brain

We are given a sequence that already has a very specific shape: it first strictly descends until a single lowest point, and after that point it strictly ascends.

We are given a sequence of integers representing cards placed one by one into a line. As we process the sequence from left to right, we maintain another sequence that behaves like a stack with a special cancellation rule.

We are given a toroidal grid, meaning moving off one edge wraps around to the opposite side. Each cell of this grid behaves like a probabilistic state machine: from a position $(x, y)$, Putata moves left, right, up, or down with probabilities determined by four local…

We are dealing with a hidden tree on vertices labeled from 1 to n. The tree is guaranteed to be in one of only two shapes: either it forms a simple path, where every vertex has degree at most two and exactly two vertices have degree one, or it forms a star, where there exists…

We are given a collection of children, each starting with some amount of sugar. Alongside them is a set of events, one event per child. Each event refers to two children: the event’s owner and another fixed “reference” child.

We are working on a weighted tree where each vertex carries a unique non-negative integer label. For each query, we are given a starting vertex and a distance limit, and we look at all vertices that lie within that distance from the start.

We are given a grid with blocked and free cells and an initial configuration of a snake whose body occupies a simple path of length $k$. The head is the first coordinate, the tail is the last, and every consecutive pair of segments is adjacent in the grid.

We are given a sequence of strings, and we are allowed to freely permute characters inside each individual string.

We are given a set of lamps placed on a number line. Each lamp has a fixed position and a color label. For each query distance $d$, we want to find a lamp $u$ with the smallest index such that if we move exactly $d$ units to the right, there exists another lamp at that…

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

We are given a sequence of students, each requesting a certain number of pizza slices in a fixed order. There are K identical pizzas, and each pizza has the same unknown capacity S slices.

We are given an initial string that represents a “botched burger”, and multiple target strings representing correctly assembled burgers.

We are given two sets of points in the plane. The first set represents “topping” points that we want to count, and the second set represents “crust” points that define a geometric structure around the origin.

We are given a set of points on a huge 2D grid. Each point represents an enemy located at coordinates $(xi, yi)$ and carries a weight $si$. No two enemies share the same coordinates.

We are given a set of points on the integer grid, each representing a pepperoni slice. The task is to construct a straight line in the plane such that at least a fixed fraction of the points, specifically at least one eighth of them, lie on or extremely close to this line in…

We are given a fixed queue of people before Shelly arrives, and a shared supply of pizza slices, tacos, and sauce portions.

We are given a set of pizzas labeled from 1 to n, where each label is also its radius. We must arrange all pizzas in a single vertical stack, which is equivalent to choosing a permutation of numbers from 1 to n.

We are given a large rectangular grid. Some grid cells contain a plant, and each plant has a fixed length. From each plant, we may choose to “tie” it in at most one of four directions: up, down, left, or right.

We are moving through a road that is split into consecutive segments between terminals. Each segment can be traveled in several parallel lanes, and each lane has its own travel cost for that segment.

We are given a line of buildings, and each building must receive a sensor placed at some integer height. For building $i$, the chosen height $di$ is restricted to lie inside its own interval $[ai, bi]$.

We are given a binary sequence and an expression formed by chaining them with the logical implication operator. If we evaluate it strictly left to right, every step combines the current accumulated value with the next bit using implication.

We are given three integer positions on a number line, representing three points. One operation lets us pick an ordered pair of these points and move one unit of value from the second chosen point to the first chosen point.

We are given a collection of strings, and for each string we must choose another string from the same collection that is “closest” under a custom distance.

We are given a rectangular plaza made of unit squares, with width $w$ and height $h$. Some of these unit cells are cracked and must be removed entirely during construction.

We are given a collection of apartments, each with an initial area. Daniil can reduce the area of any apartment by repeatedly applying an operation: choose a factor $y$, pay $y$ coins, and divide the current area by $y$, but only if the result remains an integer.

We are given a sequence of rain intensities over time, where each minute has a non-negative amount of rain falling. Alongside this timeline, there are multiple students, and each student performs a single continuous outdoor interval.

We are given a grid with $n$ rows and $m$ columns. Each cell represents a window that can either be lit or dark. The configuration is constrained in two ways. First, every row $i$ has a fixed number $ai$ of lit windows.

We are given a directed graph where each vertex already has at most one incoming and at most one outgoing edge. This restriction means every connected component is structurally very simple: it is either a directed path or a directed cycle, and no vertex is ever a branching point.

We are given an undirected graph of towns connected by roads, where Womais can traverse edges any number of times and even revisit vertices freely. He starts at a fixed town $s$ and wants to reach a destination $t$.

We are given a mutable string $s$ and a pattern $p$ that contains lowercase letters and wildcard stars. A star can be replaced by any (possibly empty) string, independently of other stars.

We are given a set of contestants, each described by two integers. Think of each contestant as carrying two “moves”: the first move is $ai$, and the second move is $bi$.

