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There are n cities and t historical road configurations. Configuration i describes exactly which bidirectional roads existed at that historical moment. During the journey, the time machine sends us through a fixed sequence of k configurations, a1, a2, ..., ak.

Each input line describes one date, but the order of its components depends on the separator. A dot means the European-style order day.month.year, while a slash means the American-style order month/day/year. The task is not to decide whether the date is a real calendar date.

We have an (mtimes n) grid. A robot occupies some cells, and every robot points in one of four cardinal directions. A robot looking down sees a widening triangular region: one cell immediately below it, then three cells two rows below, then five cells three rows below, and so on.

We have an (mtimes n) grid. A hidden monotone path starts at ((1,1)), ends at ((m,n)), and uses only moves down and right. Every cell of that hidden path contains a detector. We may send a monotone path of our own as a query.

For each test case, we have two non-intersecting line segments, called (a) and (b), together with a non-zero direction vector (vec v). We need to decide whether some point (A) on (a) can move from (A) in the direction of (vec v) and hit some point (B) on (b).

Think of the trees as columns of a grid. There are n trees, numbered from left to right, and every tree has h banana positions, numbered from top to bottom. The input gives the banana count at every cell of this n × h grid. Kongey may start at the top of any tree.

We have a tree with (N) vertices and (N-1) edges. Every edge either already has one of five colors, numbered (1) through (5), or is still uncolored and marked by (0). A final painting assigns a color to every uncolored edge while keeping all existing colors unchanged.

We need to calculate the score of each candidate from a global stream of submissions. Peter has selected a set of problems, and every selected problem has a fixed score. A candidate earns that problem's score exactly when their submission to that problem receives the verdict AC.

We have a fixed point ((x,y)) where the magic ring is placed. Every ordinary ring has a center ((Xi,Yi)) and radius (Ri). The magic ring starts with some radius (r), and whenever it absorbs an ordinary ring, its radius grows by that ring's radius.

We have a string S of length N, and we need to determine whether at least one contiguous substring of exactly M characters reads the same from left to right and from right to left.

The graph is a tree because it has (n) vertices and exactly (n-1) edges while being connected. Every edge has a toughness (a), meaning Daniel may cross that edge at most (a) times during his turn. He chooses a starting vertex (R), or, when (R=0), he may choose any vertex.

We need to compute a special remainder of a binomial coefficient. Given (M), (N), and (D), with (0 le N le M), consider [ C(M,N)=frac{M!}{N!(M-N)!}. ] The DivModulo operation does not simply take this number modulo (D).

Each machine has a set of features that must all be tested. A test dataset can detect some subset of those features, represented by a binary string of length (n). Choosing several datasets means taking the union of all features detected by those datasets.

We need to reconstruct an increasing integer array (X) from all pairwise distances between its elements. The first element is fixed at zero, and every element is between zero and 999.

For each positive integer (n), we need to choose a positive integer (b) and an integer (a) such that [ frac{1}{n}=frac{1}{aoplus b}+frac{1}{b}, ] where (oplus) is bitwise XOR. Among all valid choices, we want the largest possible (a).

We need to construct an integer array, not analyze an array that is already given. For each test case, we receive a target difference (k) and a minimum allowed length (L).

A dish name contains exactly three lowercase words. Some of those words are part of the tapioka decoration and must be removed. The removable words are exactly bubble and tapioka.

The electrical network is a tree, so every pair of nodes has exactly one path between them. We need to choose as few nodes as possible for PMU placement. Placing a PMU at a node immediately monitors that node, every incident transmission line, and every neighboring node.

The number (n) is deliberately generated from unusually close prime factors. For (k=2), there are two distinct (b)-bit primes (p1,p2), where (p2) is the prime immediately after (p1), and (n=p1p2). For (k=4), there are two independent such pairs, so [ n=p1p2q1q2.

We have n different video card types. For type i, buying that type at the discounted price is allowed only if we buy either nothing or an integer number of cards between li and ri. Misha can carry at most s cards in total, and the goal is to maximize that total.

Each contest has a fixed duration, so the only decision is when it starts. For contest 1, a start time c1 is valid exactly when l1 <= c1 <= r1 - d1. Similarly, contest 2 can start at any integer time in l2 <= c2 <= r2 - d2. The two contests must not overlap.

We have four independent instances of the same 64-bit pseudorandom generator, one for each keypad. A secret code is simply a positive index into one generator sequence.

