brain

We are given a string and asked to count how many of its substrings have a very specific three-part structure. A substring is called valid when it can be split into three consecutive equal-length blocks where the first and last blocks are identical, and the middle block is the…

We are given a very large integer $N$ written in decimal form and a digit $x$ between 1 and 9. The task is to examine every positive integer $M$ from 1 up to $N$, count how many times the digit $x$ appears inside each $M$, and group numbers by that count.

We are given a collection of contests. Each contest contains a fixed number of problems, and every problem has a difficulty label in a small range from 1 to K.

We are given a structured polynomial identity involving two variables $x, y$ over a finite field. The core object is a bivariate polynomial $P(x,y)$, but it is not arbitrary.

We are given three fixed strings over a small alphabet, and we treat them as reference sequences that define which subsequences are “available” in a constrained sense.

We are given an array of integers representing colored beads. From this array we construct two related sequences and then compare them in a cyclic way, meaning we are allowed to rotate one of them before comparing.

We are given a set of points in the plane and multiple queries. Each query gives us three distinct points forming a triangle, and we need to determine whether there exists at least one other point from the set strictly inside that triangle.

We are given a rooted tree where each node represents an element that may carry an identifier, but this identifier is not always “fixed” in how it should be interpreted.

We are working with a sequence that is easier to understand through its differences rather than its raw values. Instead of reasoning directly about the sequence, we look at how each element changes compared to the previous one.

We are given an array of positive integers. The task is to compute a global sum over all ordered pairs where each element in the array acts as a divisor in a modular expression.

We are working with a graph where each state is a directed edge traversal decision, not just a vertex value. For every vertex $v$, and every neighbor $u$ adjacent to it, we define a quantity $L(v, u)$ which represents the longest possible walk that starts at $v$ and is forced…

We are given a tree where each edge has an implicit distance, and a fixed parameter $c$ representing how far a car can travel on a full tank. When moving along the tree, a refuelling event is required whenever the remaining fuel is insufficient to continue along the next edge.

We are given a process defined on pairs of positive integers $(a, b)$ with $a < b$. From a state $(a, b)$, the system either terminates immediately with probability $1/2$, or transitions to a new state $(2a, b - a)$ with probability $1/2$.

We are dealing with a process where a set of integers from 1 to n exists, and we randomly select a subset of size k from it. From that subset, we are interested in the maximum element.

We are given a rooted tree, and the task is to imagine drawing it on a grid under a very specific geometric rule. Each subtree is drawn inside a rectangular bounding box, and different subtrees attached to the same parent must be placed inside disjoint bounding boxes.

We are given a list of athletes, each with a strength value, and a sorted list of coffee cups, each with an increasing strength requirement. A cup can only be drunk by an athlete whose strength is at least the cup’s requirement.

We are given a collection of pokémons, each described by two numbers: its initial strength and its price. We are allowed to pick exactly one pokémon as our champion before the tournament starts.

We are given two arrays of the same odd length, and every value appearing in either array is unique globally. Because of this uniqueness, sorting each array produces a well-defined ordering with no ties, and the median is simply the element at position $(n+1)/2$.

We are given a one-dimensional cave split into n vertical columns. Each column has a ceiling height h[i]. Over time, the ceiling in every column drops uniformly: at each second, every positive height decreases by one.

We are given a permutation of numbers from 1 to n, but it is hidden. Instead of the permutation itself, we only know a derived array of length n − 1 where each value is the maximum of two consecutive elements in that hidden permutation.

We are given a length $n$ array, and from it we build an $n times n$ grid where every row is a cyclic shift of the previous one. The first row is the array itself, and each next row shifts all elements one position to the left, wrapping the first element to the end.

We are given a tree, meaning a connected graph with no cycles. Two special vertices are marked, a starting point $s$ and a target $t$. A fox moves on this tree, but its movement rules are stronger than normal adjacency.

We are working on a dynamic grid where each cell stores an integer height. A cell is considered land only if its current height is strictly positive. Connectivity is defined in the usual grid sense, where movement is allowed only between cells sharing a side.

We are given a very large grid that is never explicitly revealed. Each cell of this grid is painted with one of three colors, and we are guaranteed that no two side-adjacent cells share the same color.

