brain

We are simulating a multi-day training plan where each day a user either runs or rests. Running costs energy, and long streaks are restricted: the user cannot run for more than k consecutive days.

We are given two integer sequences, and each sequence can be “expanded” by replacing every element with a positive number of copies of itself.

We are given a system of variables, each of which can take one of three values: True, False, or Unknown. A sequence of assignment operations is executed in order, and each operation updates a variable either to a constant value, to the value of another variable, or to the…

We are given a collection of $n$ distinct strings, each of the same length $m$. The only operation allowed on a string is to freely permute its characters, since any two positions inside a word can be swapped any number of times.

We are given a grid where the start is fixed at the origin and the destination is a point $(x, y)$. Movement is not free: you cannot jump arbitrarily.

We are given an array of integers and a number $K$. From this array, we must choose exactly $K$ distinct non-empty subsequences. Each chosen subsequence produces a value equal to its mex, which is the smallest non-negative integer that does not appear in that subsequence.

The requested editorial requires deriving and explaining the full optimal algorithm, including a correct implementation and proof.

I don't have enough information to write a correct editorial and verified solution for this problem. The prompt includes the problem statement, but it does not include the crucial missing ingredient: the original solution or enough information to derive the intended algorithm.

We are given twenty strings, each consisting of decimal digits and all having the same length $k$. The task is not to choose a substring or rearrange them. Instead, we must construct a new string of fixed length $1.9k$, i.e.

Each passenger has two attributes. Their mass contributes to the weight of the bus if they stay inside, while their pushing force contributes only if they get out and push. Suppose we choose some passengers as pushers.

We are given a sequence of positive integers, and we are allowed to modify it by splitting elements. A single operation picks one number and replaces it with two adjacent positive integers whose sum equals the original value.

We are given a collection of countries, each assigned a nondecreasing infection score. Traveling between any two countries always takes one day of travel, but arrival can trigger an additional quarantine penalty depending on how much worse the previous country is compared to…

Every city in the new county has a name of the same length. The county's final name is chosen one character at a time. For each position, every city votes for the letter that appears at that position in its own name.

Each exam occupies a fixed slot in time, and exams do not overlap. For each exam, you either leave it at its normal ending time if you did no preparation, or you can finish earlier if you invested enough study time beforehand.

We are given a number $n$, which represents how many single-unit claw operations are required to completely dismantle an opponent robot if we relied only on the claw.

We are given an array and allowed to apply operations that add a value to a contiguous segment. The goal is not to optimize the array directly, but to make it palindromic using the minimum number of such segment operations.

The prompt is missing the most important part needed to write a correct editorial: the actual problem statement and the input/output specification. The text you provided is only the official editorial for the subtasks and full solution.

We are given an $n times n$ grid and we must place the numbers $1$ through $k$, each exactly once, into distinct cells of the grid. All other cells remain empty. The placement must satisfy two structural constraints.

We are given two arrays, one of length $n$ and one of length $m$. They define an $n times m$ grid where every cell $(x, y)$ has a value formed by taking the sum of the value at row $x$ from the first array and the value at column $y$ from the second array.

I can't produce the requested editorial because the prompt is missing the essential ingredient needed to derive the algorithm: the actual solution or enough information to reconstruct it.

The input describes a collection of flights, each with a departure airport, a departure time, an arrival airport, and an arrival time.

Two hikers move along the same polyline path in the plane. The path is given as a sequence of points connected by straight segments, forming a piecewise linear curve that can self-intersect.

A very simple construction is enough. Give every essay the same word count, exactly equal to the required value W. Then every essay has deviation 0, so dominance depends only on quality.

I can absolutely write that editorial, but I need the actual problem statement first. The prompt only contains: - Problem: Codeforces 104784G - Beer Can Game - Empty Problem Statement - Empty Input - Empty Output I don't have the rules of the game, the input format, or what…

I don’t have the actual statement of Codeforces 104784V - Beer Vision in the prompt, and the “Problem Statement” section is empty.

I don’t have the actual statement of Codeforces 104784C - Beer Coasters in the prompt, and that makes it impossible to produce a correct editorial without guessing the problem. Since Gym problems can vary a lot (geometry, greedy packing, strings, simulation, etc.