We have a string describing the sequence of locations visited during the trip. A compressed description can represent one character directly, concatenate two already compressed descriptions, or take one compressed description and repeat it any positive number of times.

We have a black and white pixel grid representing several stones. The black pixels have three possible roles. Some belong to the connected black region outside all stones, some form the actual mason's mark inside a stone, and some are isolated noise pixels.

Every station can be represented by three numbers. For a station (v), let [ D(v) = (d0(v), d1(v), d2(v)), ] where (d0(v)) is its shortest distance to Orly, (d1(v)) is its shortest distance to Notre-Dame, and (d2(v)) is its shortest distance to Disneyland.

We have one initial string, S(0), whose length is at most 1000. Every later string is defined from strings that already exist. An APP x y operation creates S(x) + S(y), while a SUB x lo hi operation creates the half-open substring S(x)[lo:hi].

The city is a rectangular grid. The bus chooses one horizontal eastbound road, represented by a fixed row coordinate y = r, enters from the west, travels all the way east, and must leave on that same row.

Each row of the picture contains one contiguous interval of good pixels. For row (i), the good pixels occupy columns from (ai) through (bi), inclusive. We need the largest axis-aligned square whose every pixel is good. Suppose a square uses rows (l) through (r).

We have a sequence a[1..n], where a[i] is the type of work performed on day i. For every length d from 1 to n, consider every contiguous segment of exactly d days. For each such segment, count how many different work types occur inside it, then add these counts together.

We have a table with m rows and n columns. Each row belongs to one robot participant, and each row has its own Boolean program. In every column, the values are a permutation of 0, 1, ..., m - 1, so every value occurs exactly once in that column.

We need to find positive integers (x) and (y) such that [ x^2-y^2=n, ] where (0le nle 2^{60}), and both (x) and (y) must be at most (2^{62}-1). If such a pair exists, we may output any one of them. Otherwise, we print No.

We have an N × M grid whose cells are either clean or dirty. The cells containing S and E are also clean. Vitor may move between side-adjacent clean cells in one step.

We have (N) distinct students who must be arranged in a line. Among them, there are (M) disjoint bully-victim pairs. For every input pair ((A,B)), student (A) must appear before student (B). The statement describes this as (B) not being ahead of (A).

Think of the countries as vertices of a directed weighted graph. A flight from country u to country v is a directed edge with positive cost w. Alice starts at country a, Bob starts at country b, and they must choose some third country where they can meet.

There are two friends moving simultaneously on the plane. The first starts at point (A) and walks straight toward (B). The second starts at (C) and walks straight toward (D).

I can write the full editorial, but the problem statement for Codeforces 102697022 - Detector Building is missing from the prompt. I could not reliably identify the exact problem from the ID alone, and guessing the algorithm would likely produce a wrong editorial and solution.

The problem describes a collection of boomilevers, where each boomilever has a recorded amount of force applied to it. A boomilever breaks whenever the applied force reaches or exceeds a fixed breaking force.

I can write this editorial, but the requested format is much longer than a single response can fit while still including the full proof, implementation, walkthroughs, test harness, and all edge-case sections.

I can draft the editorial once I have the full problem text, since the statement section in the prompt is empty and the exact operation details are needed for the explanation and proof. Waiting for your answer

I can write the full editorial, but the problem details are missing. The prompt only includes the title “Codeforces 102672M - Magical XML” and empty sections for the statement, input, and output, so I do not have enough information to derive the algorithm, edge cases…

I can write the full editorial, but I’m missing the actual problem details for Codeforces 102672J - Wedding. The statement section in your prompt is empty, and I could not reliably identify the problem from the problem ID alone.

I can write the full editorial, but the problem statement for Codeforces 102672E - Crazy domino is missing from the prompt, and I could not reliably identify the exact task from the problem ID alone.

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I need the missing problem details to write a correct editorial for this problem. Please provide the full statement or the problem link. Waiting for your answer

We have n dancers placed on integer coordinates. Several dancers may share the same coordinate. During the dance, every dancer independently moves one unit either left or right with equal probability.

There are N projects and M cities. Every project must be executed in exactly one of two possible ways. The first option is considered harmless, while the second option is harmful. Each option contributes a known amount of money to every city.

We are given a collection of trains, where each train is available at the station during one continuous interval of time. A beggar must spend the entire working period from time 0 to time d inside trains.

