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Given integers l, r, and k, determine how many distinct greatest common divisors (GCDs) can be formed by selecting any k numbers from the range [l, r]. The constraint is: 1 ≤ l ≤ r ≤ 10^12, 2 ≤ k ≤ r - l + 1. Rather than computing all possible GCD values directly, we count all potential divisors. Any integer m = i × j that divides two numbers x ...

Problem 1: Counting Valid Pairs with LCM Condition Problem Description Given an integer n where 1 ≤ n ≤ 10^8, determine the number of pairs (x, y) such that 1 ≤ x, y ≤ n and the following inequality holds: lcm(x, y) / gcd(x, y) ≤ 3 Note that lcm(x, y) = (x * y) / gcd(x, y)^2. Solution Approach This is a straightforward number theory problem. L ...

Subtree Budget Allocation and Optimal Path Reconstruction The core challenge revolves around allocating a fixed budget across a binary tree structure to maximize a specific threshold value. The solution begins by analyzing the root's contribution and propagating constraints downward. We define a dynamic programming state dp_max[u][b] representi ...

The Ternary Goldbach Conjecture asserts that any odd integer greater than 7 can be represented as the sum of three prime numbers. While proven for sufficiently large numbers, verifying this for smaller integers requires an effiicent computational approach. Given an odd integer n (9 < n < 20,000), our objective is to find a triplet of prim ...

A straightforward Rust implementation for finding all factors of a number is shown below. This approach iterates up to the square root of the input value. fn compute_factors_simple(num: u64) -> Vec<u64> { let sqrt_val = (num as f64).sqrt().floor() as u64; let mut factors = Vec::new(); for divisor in 1..=sqrt_val { i ...

Problem A: Counting Digit Holes The task requires calculating the total number of closed loops (holes) in a sequence of digits. Digits '0', '6', and '9' contain one loop each, while '8' contains two loops. The solution involves iterating through the string and accumulating the count based on the digit encountered. #include <iostream> #inc ...

Fast Exponentiation Recursive Approach function fastExponentiation(base, exponent, modulus) { if (exponent === 0) return 1; const halfExp = fastExponentiation(base, Math.floor(exponent / 2), modulus); let result = (halfExp * halfExp) % modulus; if (exponent % 2 === 1) { result = (result * base) % modulus; ...

Introduction Consider the following geometric problem: mark all vertical lines x = c and horizontal lines y = c where c ∈ ℤ on a plane. Now consider a line y = (px + r)/q where p, r ∈ ℕ and q ∈ ℕ₊. Since the residue class of r modulo q determines the behavior, we can assume r < q without loss of generality. Imagine a moving point traveling a ...

Problem A: Strategic Allocation This challenge involves selecting a subset of items to meet a weight capacity requirement with the minimum count. The optimal approach utilizes a greedy strategy. By prioritizing larger weights first, we minimize the number of items required to reach the target threshold. void processAllocation() { int itemCo ...

To solve this problem, we scan every possible 2x2 submatrix in a n × m grid and check whether the four characters within it collectively contain at least one 'y', one 'o', and one 'u'. The order does not matter—only the presence of all three required characters. The algorithm iterates over all valid top-left corners of 2x2 blocks (from row 1 to ...