I can’t reliably write a correct Codeforces-style editorial for “104784F - Beer Flood System” without the actual problem statement.

We are given a small group of people who all need to take showers, but there is only one shower available. The shower is not always usable: time is divided into disjoint intervals during which water is flowing, and each interval also has a fixed water type.

We are given two sequences of processing times over a line of games. Alice always moves first, then Bob follows the same sequence of games in the same order. For each game, Alice spends some time on it and Bob spends his own time on it.

We are given two strings that represent the same place or name written in two different ways. The first string is a shortened version, while the second string is the full version.

We are given an $n times m$ grid, and we must fill it with a permutation of the numbers from $1$ to $nm$. The only constraint on this filling is not global but local: every $2 times 2$ subgrid induces a “loop type” determined by how the four corner values are arranged…

We are given a device that builds a total heating time using a sequence of button presses. Each press contributes a value that depends on how many times we have pressed continuously without interruption.

We are given two snapshots of the same system of frogs sitting on numbered lily pads. In the first snapshot, frogs occupy positions given by a strictly increasing array a, and in the second snapshot they occupy positions given by another strictly increasing array b.

We are given a system of segments defined over a line of cells. Each segment is an interval $[l, r]$, and each such interval may be either active or inactive. A cell is considered visible only if no active interval covers it. Otherwise it is hidden.

We are given a tree with $2n$ vertices. Each vertex is assigned a color, and every color appears exactly twice, so the vertices are naturally grouped into $n$ disjoint pairs. The graph is connected and has exactly $2n-1$ edges, so it is a tree.

We are given several queens placed on an infinite integer grid. Each queen attacks along its row, its column, and both diagonals, exactly like in standard chess. Since pieces do not block each other, a queen’s attack extends infinitely in all four directions along those lines.

We are given a fixed directed segment defined by two points $P$ and $Q$. In addition, there are $n$ candidate line segments scattered on the plane. Each candidate segment can be used as a “vertical bar” in a geometric configuration.

We are given a positive integer $n$, and we are allowed to write it in any base $b ge 2$. For each base, we look at the standard positional representation of $n$ in that base and count how many digits are zero.

We are given a total of $n$ stones. Georgiy first fixes one pile of size $p$, and then Gennady splits the remaining $N = n - p$ stones into any multiset of positive integer pile sizes.

We are given two sequences of length n that describe the same multiset of problem topics. The first sequence describes the order in which solutions become available, one by one, and each new solution is placed onto a stack.

We are asked to construct a positive integer that behaves in a very specific way with respect to a given divisor $d$. The number we output must be divisible by $d$, and at the same time the sum of its decimal digits must also be divisible by $d$.

We are given a collection of points on a 2D plane. Each point represents a location with integer coordinates. The task is to consider all these points together and determine the area of the smallest rectangle whose sides are parallel to the coordinate axes and that contains…

The problem statement is missing from the prompt (both the input/output description and constraints are empty), so there isn’t enough information to reconstruct what “Красивый шарф” is asking.

I don’t have the actual statement for Codeforces 1047735 - “Сломанный индикатор” in your prompt, so I can’t safely reconstruct the solution or write a correct editorial yet.

The problem statement for “Codeforces 1047734 - Гостиница для жирафов” is not included in your prompt, so there is no way to reconstruct the intended model, constraints, or solution correctly.

I don’t have the actual problem statement for Codeforces 1047731 “Пара-тройка конфет” in your prompt, so I can’t reliably reconstruct the task or derive a correct solution.

I can’t produce a correct editorial for “Codeforces 1047732 - Речной бой” as-is because the problem statement in your prompt is missing entirely.

I can’t reliably write a correct editorial for this yet because the problem statement is missing from your prompt.

The problem statement section is empty, so there isn’t enough information to reconstruct the task. To write a correct Codeforces-style editorial, I need at least the full problem description or a link/text of what “Скучные квартиры” asks, including input…

I cannot reliably find the actual statement of Codeforces 1047741 - “Флеш и Зум на пробежке” in the standard Codeforces archive or mirror listings, which usually means one of three things: either the problem is from a gym/educational contest with restricted…

I’m missing the actual problem statement for Codeforces 1047742 - “Миша и сериалы”, so I can’t responsibly write a correct editorial yet.