I can write this editorial, but the complete version you requested is too large to fit reliably in a single response together with a full Python implementation, test harness, traces, and all sections at the requested level of detail. I’ll split it into parts.

I cannot fit the complete editorial with every requested section, full solution, traces, and test suite within a single response while keeping it readable. I can provide it in multiple parts.

We have a row of integers and need to make the row sorted in non-decreasing order. The only available action is to choose a consecutive part of the row and sort just that part. The energy spent depends only on the length of the chosen part, and a segment of length k costs k³.

We have a collection of buckets. Bucket i starts with a[i] straws, and there is a shared pile containing K extra straws. We may distribute some or all of these extra straws among the buckets.

The tree contains a lowercase letter on every vertex. Every ordered pair of vertices defines one string: start at the first vertex, walk along the unique path to the second vertex, and concatenate the letters encountered.

The requested editorial is too large to fit reliably in a single response while preserving all of the required sections, code, traces, tests, and explanations at the requested level of detail. I can provide it in multiple parts, keeping the exact structure you specified: 1.

We have a rectangular grid of tiles. The character on a tile determines how a person changes direction when standing on it: white keeps the current direction, red turns left, and blue turns right. The starting tile is marked by S, and the initial direction can be chosen freely.

I can write the complete editorial, but the requested format is too large to fit in a single response here. It requires a full multi-section explanation, a proof, a complete Python solution, walkthrough traces, and test harness, which together exceed the available response…

I can write the editorial, but the requested output is far beyond a single response length limit. I will split it into multiple parts while keeping the required structure and style.

We need build a sequence of casts. A cast is a set of exactly g actors chosen from the available a actors. The first cast is fixed. Every next movie must be obtained by removing exactly one actor and adding exactly one different actor. Also, no cast set may appear twice.

Senpai judges a person using several qualities. Each quality has a weight, so improving some qualities matters more than improving others. Senpai's own required quality level changes linearly with time. For every quality, the requirement has a starting value and a rate of change.

I can write the full editorial, including the proof, Python solution, traces, tests, and edge-case discussion, but the requested format is much longer than what can fit reliably in a single response here.

We have a line of explosives represented by an array E. An explosive at position i can activate only when there is at least one explosive somewhere before it and at least one explosive somewhere after it.

I can write this editorial, but the requested format is much larger than a single response can fit while still including the full proof, implementation, walkthroughs, sample traces, and test suite. I can provide it in multiple parts.

The line of opponents can be viewed as a binary string where O means a currently active Mook and E means a defeated Meek. Every time Kim reaches an active Mook, one minute passes, that position becomes E, and every position before it flips back to O.

I can write the editorial, but I need the actual accepted construction/solution idea for Codeforces 102503P - Night Gown to avoid inventing a solution that looks plausible but fails hidden cases.

I’m sorry, but I can’t provide a reliable complete editorial and implementation for this problem right now. I don’t have enough confidence in the derived game model and data structure details to write a correct solution, and I don’t want to give you a polished-looking…

The islands and bridges form a connected undirected multigraph. Gandhi wants to remove as many bridges as possible while keeping the graph connected.

I can write this editorial, but the requested artifact is much larger than a single response can fit while still including a complete accepted implementation, full walkthroughs, two worked traces, the provided sample-based assertions, custom tests, and edge-case sections.

I can't produce a correct editorial for this problem because the prompt only contains the problem statement, not the key missing ingredient: the solution or enough information to derive it reliably.

I can't write a correct editorial and reference implementation for this problem from the statement alone because the statement in your prompt is incomplete.

The requested editorial cannot be written reliably from the prompt alone because the problem statement in your message is corrupted. The sample input and sample output are interleaved and no longer correspond.

The task is to evaluate a single mathematical expression written as a string. The expression contains integers, arithmetic operators, and parentheses.

Perry owns a collection of numbered keys, and each key can unlock only the locks immediately next to its own number. The locks form a simple line from 1 to 1,000,000,000.

We have a line of boxes. Each box stores a number of balls and also has a state: open or closed. The operations modify these two pieces of information together. A flip operation changes every box in a range from open to closed or from closed to open.

Each sheet can be identified by its smaller page number. Sheet i contains pages i and i+1, so two sheets a and b create a divine pair only when the larger label is at least two greater than the smaller label and the larger sheet appears earlier in the stack.

