AFAIK

Valar Morghulis

我最近又准备小翻新一下博客了。虽然东西没写多少,但是主题更新的快呀哈哈哈哈哈毕竟差生文具多

Read more »

一个集合 \(S\) 被称为 sum-free set 当且仅当 \(\forall a, b \in S, a + b \not\in S\)

试证明:对于任何一个集合 \(A \subseteq \ZZ\backslash\{0\}\),其最大的 sum-free subset 大小至少为 \(|A| / 3\).

(有一个和题面差不多长度的解答)

解答

对于 \(t \sim U[0, 1]\),令 \(A_t := \{x \in A: 1/3 < xt \bmod 1 < 2/3 \}\)。显然 \(A_t\) 为 sum-free set. 最后再注意到 \(\max_t |A_t| \geq \mathbb{E}_t [|A_t|] = |A|/3\) 即可。

解法来自 Erdos。

Recently I'm playing with a new code editor, Zed. I used it for about ten days and here write a review to share my experiences with it. This review is based on the latest version (v0.186.9).

TL;DR

My feelings about Zed are mixed: as an editor, it's pretty good; as a (Python) code editor, it's pretty bad; as an AI code editor, it's reasonable. As of its current state (v0.186.9), it's hard for me to recommend it to colleagues due to its poor Python experience, but I think I'll keep an eye on it.

Update (@2025-05-19): After tuning LSP configurations, the Python experience is much better. See below.

Read more »

In this post, we are interested in finding all integer and rational solutions for a general binary quadratic equation: \[ \begin{equation} ax^2 + bxy + cy^2 + dx + ey + f = 0, \label{eq:general} \end{equation} \] for any integer coefficients \(a, b, c, d, e, f\). The equation might degenerate; we only point it out when it does but do not solve it, as it's typically easier to solve a degenerate equation.

This post mainly serves as an entry point for a series of posts and also contains a little bit of the story behind the posts. That is why this post has index 0 in the title.

Read more »

In this post, we are interested in finding all integer solutions of the following equation: \[ \begin{equation} ax^2 + bxy + cy^2 = n. %\label{eq:ibqf} \end{equation} \] where \(acn \neq 0\), \(b^2 - 4ac \neq 0\). This equation can certainly be solved using methods from the generalized Pell equation, but here, we explore other methods.

Read more »

In this post, we are interested in finding all integer solutions of Pell equation \[ \begin{equation} x^2 - d y^2 = 1, \label{eq:pell} % \tag{Pell Equation} \end{equation} \] and generalized Pell equation \[ \begin{equation} x^2 - d y^2 = n, \label{eq:generalized-pell} % \tag{Generalized Pell Equation} \end{equation} \] for a non-square integer \(d > 0\) and \(n \neq 0\).

Read more »

In this post, we're interested in finding any or all rational solutions of the following equation: \[ ax^2 + by^2 + c = 0, \] or alternatively, all integer solutions to the Legendre equation: \[ \begin{equation} ax^2 + by^2 + cz^2 = 0, \label{eq:legendre} % \tag{Legendre Equation} \end{equation} \] where \(abc \neq 0\).

Read more »

转眼间 2024 年已经过去了……隔了好久没写娱乐相关的文章了,随便写几个作为去年的年终总结得了……(然后震惊地发现我上次写游戏已经是四年前了……)

  • Return of the Obra Dinn
  • The Case of the Golden IdolThe Rise of the Golden Idol
  • Patrick's Parabox
  • COCOON
  • ANIMAL WELL

(可能有略微剧透,手动狗头)

Read more »
0%