题目背景 完整题解来自ox帅气lecturer, 我只能证明F<\mathbb C的情况 ;( ;( ;() 题面 V是域F上的一个线性空间, \dim V=n, 且有T\in L(V). 现取\lambda_1,...,\lambda_n\in F两两不等, 且任意\lambda_i都不是T的特征值. 试证明存在\alpha_1,...,\alpha_n\in F使得 \……

题目背景 人见人爱的代数小题 :] 惊人的是竟然在Hungerford II.2.8.9才出现, 还标了名字 😮 题面 A^ * is a normal subgroup of A, B^ * is a normal subgroup of B. Prove that: (i) A^ *(A\cap B^ *) is a normal subgroup of A^ *(A\cap B) (ii) B^ *(A^ *\cap B) i……

题目背景 本题竟然与xgn目前正在学习的序数有关, 真是可喜可贺!(crim boi在thu里有包含序数的课程吗?) 题面 If you don't remember what a topology is, please refer to this passage. Preliminary: Open sets in \mathbb R are just union (can be infinite) of som……

Solved! Difficulty: Div2B 1 problem for div3 participants. Beautiful Dice Painting When Epis is bored, she will draw dices. The canvas contains n*n dots. Initially they are all white. You need to paint exactly v dots black so that the final c……

题目背景 Rudin Real and Complex Analysis第一题, 真是开门红. 题面 We call a pairing (X,\mathfrak{M}) \sigma-algebra if \mathfrak{M} is a subset of P(X) satisfying the three properties below: (1) X\in\mathfrak{M} (2) If A\in\mathfrak{M}, A^c\in\mathf……

trivial 思考题 I am Banach This is a classic problem by Banach nearly without any change. There are N boys and M girls. Each boy loves a single girl and each girl loves a single boy. Your task is to divide the boys into two groups A_0 and A_1 and t……

Starting from today, we will try to write tutorials and rate difficulties for already solved Orange Boy Can You Solve It Out?(OBCYSIO) problems and separate them from unsolved/unsolvable ones. There's a new category called Solved OBCYSIO and you ca……

思考题 during OI recovery Rubbish Problem Maybe you have heard of the evil problem Hilbert's Hedge Maze and here's another very fun problem! You have n items arranged in rows. Each row contains r items and the items are arranged from left to right,……

题目背景 南大几把选拔中最水的题(T3), 适合出在初中竞赛里. 另外这期写解答方便以后想考南大几把的学生😇, 篇幅太长除了题面不写中文了. 不过题面也是pmt提供的, 不一定完全准确. 题面 记排列P的置换环数量为c(P),排列P,Q的复合记为P\oplus Q，已知长度为n的排列f_1,f……

（提醒：本游记体量较大，字数超过10000，图片数量超过100，可以考虑在WiFi环境下阅读）   “晕眩于它的华美，在放纵中失魂落魄。于是，欢乐、惊奇、悲伤，我感到一种无可名状的激动”，上世纪二十年代，日本作家村松梢风把上海称作“魔都”，而一个世纪之后的今天，虽然……