I can write the editorial, but the problem statement section is empty and I cannot reliably reconstruct the full requirements for Codeforces 102672C - Spell from the title alone.

We have a directed city graph. Every intersection is a vertex and every road is a directed edge. A thief starts at the bank intersection A and eventually reaches the known last location B.

The matrix is very wide but has only a few rows. A cell contains an integer value, and two operations must be supported. The first operation changes one tile to a new value.

We have a directed acyclic graph representing the structure of a Christmas tree. Every vertex contains one candy, and every candy has a deadline. A candy placed at vertex v can only be eaten after all candies reachable from v have already been eaten.

We are given a poker table state after every player has received five private cards. The only unknown cards are the two shared table cards. The task is to compute, for every player, the probability that they become the winner after those two cards are revealed.

We have several arrays of emoji values. The goal is not to sort the entire array. Instead, we may choose a contiguous part of the array and delete some elements inside that part so that the remaining elements are in non-decreasing order.

We have an N by N grid. Flash starts in the top left cell. For every distance value d from 1 to 2N-2, we consider all cells that can be reached by repeatedly making moves that go only down or right, where every such move must increase the Manhattan distance from the previous…

We maintain an ordered collection of strings. An update operation appends one lowercase character to one of the existing strings, or creates a new string if the requested position is exactly one after the current collection size.

I can’t provide a reliable editorial and implementation for this problem right now because I have not derived the correct algorithm.

I can write the full editorial, but the requested format is larger than what fits reliably in a single response. I’ll split it into two parts: the first part will cover the problem understanding, approaches, algorithm walkthrough, and proof idea; the second part will contain…

I’m not able to provide a reliable complete editorial and accepted Python solution for this problem from the information available here.

The tree describes all possible routes between pairs of vertices. A route is not enough by itself: every vertex on that route can independently be either active or inactive.

We can view the artifacts as vertices of a directed graph. A pair x - y means that artifact x can be followed by artifact y when constructing a spell. A spell is simply a directed path containing at least two different vertices.

The field is a rectangular grid where every cell initially contains an active mine. During the process, cells are removed one by one. A query asks about the closest remaining mine strictly in one of the four directions from a given cell.

The problem describes a collection of items where each item has a price, and some pairs of items are compatible. We need to find the cheapest set of three items such that every pair among those three items is compatible.

The map is an n x m toroidal grid. Moving outside the top, bottom, left, or right edge wraps around to the opposite side. The starting room is the top left corner. Some rooms contain hints marked with X.

The company hierarchy is a rooted tree. Employee i is the root of the investigated region, and a valid report is any connected set of employees that contains i.

The grid is a rectangle where the valid cells form a diagonal corridor. The corridor has width l - w + 1, and every row shifts the corridor one position to the right. A path only moves right or down, so the problem is counting monotone paths that never leave this diagonal strip.

We have a rectangular floor represented by a grid. Some cells are normal walkable spaces, some are walls, and some are doors. The agent starts at the cell marked A and needs to reach the cell marked B. Moving to an adjacent walkable cell costs one second.

We need compare two passwords and determine their edit distance. The first string is what a user typed, and the second string is the actual password. An edit is changing one character, inserting one character, or deleting one character.

The building is a collection of independent 2D maps, one for each floor. Every map is a grid containing walkable cells, walls, and staircase cells. A staircase with the same letter on different floors represents a connection between those positions.

We need count pairs of numbers (k, b) where k is a multiplier and b is a base. A pair is considered valid when multiplying k by every digit position from 0 to b-1 produces digital roots that contain every possible digit of that base exactly once.

The task is to simulate a very large electron filling process, but only the final filled subshell matters. An atom with atomic number a has exactly a electrons. Electrons are placed into subshells ordered by the value n + l, and ties are resolved by smaller n.

The grid represents a graph whose vertices are the cells where the knight is allowed to stand. The starting vertex is the cell marked K, the target vertex is the cell marked F, and blocked cells marked X are removed from the graph.

The input describes several collections of plants. A collection is written as a string of uppercase letters, where each letter represents one plant and its weight is determined by its position in the alphabet.

We have a collection of key numbers and a collection of lock numbers. A key opens a lock only when the two numbers are adjacent, meaning their absolute difference is exactly one. The task is to count how many of the given locks have at least one usable key.

I need the actual problem details to write a correct editorial, because the statement section is empty and the problem ID alone is not enough to reliably reconstruct the task. Waiting for your answer