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ARC 219 E Equal Distribution

0 comments Problem-solving notes Puzzle
Problem StatementThere is a shortcake in the shape of a $2H \times 2W$ grid. The cell at the $i$-th row from the top and the $j$-th column from the left is denoted as cell $\def\mydef{\def}\my...

Second-order Differential Equation

0 comments Study Notes Mathematics
Second-order Linear Differential Homogeneous EquationsConsider operator$$ \gdef\mydef{\gdef}\mydef\s#1{\left(#1\right)}\mydef\m#1{\left[#1\right]}\mydef\b#1{\left\{#1\right\}}\mydef\abs#1{\left\lve...

Discrete Fourier Transform and Fast Fourier Transform

0 comments Study Notes Polynomial FFT
ConvolutionConsider two polynomials $\gdef\mydef{\gdef}\mydef\s#1{\left(#1\right)}\mydef\m#1{\left[#1\right]}\mydef\b#1{\left\{#1\right\}}\mydef\abs#1{\left\lvert#1\right\rvert} F\s x = \sum_kf_kx^...

Riemann Integral

0 comments Study Notes Mathematics
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 comments Study Notes Mathematics 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 comments Problem-solving notes Mathematics
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

0 comments Study Notes Mathematics Matrix 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 comments Problem-solving notes Dynamic Programming Flow
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 comments Problem-solving notes 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

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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...