Hello World

「 Hello, the lovely world. 」

「 Stay hungry, Stay foolish. 」

Inline Formula

One day you are surviving in the wild. After a period of exploration, you determine a safe area, which is a convex hull with $n$ vertices $P_1,P_2,\dots,P_n$ in counter-clockwise order and any three of them are not collinear.

Now you notice that there will be $q$ airdrop supplies, and for the $i$-th supply, its delivery range is described by a circle $C_i$ , which means the supply will landed with uniformly probability among all the points with a real number coordinate inside $C_i$.

You need supplies so much that you decide to predetermine a starting point for each supply, and the starting point of two different supplies can be different. Every starting point should be inside the safe area and have the smallest expected value of the square of the Euclidean distance to the corresponding supply landing point.

Recall that On a two-dimensional plane, the Euclidean distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$. If both coordinates of a point are all integers, then we call this point an integer point.

Block Formula

$$
\int \frac{dx}{cos^2x}=\int sec^2xdx=tan x+C\
$$

$$
\int \frac{dx}{sin^2x}=\int csc^2xdx=-cot x+C
$$

$$
\sum\limits_{i = 0}^{E} w[i]{M \choose i} {N \choose iS} \frac{(iS)!}{(S!)^{i}} \sum\limits_{j = 0}^{E - i} (-1)^{j} {M - i \choose j} {N - iS \choose jS} \frac{(jS)!}{(S!)^{j}} (M - i - j)^{N - iS - jS}
$$

$$
= \sum\limits_{i = 0}^{E} w[i]{M \choose i} {N \choose iS} \frac{(iS)!}{(S!)^{i}} \sum\limits_{j = i}^{E} (-1)^{j - i} {M - i \choose j - i} {N - iS \choose jS - iS} \frac{(jS - iS)!}{(S!)^{j - i}} (M - j)^{N - jS}
$$

Matrix

$$\begin{bmatrix}
(\omega_n^0)^0 & (\omega_n^0)^1 & \cdots & (\omega_n^0)^{n-1}\\
(\omega_n^1)^0 & (\omega_n^1)^1 & \cdots & (\omega_n^1)^{n-1}\\
\vdots & \vdots & \ddots & \vdots \\
(\omega_n^{n-1})^0 & (\omega_n^{n-1})^1 & \cdots & (\omega_n^{n-1})^{n-1}
\end{bmatrix}
\begin{bmatrix}
a_0\\
a_1\\
\vdots\\
a_{n-1}
\end{bmatrix}=
\begin{bmatrix}
A(\omega_n^0)\\
A(\omega_n^1)\\
\vdots \\
A(\omega_n^{n-1})
\end{bmatrix}$$

To prevent escaping, you need to use \\\\ to make a new line rather than \\ .

Link

For more about the Mathjax symbols, refer to: The Comprehensive LaTeX Symbol List.

Code Highlight & Fold

>>> import this
The Zen of Python, by Tim Peters

Beautiful is better than ugly.
Explicit is better than implicit.
Simple is better than complex.
Complex is better than complicated.
Flat is better than nested.
Sparse is better than dense.
Readability counts.
Special cases aren't special enough to break the rules.
Although practicality beats purity.
Errors should never pass silently.
Unless explicitly silenced.
In the face of ambiguity, refuse the temptation to guess.
There should be one-- and preferably only one --obvious way to do it.
Although that way may not be obvious at first unless you're Dutch.
Now is better than never.
Although never is often better than *right* now.
If the implementation is hard to explain, it's a bad idea.
If the implementation is easy to explain, it may be a good idea.
Namespaces are one honking great idea -- let's do more of those!