今年的 SIGYAO30 就决定放在 Denver 啦，
2025 年的第一场 SIGYAO30，比以往时候鸽的更多一些，最后只有十一个人过来了……东岸的人尤为鸽，为什么鸽也能出现了人传人现象？汤姐一如既往的鸽，她吐槽为什么每次 SIGYAO30 都能精准选在她生理期。胡老师本来非常积极，但是出发前一天，胡老师吃坏肚子了……令人惊奇的是鸽王 xpd 居然来了，另外一个亮点是毕老师不远千里迢迢从 HK 赶过来了，第一次参加线下的 SIGYAO30。
几个星期前，一位小伙伴 wzc 问我有啥娱乐活动。聊着聊着，我们就约起了 Yosemite。于是几天前，我们终于趁着还不需要预约的时候，赶个 ddl 去了趟 Yosemite.
TL;DR 实在是妙啊！不愧是国家公园啊！瀑布量大管饱。
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
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).
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.
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.
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.
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 \(n \neq 0\).
In this post, we're interested in finding all integer solutions of a binary quadratic equation: \[ \begin{equation} ax^2 + by^2 = n, \label{eq:ellipse} \end{equation} \] for \(a, b, n > 0\).
In this post, we're interested in finding any or all integer solutions of a binary quadratic equation: \[ x^2 - ny - a = 0. \] We assume that the equation is non-degenerate in the sense that \(n \neq 0\).