The road can be thought of as a single journey of length $d$ kilometers, while the environment changes over time due to events involving other cars.

We are given a sequence of daily croissant prices for the next $n$ days. Each day, exactly one croissant must be eaten, and croissants are perishable: any croissant remains edible only for 7 days after purchase, including the purchase day, and becomes useless afterward.

Working

We are given a directed system of relationships between beavers. Each input edge represents a pair of beavers where one currently monitors the other.

We are given a one-dimensional board made of cells, each cell being either red, green, or white. Two players alternate turns, and on each turn a player tries to “activate” a small cluster of white cells that are close to each other.

We are choosing an integer value for each of $N$ problems, with each value constrained to lie in its own interval $[lk, rk]$. Once the values are fixed, they define a prefix-sum sequence: the $i$-th beaver’s score is the sum of the first $i$ chosen values.

We are given an initial sequence a that must appear as a prefix of a longer sequence b. The values in b are integers between 1 and M. After this prefix, we are allowed to append more elements freely.

We maintain a growing sequence where each position stores a multiset of positive integers. Initially the sequence is empty, and each operation appends a new multiset derived from earlier ones. There are four ways to construct a new multiset.

We are choosing integer values for an array of length $N$, where each position $k$ has its own allowed interval $[lk, rk]$. Once we fix a full assignment of values, we compute two families of prefix sums: one that accumulates from the left and one that accumulates from the right.

We are given a very large number written as a string, and every digit is either 1 or 2. From this number we are allowed to remove at most one digit, keeping the relative order of the remaining digits unchanged.

We are given a sequence of numbers, and we look at every contiguous segment of length at least two. For any such segment, we consider all pairs of distinct indices inside it and compute their bitwise XOR.

We are given a graph where each node is a building and each edge is a tunnel between two buildings. Every tunnel has two values attached to it: a cost in coins required to enter it, and a reward in coins received after traversing it.

We are given several uppercase strings, and for each one we must decide whether it can be decomposed into three consecutive parts that follow a very specific pattern.

We are given several independent exam scenarios. Each scenario describes a fixed total exam duration and a list of problems.

We are working with a circular keypad labeled from 0 to 9. You start with a cursor positioned on digit 1, and your task is to type a fixed 4-digit PIN. At any moment, you can either press the current digit under the cursor, or move the cursor to an adjacent digit on the circle.

We start with a single integer, and two players alternately modify it. On each move a player can increase or decrease the current value by exactly one. Vanya moves first. The game ends early if, immediately after Vanya makes a move, the resulting number is divisible by 3.

We are given a small collection of single-digit numbers. For each test case, we are allowed to pick exactly one of these digits and increase it by one.

The carpet is a small grid of lowercase letters. You read it column by column from left to right, but you are not forced to take every character. From each column you may either pick exactly one letter from that column or skip the column entirely.

We are given two sorted sequences of positions on a number line. One sequence represents garbage bags located along a street, and the other represents exits where a truck can leave the street.

We are given coordinates of three fixed points on a grid, which we can think of as three stones connected in a fixed cycle of movement, and a fourth point representing another cat starting somewhere else on the same grid.

We are given a route written in absolute compass directions, where each step is one of four values: North, East, South, or West. The platform executing the route starts with a fixed orientation, initially facing North.

The problem statement section is empty, so there isn’t enough information to reconstruct what “Air Reform” (CF 104896D) is actually asking.

I can write the full editorial in the requested style, but I’m missing the actual problem statement for Codeforces 104896B - Integral Array.

We are given a hidden multiset of points inside a 256 by 256 grid. There are $n$ items, each occupying some cell, and multiple items may share a cell. We do not know any coordinates initially.

We are given a setup with two interacting cooldown systems that behave like reusable actions in time. One action is a gold-generating ability that can be used repeatedly, but after each use it becomes unavailable for a fixed duration.

We are given a system of rooms arranged in a cycle from 0 to n-1. Each room has a single integer a[i] shown on a circular dial. From room i, Bobo has two possible actions, each costing exactly one unit of time.

We are given a rooted tree on vertices labeled from 1 to n, where every node except the root has exactly one outgoing edge pointing to a smaller-indexed parent. This orientation makes every vertex have a unique path to the root. A sequence of n drivers arrives one after another.

We are given a rectangular region in the plane, aligned with the axes. Inside this rectangle, we want to compute the area of a subset of points defined by a logical formula over linear inequalities of the form $ax + by + c ge 0$.

We are given a system of parity constraints over positions arranged in a line. Each constraint describes the parity relationship between two prefix positions, typically expressing whether the number of “active” elements between two indices is even or odd.

We are given a simple undirected graph where every edge is labeled either red or blue. The underlying graph is sparse in the sense that every vertex touches at most three edges in total, regardless of color. From this graph we choose a nonempty subset of vertices.

