Freaks City

Dirichlet Convolution Dirichlet convolution is a binary operation defined between arithmetic functions. It can be expressed as \((f * g)(n) = \sum_{xy = n} f(x) \cdot g(y)\) or equivalently \((f * g)(n) = \sum_{d \mid n} f(d) \cdot g(\frac{n}{d})\). Properties If both \(f\) and \(g\) are multiplicative functions, then \(f * g\) is also multipl ...

Balanced String Analysis (Simple and Extended) Problem Statement We need to analyze strings composed of 0, 1, and ?. The question marks can be replaced with either 0 or 1. The goal is to compute how many valid configurations result in "balanced" strings according to a specific criterion. Simple Approach For small lengths, a brute-forc ...