We are given four short lowercase strings and need to determine whether it is possible to place all of them into a fixed grid as two horizontal words and two vertical words.

Codeforces 105863B: Numbers on the Blackboard

We are given a multiset of integers. The task is not about sorting or selecting elements directly, but about understanding how these values interact when forming pairs whose sum is fixed.

We are working on a coordinate system where each move is not a standard step but a reflection-like transformation.

We are given a fixed number of positions and a range of values, and we conceptually assign values from this range to each position.

We are given an array of positive integers and we need to evaluate a function that depends on splitting this array into contiguous blocks.

We are given an undirected graph structure implicitly described through distances from a fixed root node, node 1.

We are asked to count labeled undirected graphs on $n$ vertices where each possible edge is either absent or assigned a weight in $[1, k]$, under a specific structural constraint about minimum spanning trees.

The task revolves around applying a precise rounding rule to numeric inputs. Each input value is given in a textual form, and we are required to transform it into an integer according to standard rounding behavior, where the fractional part determines whether we move the value…

I can’t write a correct editorial from just “H” and an empty statement. Codeforces 105851H (“Link与矩阵除法”) clearly has missing problem content in your prompt, so there’s no way to reconstruct: - what the input actually represents (matrix operations, graph…

I can’t write a meaningful editorial for this because the actual problem content is missing. Right now the “Problem Statement” section only contains the single character G, and there is no description of the input, output, or rules.

The problem statement for Codeforces 105851E - 布置WAP is missing in what you provided. Right now there’s no description of the task, inputs, outputs, or constraints, so any attempt to write a correct editorial would be guesswork.

The task is about maintaining and querying ancestry relationships in a rooted tree. We are given a connected acyclic graph, and we interpret one node as the root of the structure.

The problem statement in your prompt is effectively empty (only the title “删除01串” without any input/output definition or constraints), so there isn’t enough information to reconstruct what is being solved.

We are given an array of positive integers and many range queries. For each query, a segment from index $l$ to $r$ is chosen. Inside this segment, we are allowed to pick a split point $k$ such that $l le k < r$.

We are given a tree with $n$ nodes. Each node represents a region, and each region has a value that depends on whether it is currently marked as colored or not. Initially all nodes are uncolored and each node $i$ has a base value $ai$.

We are given a positive integer written as a decimal string. In one move, we are allowed to choose a single pair of adjacent digits and swap them. We may perform at most one such swap, or choose to do nothing.

We are given a connected undirected graph with $n$ islands and exactly $n$ roads. Every road has cost 1, and the same road can be traversed multiple times, paying its cost each time.

We are given a rooted tree where node 1 is the root. Each node represents a skill, and each skill has a value called its power.

We are given a very large interval of integers, from $L$ to $R$, where $R$ can be as large as $10^{18}$. The task is to count how many numbers inside this interval satisfy a digit-based property: when you look at the decimal representation of a number, the absolute difference…

We are working on a grid of all pairs $(i, j)$ where $1 le i le n$ and $1 le j le m$. For each pair, we look at the greatest common divisor of $i$ and $j$. From that gcd value $g$, we compute the number of divisors of $g$, written as $d(g)$.

We are given a sequence $b$ whose length is $n$. This sequence was originally produced from an unknown integer array $a$, but during the process, the array was modified in a structured way. The hidden construction is the following.

We are given several independent test cases. In each test case, there are $n$ caves labeled from 1 to $n$, and a sequence of $n$ planned inspections, where each inspection targets one cave. The sequence is fixed and known in advance.

We are given a sequence of exactly three integers. We are allowed to rearrange these three values using swaps between any two positions, and after doing so we want the resulting arrangement to satisfy a very specific pattern: the first element must be strictly greater than the…

We are given a collection of bins. Each bin starts with some number of balls and has a maximum capacity. The key operation in the process is that whenever a bin becomes full, it is removed from active consideration and its entire content is transferred into a shared reserve.

The problem defines a sequence built from a very structured combinatorial recurrence. Each term is formed by considering all ways to decompose an integer into ordered or unordered collections of positive integers, and then aggregating weights derived from those decompositions.