We are given two permutations of the same set of vertices in a directed acyclic graph. One of them is the lexicographically smallest topological ordering of some unknown DAG, and the other is the lexicographically largest topological ordering of that same DAG.

We are building a growing multiset of distinct integers, where after each insertion we must compute the minimum number of spells needed to completely remove all current values. Each spell acts on all current elements at once, but the two spell types behave very differently.

We are given a small collection of constraints, each comparing two strings using either equality or strict inequality. Each string is not a number in the usual sense but a base-10 numeral where each character is either a digit or an uppercase English letter.

We are effectively dealing with a matrix where each entry contributes either $+1$ or $-1$, and we must select a subset of entries so that every row and column has a prescribed resulting sum.

We are given a number written in a mixed representation that looks like an integer part followed by a fractional part that may have a repeating pattern.

We are given a vertical cylindrical bottle of height $h$ and radius $r$. The bottle is partially filled with water up to some height $x$, measured from the bottom. The remaining space above the water is filled with air.

We are asked to count certain well-formed bracket strings of length 2n. These strings behave like standard valid parentheses sequences, meaning every prefix never has more closing brackets than opening ones, and the total number of opening and closing brackets is equal.

We are building words of fixed length $N$ using a very small alphabet consisting of exactly three symbols: $a$, $o$, and $c$. A word is simply a length-$N$ string over these three characters.

We are given a single non-negative integer $n$, and we are asked to count how many unordered pairs of non-negative integers $(a, b)$ satisfy $a + b = n$, under a special restriction: both $a$ and $b$ must be “fun”.

The statement content for Codeforces 104830I - “Transportation Vouchers (voucher)” is not actually included in your prompt (it only shows headers, with empty input/output sections and no functional description).

The statement for Codeforces 104830H (“Ore Excavation”) is not included in the prompt, so there is no way to reconstruct the actual task, constraints, or required algorithm safely. Right now we only have the title and empty input/output sections.

I can’t write a correct editorial for this yet because the actual problem content is missing. Right now I only see the title, time/memory limits, and an empty Input/Output section.

The statement for Codeforces 104830C is missing almost entirely (both the input/output description and the actual task definition).

I can’t write a correct editorial for this yet because the actual problem content is missing. Right now you’ve provided only the contest header (input/output placeholders and metadata), but there is no description of the rules, the structure of the level, or what is being…

The problem statement content is missing from your prompt, so I don’t have the actual task definition, constraints, or samples needed to write a correct Codeforces-style editorial.

I can’t write a correct editorial for this yet because the actual problem content is missing. Right now we only have the title “Farmland Management (farming)” but no description of what operations are performed, what the grid or data structure looks like, or what needs to…

I can’t write a correct editorial for this yet because the actual problem content is missing. What you pasted only contains the header metadata and empty Input/Output sections, but no description of what an “antenna” does, what the input represents, or what needs to be…

We are asked to construct an ordered list of binary strings, each of fixed length $k$, with exactly $n$ strings total. Each string represents a thread of beads, where each position is either black or white.

We are given a small collection of fuses, each of which burns completely in a known fixed number of seconds. A fuse is not just a simple timer: we are allowed to ignite either one end or both ends, and we are also allowed to start new ignitions later, but only at time zero or…

We are given a set of weighted objects, called meteorites, each with a positive mass. We are also given a collection of circular targets, craters, each identified by its center coordinates and radius.

Codeforces 104805I: Problem on array

We are given several circular shadows, each attached to a tree whose center lies on a single horizontal line. Every tree produces a circular region of shade in the plane, and the task is to compute the total area covered by the union of all these disks.

We are given a fixed interval representing Igor’s conference schedule in Moscow. This interval is defined on a weekly time axis, starting from some day and time when participants gather at the station and ending at another day and time when they leave.

We are given a connected undirected graph with exactly $N$ vertices and $N$ edges. Each edge represents a bidirectional ferry route between two islands. Even though the number of edges equals the number of vertices, the graph is guaranteed to remain connected.

We are given an array and we first form all prefix sums of that array. If the array is a, then we build a new sequence s where s[i] is the sum of the first i elements. After that, we consider all pairs of distinct elements in this prefix sum sequence and multiply them.

We are given a stream of photo records, each describing where and when a photo was taken. Every record contains a location name and four time fields: day, month, hour, and minute. The year is implicitly fixed as 2113 for all photos.

We are given three points in the plane, each described by integer coordinates. The points are guaranteed not to lie on the same straight line, so they form a valid triangle. The task is to construct the unique circle that passes through all three points and compute its radius.

We are given a bag containing a finite multiset of tokens. Among these tokens, some are special “knight” tokens and the rest are ordinary. From this bag, a player draws a fixed number of tokens uniformly at random without replacement.

We are given a graph of cities connected by roads, where each road initially contains a single resource bag with a value between 1 and 10, or zero if the bag has already been taken. Igor starts at city 1 and must make exactly k moves along roads.

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.