Freaks City

The problem models a sequential arrangement where each position $i$ offers two distinct categories of elements. Category X occupies exactly one slot, while Category Y spans two consecutive slots $(i-1, i)$. The objective is to compute the total number of valid configurations modulo $10^9+7$. A two-state dynamic programming approach efficiently ...

Tree-Based Resource Distribution When distributing a fixed quantity of resources across a tree structure where each node must eventually hold an equal amount, removing any edge partitions the graph into two independent substructures. Let the total resource sum be $S$ and the number of nodes be $N$. The target allocation per node is $k = S / N$. ...

A. Apple Tree Ring Given a sequence of integers representing apple counts on trees arranged in a circle, determine the maximum number of distinct values that can be consumed under infinite rotations. Since rotations allow arbitrary reordering over time, the optimal strategy is to consume one unique value per round. Hence, the answer equals the ...

Geometric Distribution and Expected Value When modeling scenarios with repeated independent trials where success occurs with probability $p$, the process follows a geometric distribution. The expected number of trials to achieve the first success is mathematically derived as $1/p$. Let $E(X)$ denote the expected number of draws required. Using ...

This analysis addresses the problem of decomposing a string into the minimum number of substrings such that none of the substrings are "cyclic" (periodic). A string is considered cyclic if it can be constructed by repeating a smaller substring multiple times. Given a string S of length N, we must deetrmine the minimum number of partit ...

Approach 1: Brute Force Method A straightforward solution involves checking all possible combinations of three indices using nested loops. This approach iterates through every possible triplet (i, j, k) in the array. The time compleixty is O(T × N³), which is impractical given the constraint 3 ≤ n ≤ 2 × 10⁵. This method would exceed time limits ...

Mathematical Algorithms Fast Exponentiation Reduces the time complexity of computing powers to logarithmic time by leveraging binary decomposition of the exponent. long long binary_pow(long long base, long long exp, long long mod) { long long res = 1; base %= mod; while (exp > 0) { if (exp & 1) res = (res * base) % mo ...

Competition Summary Overview Problem Details Problem 1: Prime Number Identification Problem 2: Range Maximum Queries Problem 3: Dynamic Median Finding 20 Point Solution 100 Point Solution Problem 4: Magic Stone Path Optimization Basic Dynamic Programming Approach Problem 5: Strategic Decision Making Summer Training Competition Day1 O ...

To solve this problem, find a prime number greater than \(10^9\). If the input contains 1, then there is no solution. #include <bits> using namespace std; typedef long long ll; void process() { int size; cin >> size; bool valid = true; vector<int> data(size); for (int i = 0; i < size; ++i) { ...

A number n can be expressed as the sum of k terms where each term's decimal digits are exclusively 1, 2, or 3. The goal is to find the smallest possible k such a decomposition exists. For T ≤ 1000 test cases and n up to 10^18. Define a function valid(x, m) that returns true if x can be decomposed into m terms satisfying the digit condition. A f ...