Freaks City

Circular Array Merging with Max-Cost We are given a circular array of positive integers. In one step we can pick any two adjacent elements, remove them, and pay a cost equal to the larger of the two values. The goal is to reduce the array to a single element while minimiizng the total cost. Key observation Because every merge can only increase- ...

Problem 1: Tank Battle Description Two players engage in a tank battle. Player A has n identical tanks with health a each, and Player B has m identical tanks with health b each. Each round, all surviving tanks fire simultaneously, dealing 1 damage to a target tank. Players use optimal strategies to maximize their victory. The battle continues u ...

Core Principles of Backtracking Backtracking operates as an exhaustive search mechanism built upon recursive execution. Rather than optimizing pathfinding, it systematically evaluates every viable candidate configuration until valid outcomes are isolated. Although computationally expensive, this approach remains necessary for combinatorial doma ...

1.1 Basic Sieve Approach The fundamental strategy involves iterating through integers starting from 2 and marking all their multiples as composite. Because a prime number is divisible only by 1 and itself, it remains unmarked. The total operations correspond to the sum of the harmonic series ($\frac{n}{2} + \frac{n}{3} + \dots$), yielding a tim ...

Given two points A and B and n circles, determine smallest possible radius for an unknown circle O that passes through A and B without intersecting any of the given circles. The constraint are 1 ≤ n ≤ 10^5, with point coordinates and radii in [-10^5, 10^5]. The solution must achieve a relative or absolute error ≤ 1e-3, and the answer is bounde ...

Counting Valid Mountain SequencesTo solve the mountain sequence counting problem, we need to determine the number of permutations that satisfy the mountain property. This can be efficiently computed using combinatorial mathematics by recognizing patterns in the sorted sequence.#include <iostream> #include <vector> #include <algor ...

Problem Analysis Given an n x 3 grid, each cell must be colored with one of three colors (red, yellow, green) such that adjacent cells (sharing horizontal or vertical edges) have different colors. The goal is to count the total number of valid colorings modulo 10^9 + 7. For a single row (n = 1), there are 12 valid colorings. As n increases, the ...

AGC010F Tree Game Let \( val[u] \) represent the number of stones located at node \( u \). A useful heuristic arises when considering a node \( u \) with a single child \( v \). If \( val[u] \le val[v] \), and the first player moves the token to \( v \), the second player can immediately move it back to \( u \). This traps the first player, lea ...

OSU! Let $E_i$ denote the expected score after the $i$-th operation. The key observation is that $(x+1)^3 - x^3 = 3x^2 + 3x + 1$. To compute this increment in expectation, maintain: $$ \text{exp1}i = (\text{exp1}{i-1} + 1) \cdot p_i $$ $$ \text{exp2}i = (\text{exp2}{i-1} + 2 \cdot \text{exp1}_{i-1} + 1) \cdot p_i $$ Then update the total expect ...

Problem 1: Counting Valid Sets In this problem, the floor operation does not affect the correcntess, so a brute-force DFS approach is applicable. A critical observation: if the array contains the element \(1\), it must be skipped immediately. Otherwise, \(1\) could be selected infinitely many time, leading to incorrect results. Solution Code in ...