sum_{i=1}^{n-1}(n-i)(k-i)\log(k-i)\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^2 k \log k)}
The figure \( F \) is the intersection of \( n \) circles (\( n \geq 2 \), radii are not necessarily the same). What is the maximum number of curved "sides" that the figure \( F \) can have? (A curved side is a segment of the boundary of \( F \) that belongs to one of the circles and is bounded by intersection points with other circles.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n-2}
The side surface of a cylinder unfolds into a rectangle with side lengths of $6\pi$ and $4\pi$. The surface area of the cylinder is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24\pi^2 + 8\pi}
A five-digit positive integer is created using each of the odd digits $1, 3, 5, 7, 9$ once so that the thousands digit is larger than the hundreds digit, the thousands digit is larger than the ten thousands digit, the tens digit is larger than the hundreds digit, and the tens digit is larger than the units digit. How many such five-digit positive integers are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Find the limit of the sequence \((b_n)_{n \in \mathbb{N}}\) defined by: \[ b_n = \sum_{k=1}^n\frac{k^3}{n^4+k^2} \] as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Let $\alpha, \beta$ be the roots of $x^2-2$ and $x^2+2$, respectively, over an algebraic closure of $\mathbb{F}_{2023}$. Determine whether $\alpha+\beta$ has degree $2$ or $4$ over $\mathbb{F}_{2023}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A supermarket has 2355 loaves of bread at the start of the day. By mid-morning, they sell 30% of their loaf stock, and a bakery offers them a 20% discount on a bulk purchase of 750 loaves. The supermarket also purchases 489 loaves with a 15% discount that evening. If the supermarket takes advantage of both discounts, how many loaves of bread are there at the end of the day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2888}
From the seven digits $l$, $2$, $3$, $4$, $5$, $6$, $7$, the number of seven-digit positive integers with exactly two even numbers adjacent to each other is $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2880}
Given that $|\overrightarrow{a}|= \sqrt {2}$, $\overrightarrow{b}=(-1,1)$, $\overrightarrow{c}=(2,-2)$, and $\overrightarrow{a}\cdot ( \overrightarrow{b}+ \overrightarrow{c})=1$. Find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac {2\pi}{3}}
On a square board with a side length of 2017, there is a chess bishop in the lower-left corner. Alexey and Danil take turns making moves with this bishop, with Alexey going first. The players are forbidden from moving the bishop to any cell it has already visited (the bishop moves diagonally for any distance). The player who cannot make a move loses. Which of the boys can guarantee a win regardless of the opponent's actions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Danil}
Compute the sum of all positive integers $a \leq 26$ for which there exist integers $b$ and $c$ such that $a+23 b+15 c-2$ and $2 a+5 b+14 c-8$ are both multiples of 26.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Cube $ABCDEFGH,$ labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\overline{AB}$ and $\overline{CG}$ respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$
[asy]import cse5; unitsize(10mm); pathpen=black; dotfactor=3; pair A = (0,0), B = (3.8,0), C = (5.876,1.564), D = (2.076,1.564), E = (0,3.8), F = (3.8,3.8), G = (5.876,5.364), H = (2.076,5.364), M = (1.9,0), N = (5.876,3.465); pair[] dotted = {A,B,C,D,E,F,G,H,M,N}; D(A--B--C--G--H--E--A); D(E--F--B); D(F--G); pathpen=dashed; D(A--D--H); D(D--C); dot(dotted); label("$A$",A,SW); label("$B$",B,S); label("$C$",C,SE); label("$D$",D,NW); label("$E$",E,W); label("$F$",F,SE); label("$G$",G,NE); label("$H$",H,NW); label("$M$",M,S); label("$N$",N,NE); [/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
Determine the condition under which the space \( l^p \) is a subset of the space \( l^q \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p \leq q}
Given the basis $B = \{(1, i), (-i, 2)\}$ of $\mathbb{C}^2$ and the linear map $T: \mathbb{C}^2 \to \mathbb{C}^2$ defined by $T(x, y) = (x, 0)$, find the matrix representation of $T$ with respect to basis $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\begin{pmatrix}2&-2i\\-i&-1\end{pmatrix}}
In triangle $ABC$, points $D$, $E$, and $F$ are on sides $BC$, $CA$, and $AB$ respectively, such that $BF = BD = CD = CE = 5$ and $AE - AF = 3$. Let $I$ be the incenter of $\triangle ABC$. The circumcircles of $\triangle BFI$ and $\triangle CEI$ intersect at $X \neq I$. Find the length of $DX$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the flux of the vector field $F(x, y, z) = (x^2, y^2, z^2)$ through the closed surface formed by the paraboloid $z = x^2 + y^2$ and the plane $z = 4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128π/3}
Jordan noticed that there are 2 cars in his driveway that each have 4 wheels, but one of them has a flat tire, so it's using a spare wheel temporarily. There are 3 bikes; one of them is missing a wheel, and the other two have 2 wheels each. In addition, there is a trash can with 2 wheels, a tricycle, a pair of old roller skates, a wheelchair with 2 additional small front wheels, and a wagon with 4 wheels. Of these items, two of the bikes have a wheel lock installed, and one roller skate is missing a wheel. How many wheels are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
如果一个十位数 $F$ 的各位数字之和为 81 ，则称 $F$ 是一个 "小猿数". 则小猿数的个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48619}
A tank can be filled by a tap in 20 minutes and by another tap in 60 minutes. Both the taps are kept open for some minutes and then the first tap is shut off. After this, the tank will be completely filled in 20.000000000000004 minutes. For how long were both taps kept open before the first tap was shut off?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine the radius of convergence for the power series \(\sum (n^2 \sin(2/n^2))^{n^2+2} x^{n^2}\). Use the formulas \(R=\frac{1}{\limsup_{n\to \infty}\sqrt[n]{|a_n|}}\) and \(R=\lim_{n\to \infty}\left |\frac{a_n}{a_{n+1}} \right |\) to find the radius of convergence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
In a football league, whenever a player is transferred from a team  $X$  with  $x$  players to a team  $Y$  with  $y$  players, the federation is paid  $y-x$  billions liras by  $Y$  if  $y \geq x$ , while the federation pays  $x-y$  billions liras to  $X$  if  $x > y$ . A player is allowed to change as many teams as he wishes during a season. Suppose that a season started with  $18$  teams of  $20$  players each. At the end of the season,  $12$  of the teams turn out to have again  $20$  players, while the remaining  $6$  teams end up with  $16,16, 21, 22, 22, 23$  players, respectively. What is the maximal amount the federation may have won during the season?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
$ S = \{1,2,\dots,n\}$ is divided into two subsets. How the set is divided, if there exist two elements whose sum is a perfect square, then $ n$ is at least ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
15
}
Consider three different types of computer components with lifetimes following exponential distributions with parameters 1, 2, and 3, respectively. The probability density functions are given by:

\[ f_1(x_1) = e^{-x_1}, \quad f_2(x_2) = \frac{1}{2}e^{-\frac{x_2}{2}}, \quad f_3(x_3) = \frac{1}{3}e^{-\frac{x_3}{3}} \]

where the support sets are \(x_i \geq 0\) for \(i = 1, 2, 3\). One component from each type is randomly sampled, and their lifetimes are recorded. Calculate the probability that \(X_1 > 5X_2 > 10X_3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{374}}
Given the function f(x) = x^3 + ax^2 - a^2x, it is known to be a decreasing function in the interval (0, 1) and an increasing function in the interval (2, +∞). Determine the range of values for the real number a.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-2, -1] \cup [3, 6]}
Find the number of real roots of the equation $x^3 - 6x^2 + 9x - 4 = 0$. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a casino, you and an infinite number of friends each start with $1. The slot machine costs $1 to play and has three possible outcomes: a win, a loss, or a null. The probabilities are: winning (and gaining $2) with $P(Win) = \frac{1}{5}$, losing with $P(Lose) = \frac{1}{5}$, and getting your $1 back with $P(Null) = \frac{3}{5}$. Winners pass their extra dollar to a friend who starts playing, while losers and null outcomes continue playing with the same dollar. Successive plays are independent, and the casino offers an infinite number of machines. Given that the probability of going broke with $2 is the square of the probability of going broke with $1 (i.e., $P(Broke|2\text{ dollars}) = P(Broke|1\text{ dollar})^2$), find the probability that all of you eventually go broke.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
We color some unit squares in a $ 99\times 99 $ square grid  with one of $ 5 $ given distinct colors, such that each color appears  the same number of times. On each row and on each column there are no differently colored unit squares. Find the maximum possible number of colored unit squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1900}
The intersection of plane $\alpha$ and plane $\beta$ is line $c$. Line $a$ is parallel to plane $\alpha$ and intersects plane $\beta$. What is the positional relationship between line $a$ and line $c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{skew}
Determine the total number of ways to achieve four in a row in a 4x4x4 tic-tac-toe game.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76}
Calculate the limit: $$\displaystyle \lim_{\alpha \to -1}\left(2\Gamma(-\alpha-1)+\Gamma\left(\frac{\alpha + 1}{2}\right) \right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3\gamma}
Given proposition P: ∃x₀ ∈ ℝ, x₀² + 2x₀ + 2 ≤ 0, what is the negation of proposition P? Express your answer as a logical statement using proper notation for universal and existential propositions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{∀x ∈ ℝ, x² + 2x + 2 > 0}
Given that the Green Park Middle School chess team consists of three boys and four girls, and a girl at each end and the three boys and one girl alternating in the middle, determine the number of possible arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
In quadrilateral $EFGH$, $EF = 6$, $FG = 18$, $GH = 6$, and $HE = x$ where $x$ is an integer. Calculate the value of $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
9.4. Vasya cut out a triangle from cardboard and numbered its vertices with the digits 1, 2, and 3. It turned out that if Vasya's triangle is rotated 15 times clockwise around its vertex numbered 1 by the angle at this vertex, the triangle will return to its original position. If Vasya's triangle is rotated 6 times clockwise around its vertex numbered 2 by the angle at this vertex, the triangle will return to its original position. Vasya claims that if his triangle is rotated $n$ times around the vertex numbered 3 by the angle at this vertex, the triangle will return to its original position. What is the smallest $n$ that Vasya could have named so that his claim is true for at least some cardboard triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Tyrone had $97$ marbles and Eric had $11$ marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$25$}
Let $X_1, \dots, X_n$ be independent and identically distributed (i.i.d.) random samples from the uniform distribution $Unif(\theta, 6\theta)$, where $\theta > 0$. Apply the Rao-Blackwell theorem to find an unbiased estimator of $\theta$, denoted by $\hat{\theta}$, such that its variance is minimized. What is the expression for $\hat{\theta}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{X_{(1)} + X_{(n)}}{7}\)}
Determine the image of the line $y = mx$ under the exponential transformation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r = e^{\theta/m}}
Find the radical ideal \( \sqrt{I} \) where \( I = (x^2y, y^2z) \) in the polynomial ring \( \mathbb{C}[x,y,z] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(xy, yz)}
For which values of integers \(r\) and \(k\) does the polynomial \(x^{p^k} - x\) divide \(x^{p^r} - x\) in \(\mathbb{Z}[x]\), where \(p\) is a prime number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k \mid r}
Let $F_{1}$ and $F_{2}$ be the two foci of the hyperbola $C$: $\frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$), and let $P$ be a point on $C$. If $|PF_{1}|+|PF_{2}|=6a$ and the smallest angle of $\triangle PF_{1}F_{2}$ is $30^{\circ}$, then the eccentricity of $C$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Let $ C$ be the circle with radius 1 centered on the origin. Fix the endpoint of the string with length $ 2\pi$ on the point $ A(1, 0)$ and put the other end point $ P$ on the point $ P_0(1,2\pi)$. From this situation, when we twist the string around $ C$ by moving the point $ P$ in anti clockwise with the string streched tightly, find the length of the curve that the point $ P$ draws from sarting point $ P_0$ to reaching point $ A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$2 \pi ^2$
}
Solve the Cauchy problem for the heat conduction equation given by:

\[
\begin{gathered}
u_{t}=u_{x x}, \quad x \in(-\infty,+\infty), \quad t \in(0,+\infty) \\
u(x, 0)=e^{-x} \quad x \in(-\infty,+\infty)
\end{gathered}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{t - x}}
In the land of Draconia, there are red, green, and blue dragons. Each dragon has three heads, and each head always tells the truth or always lies. Additionally, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said:

- 1st head: "To my left is a green dragon."
- 2nd head: "To my right is a blue dragon."
- 3rd head: "There is no red dragon next to me."

What is the maximum number of red dragons that could have been at the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{176}
In the arithmetic sequence $\{a\_n\}$, $a\_1=2$, $a\_2+a\_5=13$, find $a\_5+a\_6+a\_7=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Given the equation $2mnp=(m+2)(n+2)(p+2)$, where $m$, $n$, and $p$ are natural numbers with $m \leq n \leq p$, determine the maximum possible value of $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{130}
Determine the computational complexity of calculating the factorial of a number \( m \), given that the complexity of multiplying \( s+1 \) integers is \( O(s^{2}\log^{2}M) \), where \( M \) is the maximum of the integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(m^2 \log^2 m)}
For each positive integer $n,$ let $\sigma(n)$ denote the sum of the positive integer divisors of $n.$ How many positive integers $n \leq 2021$ satisfy$$\sigma(3n) \geq \sigma(n)+\sigma(2n)?$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
The answer is 1481.

}
In the Cartesian coordinate system $xOy$, a line $l$ passes through the focus $F$ of the parabola $y^{2}=4x$ and intersects the parabola at points $A$ and $B$, where point $A$ is above the $x$-axis. If the angle of inclination of line $l$ is $60^{\circ}$, then $|OA|=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt {21}}
There are $n \geq 2$ classes organized $m \geq 1$ extracurricular groups for students. Every class has students participating in at least one extracurricular group. Every extracurricular group has exactly $a$ classes that the students  in this group participate in. For any two extracurricular groups, there are no more than $b$ classes with students participating in both groups simultaneously. 

Find $m$ when $n = 8, a = 4 , b = 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In the plane, there are $2020$ points, some of which are black and the rest are green. For every black point, the following condition holds: there are exactly two green points that are at a distance of $2020$ from that black point. Find the smallest possible number of green dots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Determine the maximal possible dimension of an abelian Lie subalgebra of the Heisenberg Lie algebra \( \mathfrak{h}_n \) of dimension \( 2n+1 \). The Heisenberg Lie algebra \( \mathfrak{h}_n \) has a basis \( x_1, \dots, x_n, y_1, \dots, y_n, c \) with the Lie bracket defined by \([x_i, y_j] = \delta_{ij}c, [x_i, x_j] = [y_i, y_j] = [x_i, c] = [y_j, c] = 0\), where \(1 \le i, j \le n\) and \(\delta_{ij}\) is the Kronecker delta.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + 1}