We are working with a permutation of the integers from 1 to 3N, split conceptually into three consecutive segments of equal size N.

We are given a single lowercase string and we are asked to count special substrings that have a very rigid structure.

We are given five integers that describe a soccer team’s season statistics, but the order of these integers is scrambled. We know they correspond to matches played, wins, draws, losses, and total points, but we do not know which number is which.

We are given a single string consisting only of lowercase English letters. The task is to classify this string based on how many times each distinct letter appears. A string is called odd if every character that appears in it occurs an odd number of times.

We are given a circular board with positions labeled from 1 to n. A token starts at position 1. Each move consists of rolling a fair die with faces from 1 to d, and moving forward that many steps along the circle. If we go past n, we wrap back to 1. Some positions are special.

We are working on a very large integer grid where each point has four-neighbor movement, up, down, left, and right. Some grid points are blocked because construction is happening there, and these blocked points cannot be visited.

We are given a sequence of integers and a threshold value k. From the sequence, we want to pick a subsequence, meaning we keep some elements without changing their order, and we try to make it as long as possible.

We can model the system as a directed graph where each channel is a node. Each bot acts like a small forwarding rule: it listens on exactly one source channel, and when that channel receives a message, the bot forwards the message to a fixed list of destination channels.

We are given a straight path from position 0 to position L. Along this line there are special points called boost stations.

We are trying to identify one unknown “bad” restaurant among n candidates. Each week we are allowed to choose any subset of restaurants and observe a binary outcome: if the bad restaurant is inside the chosen subset, Mikey gets sick that week, otherwise he does not.

We are given a list of students, each with a distinct knowledge level. The goal is to partition them into contiguous groups after sorting by knowledge level, so that each group satisfies two constraints: the range inside the group, defined as maximum minus minimum knowledge…

We are given a skyline-like city made of axis-aligned rectangular buildings placed on top of the x-axis. Each building starts at some x-coordinate, extends to the right for a fixed width, and rises vertically to a given height.

We are given a starting pair of single-digit numbers, where at least one is non-zero. From this pair, we generate a sequence where each next value is formed exactly like Fibonacci: sum of the previous two values.

We are given a permutation-like array where inversions matter, and the central object of interest is the number of inversions inside a subarray. For any segment $[i, j]$, let $f(i, j)$ be the number of pairs $i le a < b le j$ such that $p[a] p[b]$.

We start with a multiset of exactly $k$ numbers. Out of these, $k-1$ copies are equal to some value $x$, and a single element equals $y$. So initially the structure is extremely uniform except for one “disturbance”.

We are given multiple independent test cases. Each test case describes a set of points on an infinite 2D grid. Every point represents a tower placed at integer coordinates.

Each test case describes a group of candidates being evaluated on two independent dimensions: performance in a series of training contests, and performance in a problem-solving assessment outside contests.

I can’t reliably write a correct editorial for this because the actual problem statement is missing. Right now you’ve provided only the title (“最小LCM”) and empty Input/Output sections, which isn’t enough to reconstruct the intended task.

The statement content is missing from your prompt (only the title “腐蚀与膨胀” is present, with no input/output or description). Without the actual rules of the process, there’s no way to reconstruct the intended algorithm or write a correct Codeforces-style editorial.

I’m missing the actual problem content for Codeforces 105851C - 砝码. The statement section you provided is empty, so there’s no way to reliably reconstruct the task, constraints, or required solution without guessing.

We are given a sequence of numbers and a sequence of modulo operations that are applied to them. Each time a number is processed with a modulus value, it is replaced by its remainder.

We are given a tree, meaning a connected graph with no cycles. Some vertices of this tree are leaves, vertices with degree one.

We are given two arrays of the same odd length, and every number appearing anywhere in the input is globally unique. The only operation allowed is to pick an index and swap the elements of the two arrays at that position.

We are given a sequence of integers and a fixed way of turning any contiguous segment into a number using a rolling base.

We are given a line of $n$ people, each holding some number of identical items. The total number of items is divisible by $n$, so there is a well-defined target value: everyone must end up with exactly $s/n$ items. Initially, the distribution is $a1, a2, dots, an$.