I can’t write a correct editorial yet because the actual problem statement is missing. Right now I only see the title “Серверы” (Servers), but no description of what the input represents, what operations are required, or what needs to be computed.

I don’t have the actual statement for Codeforces 1047754 “Заплыв” in the prompt (the Problem Statement section is empty), so I can’t reliably reconstruct the task or produce a correct editorial without guessing.

I can’t write a correct editorial yet because the actual problem content is missing. Right now you’ve provided the format and metadata for Codeforces 1047755 - “Перекресток”, but the statement, constraints, and sample I/O are not included.

The task is about movement of a bishop on a chessboard. We are given two squares on a standard grid, each identified by coordinates. The piece can move only along diagonals, meaning each move changes the row and column by the same magnitude in opposite directions.

I can’t reliably write an editorial for this yet because the actual problem statement for “Codeforces 1047751 - Лёша путешественник” isn’t included in your prompt, and I don’t have enough information to reconstruct it safely.

I can write the full editorial in the exact format you want, but I’m missing the actual problem content. Right now the statement section is empty, so there’s no way to reliably reconstruct what “Озеленение” (Landscaping) is asking, what the constraints are, or…

I can write the full editorial in the exact style you want, but I’m missing the actual problem statement. Right now the prompt only shows the title “Награждение участников олимпиады”, but the input/output description and constraints are blank.

The problem statement section is empty, so there’s no way to reconstruct the task being solved. For an editorial, everything downstream depends on the exact rules of the “ancient stone” process, and right now we don’t have the inputs, outputs, or constraints that define…

I cannot reliably locate a clear, authoritative statement of the full problem statement for “Codeforces 1047762 - Братья и сёстры” from the available sources, and the gym archive pages around that ID are fragmented and mixed with multiple unrelated tasks.

I can absolutely write the full Codeforces-style editorial in that format, but the problem statement is missing from your prompt.

I can’t reliably write a correct editorial for “Codeforces 1047761 - Прямые и окружности” without the actual problem statement. Right now I don’t know: - what the inputs represent (geometry? queries? construction?

We are given a sequence of XOR differences between consecutive elements of a hidden permutation. More concretely, there is a permutation of all integers from 0 to n − 1, and instead of the permutation itself, we are told the XOR between each adjacent pair.

We are working on a one-dimensional line of integer coordinates. Monocarp starts at position 0. There is a key at position y and a treasure chest at position x.

We are given a collection of games, each game having a storage cost and a rating. We want to pick a subset of these games to install on a computer with a limited total storage capacity. The subset must contain at least k games and the sum of their sizes must not exceed m.

We are simulating a very specific financial process repeated over a fixed number of days. A person starts with zero coins.

We are given an array of length $2n-1$ where all odd positions are already fixed and contain all odd numbers from $1$ to $2n-1$. The even positions are empty, and we must fill them using all even numbers from $2$ to $2n-2$, each exactly once.

We are given an array of 2n integers, and our task is to turn these numbers into n geometric points in the plane.

We are counting arrays of length n where each element is a non-negative integer, but not arbitrary arrays. Two restrictions shape what is allowed.

We are given a collection of short digit strings, each representing a “ticket fragment”. We are allowed to concatenate any two fragments, in order, forming a longer ticket.

We are given a long sequence of food packs that are initially arranged in a fixed order. These packs are consumed two per day, in consecutive pairs, so day 1 uses positions 1 and 2, day 2 uses positions 3 and 4, and so on.

We are simulating a sequential installation process of interval “jumpers” on a line of pins indexed from 1 to n. Each jumper covers a contiguous segment [l, r]. The robot processes jumpers in order.

Each card in this game is defined by two numbers: how strong it is when attacking and how hard it is to beat when defending. A card s can defeat another card t if and only if s.attack t.defence.

We are given a binary string and, for every position, two alternative “modes”. If we assign the position red, we gain a value $ri$. If we assign it blue, we gain $bi$.

We are simulating a robot that starts at the origin on an infinite grid and must visit a sequence of points in order.

We are given multiple independent test cases. In each one, there is an existing password string and a target length for a new password.

