Freaks City

T1 Three Integers Problem Statement Given three integers A, B, and C, there are two operations: Operation 1: Choose two numbers and decrement each by 1. Operation 2: Choose all three numbers and decrement each by 1. The goal is to reduce all three numbers to 0. If impossible, output -1. Solution Approach A key insight is that any operation sh ...

We examine the problem of merging \(n\) initial unit vectors \(\mathbf{e}_1,\ldots,\mathbf{e}_n\) via operations that combine two vectors \(\mathbf{u},\mathbf{v}\) into either \(2\mathbf{u} - \mathbf{v}\) or \(2\mathbf{v} - \mathbf{u}\), depending on which one "wins". Each final vector’s \(i\)-th component is of the form \((-1)^{c_i}2 ...

Optimization Approaches State Reduction: Leverage problem properties to minimize state space Model Adaptation: Apply known algorithmic patterns to improve transition efficiency Contribution Decomposition: Use data structures to manage partial contributions Standard Optimizations: Utilize techniques like monotonicity, convex optimization, or sl ...

This document provides solutions for problems from the SMU Summer 2023 Contest, Round 6. A. Burger Optimization This problem involves maximizing profit from selling two types of burgers with different ingredients and prices, given a limited number of buns. The strategy is to iterate through all possible counts of the first burger type, up to th ...

Problem DefinitionThe task involves processing multiple queries where, for a given integer n, we must compute the sum of squared binomial coefficients: $\sum_{i=0}^{n} \binom{n}{i}^2$. The result should be returned modulo $10^9 + 7$. Constraints allow for n up to $10^6$, necessitating an efficient algorithm.Naive Approach: Dynamic ProgrammingFo ...

Tree to Prufer Sequence Find the leaf node with the smallest label, and add its parent to the sequence. Delete that leaf node. Repeat the above operations until a sequence of length (n-2) is obtained. The reverse process reconstructs the tree. A Prufer sequence establishes a bijection between the spanning trees of a complete graph with (n) ve ...

A cow's language, known as Cowphabet, consists of the 26 lowercase letters. However, the order in which a cow recites these letters is a permutation of the standard alphabet sequence. Bessie repeats this song, and Farmer John notes down a string of letters he remembers hearing. The task is to determine the minimum number of complete Cowphabet r ...

Month Transition Calculation Problem Statement Given an integer current_month representing a month (1 through 12), compute the subsequent month in the annual cycle. Solution Approach Months follow a cyclic pattern with base 12. Converting to zero-based indexing simplifies modular arithmetic. Implementation def calculate_next_month(m: int) -> ...

Let's consider the calculation for the number of four-vertex subgraphs with at least x specific edges, denoted as f_x. Using the principle of inclusion-exclusion, the count of subgraphs with no edges at all is f_0 - f_1 + f_2 - f_3 + f_4 - f_5 + f_6. Meanwhile, the count of subgraphs with all six edges present is simply f_6. The difference we n ...

Problem A: Treasure Chest We need to calculate the minimum time to reach the chest and return to the start, given the ability to pull the chest towards the key for a maximum distance of k. There are two scenarios based on the relative positions of the chest pos_chest and the key pos_key: If pos_key <= pos_chest: We pickup the key on the way ...