We are given a connected undirected graph where every edge is labeled either red or blue and has a distinct weight. The task is not to compute a single minimum spanning tree, but to understand how spanning trees behave when we constrain how many red edges they contain.

We are given a collection of strings that are already sorted by nondecreasing length. For any two strings where the first one is not longer than the second, we want to decide how many ways we can make them “match” if we are allowed to replace a contiguous block of fixed…

We are given a universe of K keys, and a family of subsets F of these keys. Each subset A represents the set of keys that are currently usable in a maze.

We are given a tree where some vertices contain stones, and we are allowed to move each stone along edges, one step at a time.

We maintain a dynamic collection of pairs $(v, w)$. Each operation either inserts a pair, removes one occurrence of a pair, or asks a query with a number $k$.

We are maintaining a graph on vertices labeled from 1 to n, but n itself can be extremely large, so large that we cannot even think of storing anything indexed by vertices directly. Initially there are no edges, and we receive a sequence of queries.

We are given a rooted tree with a weight on every vertex. The tree structure is described by a parent array, so every node except the root has exactly one parent and the whole graph is connected and acyclic.

We are given a set of candidates indexed from 1 to n. Each candidate i starts with ai loyal supporters. In addition, there is a pool of c undecided voters who, if nothing changes, always vote for the smallest-numbered candidate among those who are still participating.

We are given a permutation-like array, except the values are arbitrary distinct integers rather than a contiguous range.

We are interacting with a hidden integer $x$ that always lies between $1$ and $n$. We cannot see $x$ directly. Instead, we are allowed to ask up to four questions of the form “is $y x$?

We are given several login passwords, one per team, and each password is just a string composed of printable characters like digits, uppercase and lowercase letters, underscores, and hyphens.

We are given an undirected graph with n nodes and m existing connections. Because the system is damaged, these connections may include duplicates, cycles, and even useless self-loops.

We are given a rectangular grid where each cell is either empty or blocked. You start at the top-left cell and want to reach the bottom-right cell, moving in the four cardinal directions without leaving the grid. Normally, movement is only allowed through empty cells.

We are given an array of length $n$, where each position stores a Pokémon type represented by an integer. The array changes over time through a sequence of operations.

We are given a very narrow weighted grid that represents a walking map. Movement happens along grid edges, and each edge has its own travel speed, so traversing different edges costs different amounts of time.

We are given a strictly increasing sequence of coin denominations starting from 1. Using these coins, we define a deterministic way to “pay” any positive integer x: always use the largest possible coin first, take as many of it as possible, then move to smaller coins, and so…

We are given a permutation of cards numbered from 1 to n, and we want to understand how many different “cutting procedures” can produce a given target sequence. A cutting procedure is not just a final arrangement rule, it is a constructive process.

We are given a sequence of write operations applied to a sparse array of size $n$, initially empty. Each operation writes a value $x$ into position $i$. The twist is that two different systems interpret overwrites differently.

We are given an array of integers, and we need to count how many ordered triples of distinct indices form an arithmetic relation where the sum of two selected elements equals a third element in the array.

We are given a $2^n times 2^n$ grid of lattice points. Each cell is assigned a unique order using the Hilbert curve indexing, which is a recursive space-filling traversal that produces a permutation of all grid cells from smallest to largest index.

We are simulating the scoring system of a simplified rhythm game where only two note types remain relevant: TAP and BREAK. Every note is judged into one of several performance categories, and each category contributes differently to the final score.

We are given a point on an infinite integer grid and a target point. In one move, the token can change its position by stepping in a “knight-like but diagonal-biased” pattern: both coordinates change together, and the allowed absolute step pairs are (1,1), (2,2), (1,2), or…

We are looking at a system that evolves on integers from 0 to N. From a current state x, each second gives you a single decision before anything else happens. If you choose the passive option, the state simply moves deterministically from x to x − 1 after one second.

We are given two parallel sequences of rock types, each sequence being a permutation of the same set of labels. The players are allowed to trim both sequences only from the left and only from the right.

We are simulating a particle moving inside a one-dimensional corridor of length $d$. The particle starts at position $k$, measured from the left wall, and initially moves to the right with speed $v$. The motion is perfectly linear between collisions.