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Problem Statement There 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 ce…

Second-order Linear Differential Homogeneous Equations Consider operator $$ L = D^2 + pD + q \tag*{} $$ where $D$ is differential operator For the case where $P(x)$ and $Q(x)$…

Convolution Consider two polynomials $F(x) = \sum_kf_kx^k$, $G(x) = \sum_kg_kx^k$ and their product $H(x) = (F \cdot G)(x) = \sum_kh_kx^k$, where $$ h_k = \sum_{i + j = k}f_ig_j. \tag*{} $$…

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…

Statement If 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 fun…

Statement Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowed Type 1) Choose a non-empty box…

Overview Dantzig's simplex method is used to solve the following linear programming problem. Minimize $z = \mathbf c^{\mathrm T} \mathbf x$ Subject to $A\mathbf x \leq \mathbf b$…

Problem Statement Alice 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 B…

Problem Statement You 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 perf…

Overview Let 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 creat…