Freaks City

Digit dynamic programming (digit DP) is a technique used to count numbers within a range ([l, r]) that satisfy specific digit-based constraints. The standard approach involves transforming the problem into computing (F(r) - F(l-1)), where (F(x)) counts valid numbers from 0 to (x). The solution typically uses memoized recursion over the digits o ...

Introduction For a better reading experience? Prerequisite knowledge (obvious): \(\operatorname{lcm}(a,b)=\dfrac{ab}{\gcd(a,b)}\) Linear Sieve for Primes Click to view code``` bool visited[MAXN]; vector primes; long long result; void sieve(){ for(int i=2;i<=n;i++){ if(!visited[i]) primes.push_back(i); for(auto j:primes){ if(ij>n) break; v ...

Problem 1: Decrypting S Country's Military Cipher Description R and S are at war. R's spy, Xiao C, has obtained S's encryption rule: each uppercase letter maps to a unique uppercase letter (bijection). Given a known encrypted string and its original string, build the substitution table. Then translate a new intercepted encrypted string. If the ...

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 ...

Competition Overview This mock contest featured relatively weak test data, allowing suboptimal solutions to score significantly higher than expected (often 50+ points above theoretical estimates). The official solutions sometimes relied heavily on pattern recognition and handling specific subtasks rather than pure algorithmic elegance. This hig ...

Given two arithmetic progressions $A_i = a_1 i + b_1$ and $B_i = a_2 i + b_2$ for $i \in \mathbb{N}$, and integers $l \leq r$, determine how many integers $n \in [l, r]$ appear in both sequences. We seek integer solutions $(x, y) \in \mathbb{N}^2$ to the equation: $$ a_1 x + b_1 = a_2 y + b_2 $$ Rewriting gives the linear Diophantine equation: ...

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 ...

Problem A: Maximum Value Boundary Analysis In this problem, we need to calculate the sum of counts $f(k)$ for pairs $(i, j)$ that satisfy specific boundary conditions related to two arrays $A$ and $B$. Let $f(k)$ represent the number of valid pairs where the second index $j$ equals $k$. We define $last\_k$ as the index where the maximum value o ...

Codeforces 1017D - Binary String Query Complexity: $\mathcal{O}((4^n+q) \log n)$ Distinct binary strings are limited to $2^n$. Precomputing distances between pairs allows for binary search queries. Codeforces 1080F - Colorful Graph Approach: Persistent Segment Tree / Sweep Line Treat this as a data structure challenge. By sweeping the right end ...

Problem Definition Given two positive integers base and modulus where base < modulus, determine the count of non-negative integers x such that 0 ≤ x < modulus and gcd(base, modulus) == gcd(base + x, modulus). Input Constraints Number of test cases: 1 ≤ T ≤ 50 Value range: 1 ≤ base < modulus ≤ 10^{10} Mathematical Reduction Let d = gc ...