A - Antenna Analysis The mathematical expression can be decomposed into two separate components: #include <iostream> #include <queue> #include <vector> using namespace std; int main() { ios::sync_with_stdio(false); cin.tie(nullptr); int n, c; cin >> n >> c; priority_que...
Algorithmic Solutions for Competitive Programming Problems Extracting Initial Characters from Input Strings When solving problems that require extracting the first letters of words from input strings, a direct character-by-character approach can be effective: #include <iostream> #include <s...
Number Theory Fundamentals Extended Euclidean Algorithm and Linear Diophantine Equations For integers a and b, the equation ax + by = d has integer solutions if and only if the greatest common divisor gcd(a, b) divides d. This is known as Bézout's Identity. The extended Euclidean algorithm alows us...
Number Theory Min-Max Inclusion-Exclusion The core identities for min-max inclusion-exclusion over finite sets are: $$\max(S) = \sum_{T \subseteq S} (-1)^{|T|+1} \min(T)$$ $$\min(S) = \sum_{T \subseteq S} (-1)^{|T|+1} \max(T)$$ Expanding the summation shows all non-extremal values cancel out. The id...
A solution exists for the equation (\gcd(an+b, c) = 1) if and only if (\gcd(a, b, c) = 1). If (\gcd(a, b, c) > 1), the problem has no solution and (-1) is output. Given the condition (\gcd(a, b, c) = 1), we can construct a valid (n). Let (x = \gcd(b, c)). If (x = 1), then (n = 0) satisfies (\gcd(...
Efficient computation of $a^b \bmod m$ utilizes the binary representation of the exponent $b$. By expressing $b$ as $\sum_{i=0}^{k} c_i \cdot 2^i$ where $c_i \in {0,1}$, the power decomposes into a product of squared terms: $a^b = \prod_{i=0}^{k} (a^{2^i})^{c_i}$. The algorithm iterates through each...
Modular Arithmetic Basics The expression a mod b denotes the remainder when a is divided by b. Addition: (a + b) % p Subtraction: (a - b + p) % p (adding p avoids negative results) Multiplication: (a * b) % p Division: Not directly supported; requires modular inverses (discussed later) Exponentiatio...