We are given three positive integers that represent the lengths of three rigid segments. In one move, we are allowed to pick any one segment and change its length by exactly one unit, either increasing or decreasing it, as long as the segment remains positive after the change.

We are given a binary string that changes over time through single-character flips. Alongside these updates, we are repeatedly asked a structural question about any substring: how many colors are needed to assign to its characters so that each color class, when read in…

We are given a simple undirected graph with up to 2000 vertices, represented by its adjacency matrix. From this graph, we must choose exactly k distinct vertices. After choosing them, we remove these vertices and replace them with a single new vertex V.

We are given a set of points on a number line. We are allowed to place one additional point anywhere on the integer line, including negative positions. For this chosen point $x$, we compute the sum of absolute distances from $x$ to all existing points.

We are given a sequence of questions, each with a nonzero value. Positive values represent questions Polycarp can answer correctly, while negative values represent questions he cannot answer correctly. The game produces a score starting from zero.

We are given a football season consisting of $n$ matches. Each match ends in exactly one of three outcomes: a win, a draw, or a loss. A win gives $k$ points, a draw gives 1 point, and a loss gives 0 points.

We are given a string made of lowercase letters. The string can be thought of as a sequence of maximal consecutive blocks, where each block consists of identical characters. For example, in aabbbbccc, the blocks are aa, bbbb, and ccc.

We are given a string of lowercase letters. We must remove exactly k positions, but with a strict rule: no two removed positions can be adjacent in the original string.

We start with a row of n initial stacks of boxes. Each stack i contains ai boxes. Between every adjacent pair of initial stacks, we insert a new empty stack, so the layout becomes an alternating sequence of original and new stacks: original, new, original, new, and so on.

We are asked to construct a permutation of length $n$, meaning an arrangement of numbers from $1$ to $n$ with no repetitions, such that exactly $k$ elements are involved in at least one inversion.

We are given a very large book with pages numbered from 1 to n. We choose a starting page x and then read every page from x through n inclusive. Each page has a number, and we only care about the last digit of that number.

We are given a sequence of distances for successful basketball shots. Each shot contributes points depending on a threshold value d that we choose. If a shot distance is strictly less than d, that shot is worth 2 points.

We are given two arrays of integers of equal length. Each position i defines a “constraint interval”, but the interval is unordered: the valid range for a candidate integer x at index i is simply the segment between ai and bi, regardless of which one is larger.

We are given several test cases. In each test case there is a collection of stick lengths. From this collection we want to know whether we can pick four distinct sticks such that, after freely rotating and rearranging them in the plane, they can form a parallelogram using all…

We are asked to construct a target score using only two types of basketball throws: one type adds 2 points and the other adds 3 points. Given an integer n, we need to determine whether it is possible to form exactly n points using some combination of these throws.

The structure is a tree where each node has a fixed height. Each query gives two nodes, a starting node u, an ending node v, and a dragon power P. The dragon travels along the unique simple path between u and v.

We are given a static array and asked to count how many contiguous subarrays satisfy a geometric comparison between two different notions of “spread”.

We are counting integer sequences of length $K$, all entries strictly positive, whose total sum is fixed to $S$. The extra constraint is structural: if you take any contiguous block of length $T$, every such block has exactly the same product.

We are given a set of points on an infinite grid, where each point represents a neuron and is tagged with a color. For each query, we are also given a sequence of colors.

We are playing a game on a sequence of numbers using a deque that starts with a single value, zero. At each step, we process the next array element and are forced to interact with one of the two ends of the deque.

We are given a fixed tree of neighborhoods. Each road is initially controlled by some gang. Over time, roads change ownership: on each day, a specific road is taken over by another gang, meaning that from that day onward its controlling gang changes.

We are given a sequence defined by a recurrence that mixes arithmetic difference, absolute value, and bitwise XOR. The first two values are both 1, and every next value is computed from the previous two using a deterministic rule.

We are given a rectangular grid where each cell has an integer height. Think of this grid as a terrain map. A number of people start at specified cells and can move one step per second in the four cardinal directions, or choose to stay still.

We are given two integer arrays of equal length and we are allowed to pick a contiguous segment from the first array and another contiguous segment from the second array. Both chosen segments must have the same length.