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数学杂记（更新于 20241122）

3 条评论 杂事 No tags
@1 强迫症的投掷器 20240420$n$ 个东西，每次等概率随机一个东西摸一下。问期望几次能摸过所有 $n$ 个东西。显然第一次一定会摸到一个新东西。考虑期望多少次操作能拿到第二种新东西。显然有 $\frac{n - 1}n$ 的概率摸到新东西，于是期望次数就是 $\frac n{n - 1}$ 次。再摸到第三个东西的期望次数就是 $\frac n{n - 2}$ 次。于是总答案就是$$...

Riemann Integral

0 条评论 学习笔记 Math
When I was taught integration, my teacher told me that when we divide the area into strips with sufficiently small widths, the error will decrease to zero. However, there was still some confusion i...

Schröder Bernstein Theorem

0 条评论 学习笔记 Math Set Theory
StatementIf there exists an injective function $f: \mathcal S \to \mathcal T$ and an injective function $g: \mathcal T \to \mathcal S$, then there exists a bijective function $h: \mathcal S \to \ma...

IMO 2010 P5

0 条评论 做题笔记 Math
StatementEach of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowedType 1) Choose a non-empty box $B_j$, $1 \leq j \leq 5$ remo...

Dantzig's Simplex Method

1 条评论 学习笔记 Matrix Math Linear Programming
OverviewDantzig's simplex method is used to solve the following linear programming problem.Minimize $z = \mathbf c^{\mathrm T} \mathbf x$Subject to $A\mathbf x \leq \mathbf b$ and $\mathbf x \geq 0...

CF1519F Chests and Keys

0 条评论 做题笔记 Dynamic Programming Flows
Problem StatementAlice and Bob play a game. Alice has got $n$ treasure chests (the $i$-th of which contains $a_i$ coins) and $m$ keys (the $j$-th of which she can sell Bob for $b_j$ coins).Firstly,...

CF2147E Maximum OR Popcount

0 条评论 做题笔记 Greedy
Problem StatementYou are given an array of $n$ non-negative integers.You want to answer given $q$ independent scenarios. In the $i$-th scenario, you are allowed to perform the following operation a...

Boids Algorithm

0 条评论 杂事 No tags
OverviewLet us watch a demo first. Go to https://boids.rbtr.ee/ to see further.As you can see, the Boids algorithm is a method to simulate the movements of social creatures.AlgorithmThe Boids algo...

CEOI 2023 Grading Server

0 条评论 做题笔记 Dynamic Programming
Grading ServerDespite running low on oil, the opening ceremony was a great success. Now, the committee is looking forward to the first competition day. But what’s this? The chairman of the Technica...

BalticOI 2022 Vault

0 条评论 做题笔记 Dynamic Programming DP-Optimize
VaultThere are $2M + 1$ kinds of weights, whose mass are $-M, -M + 1, \ldots, M - 1, M$. However, you do not have an infinite number of weights. For each mass $\ell$, there are only $A_{\ell}$ such...

CSES 2421 Counting Reorders

0 条评论 做题笔记 No tags
Counting ReordersCalculate the number of ways you can reorder the characters of a string so that no two adjacent characters are the same.For example, the answer for aabc is $6$, because the possibl...