brain

We are given a cylindrical “tree” unwrapped into an $N times M$ grid. Each cell can hold at most one ornament. We must place ornaments so that every contiguous block of $W$ columns, taken across all $N$ rows, contains at least $S$ placed ornaments in total.

We are given a positive integer $n$, interpreted as the length of a wooden stick. We cut it into a sequence of positive integer-length pieces, and we care about the order of these pieces, because two different cutting positions produce different sequences even if they contain…

We are given a repeated experiment where each trial independently results in a win with probability $frac{p}{q}$ and a loss with probability $1 - frac{p}{q}$. We start from zero wins and zero losses, and we keep playing until one of two stopping conditions is reached.

We are given a collection of distinct lowercase strings. From these strings, we can choose any non-empty subset and arrange the chosen strings in any order.

We are working on an infinite chessboard where every integer coordinate initially contains a knight. At some moment, a single special event happens: the knight at position $(0, 0)$ becomes a “super knight”, and the cell at $(n, m)$ becomes empty because its knight is removed.

We are interacting with a hidden transformation machine that always processes a block of 8 MBTI values at once. Each MBTI is encoded as a 4-bit value, so the full machine state is exactly 32 bits. The machine applies a fixed sequence of 19 operations to these 8 values.

We are asked to count how many different labeled graphs on vertices numbered from 1 to n satisfy a set of strong structural and arithmetic constraints.

We are given a directed graph of cities where city 1 is special and reachable from every other city via directed paths. Each day, a virus starts from a chosen city and immediately spreads along outgoing roads, infecting every city reachable from that start.

We are given a string consisting of lowercase Latin letters. The string is considered “boring” if it contains any contiguous block where the same character appears at least m times in a row.

We are asked to simulate a very simple bookkeeping task over a consecutive range of years. Imagine writing down every integer year starting from 1 up to 2024, one by one, without skipping any value.

We are given a sequence of integers $a1, a2, dots, an$. From these values we conceptually build an $n times n$ table. The entry at row $i$, column $j$ is defined as the integer part of the division $aj / ai$.

We are given a string made of lowercase Latin letters. The string becomes “bad” whenever some character appears in a single contiguous block of length at least m.

We are given a positive integer $n$. We consider ways to split $n$ into two natural numbers, meaning positive integers, written as pairs $(a, b)$ such that $a + b = n$. The same applies to another pair $(c, d)$ with $c + d = n$.

We are given a binary string consisting only of zeros and ones. In one operation, we take a pair of adjacent characters and apply a XOR-like rule to compress them into a single character according to simple local interactions.

We are given two circular targets on a plane. Each target is defined by its center coordinates and a radius. From these two circles, we are asked to determine which of four “scores” from 0 to 3 are achievable, where each score corresponds to a geometric relation between the…

The task is essentially a “what if” simulation on a Connect Four board. We are given a 6 by 7 grid that represents the current state of the game, where each cell can contain a red piece, another color, or be empty.

We are given a fixed linear transformation acting on a polynomial $f(x)$. The transformation does not evaluate $f$ at a single point; instead, it evaluates $f$ at several shifted positions $x, x+1, dots, x+t$, multiplies each value by a coefficient $ci$, and sums everything to…

The provided problem statement does not define any actual computational task. There is a title suggesting some theoretical connection between lambda calculus, p-adic norms, and dynamical system stability, but there are no formal definitions of input, output, or required…

I’m missing the actual problem content for Codeforces 105615M - Oops. The input/output section you provided is empty, so there’s no way to determine what the task is, what structures are involved, or what algorithmic idea it tests.

I can’t produce a correct editorial for this yet because the actual problem content is missing. Right now I only see the title “叶的距离” (Leaf Distance), but there is no statement describing the tree structure, input format, or what “distance” is being asked.

I can’t reliably reconstruct Codeforces 105615I - 晶体化 from external sources or partial metadata alone, and I don’t want to guess a problem and give you a fake editorial. Right now I only know the title, not the actual model (grid, graph, DP, geometry, etc.

I can’t produce a correct editorial for this problem because the actual problem content is missing. What you provided contains only the title (“two-sentence statement”) and empty input/output sections.

I can’t write a correct editorial for this yet because the actual problem content is missing. Right now the statement only shows the title, time/memory limits, and empty input/output sections.

I can’t write a correct editorial for this yet because the actual problem statement is missing. Right now I only see the title and empty input/output sections, but no definition of the task (what the input contains, what must be computed, or any constraints).

I can’t reconstruct a correct editorial from what you’ve provided because the actual problem statement is missing.

The problem statement you provided is incomplete. It only contains the title “BCPC 2024 A” and placeholders for input and output, but no actual description of what the task is.

I can write a full editorial, but I am missing the actual problem content for Codeforces 105629H - 急死地. The statement you pasted only shows the title and placeholders for input/output, so there is no description of what the task is, what operations are allowed, or what…

I don’t have the actual statement for Codeforces 105629G “回文串” in the prompt, and without it I can’t reliably reconstruct the intended problem.

The input section and full problem statement are missing from what you provided, so there is no way to reconstruct what “工作调度” (Work Scheduling) is actually asking, nor what constraints or operations define the task.

I can’t write a correct editorial for this problem as-is because the actual problem statement is missing. What you provided only includes the title, time/memory limits, and empty input/output sections, but none of the essential details such as what the input describes, what…

We are given a sequence-like structure, but the statement itself is extremely compressed, so the only meaningful interpretation we can reconstruct is that the problem expects us to process one or more inputs and produce a single computed output for each case.

The problem statement for Codeforces 105629C (“不够聪明的贝贝”) is not included in your prompt, so there is no way to reconstruct a correct solution or write a meaningful editorial without risking fabrication.

We are asked to fill an $n times m$ matrix with positive integers up to $10^9$. Instead of arbitrary values, each row and column must satisfy a very specific aggregate constraint.

We are given several independent test cases. Each test case describes a sequence of integers, where each element represents a “rhythmicity value” of a slice of music. There is also a fixed lower bound L.

We are given a sequence of colored lights laid out in a line. Each position has a color label, and multiple positions may share the same color. On top of this static array, we process a sequence of update operations.

We are given an array of integers, and we need to answer multiple independent queries. Each query specifies a range in the array and a threshold value.

We are given a string that keeps changing over time, and after each change we must recompute a value that depends only on the current form of the string. For any string $t$, consider all substrings of the current string $s$.

We are given two polyline paths that evolve over a shared time axis. One path is defined at odd time steps and the other at even time steps. Each path has $m$ vertices, and at each step its position is an integer height between 1 and $n$.

The system maintains a live programming contest scoreboard. There are several teams, indexed from 0 to k, and submissions arrive in strictly increasing timestamp order.

We are given a universe of elements from 1 to n, and each element i carries a weight ai. The goal is to design a randomized procedure that produces subsets of fixed size k using independent coin flips, followed by rejection of invalid outcomes.

We are given 2n labeled players, all starting with score zero. The process runs for k rounds, and in every round we must split all players into disjoint pairs. Each pair plays a match, and exactly one point is transferred between the two participants.

We are given a sequence of points, and they arrive one by one. Each point represents a person with coordinates $(xi, yi)$.

We are given a fixed pattern of work that repeats every day forever. Each day has a timeline of k hours, and at the start of every day exactly n tasks appear. Task i of a day appears at a known hour ai within that day and requires bi hours of uninterrupted processing time.

We are given a tree where each vertex holds exactly one number, and these numbers form a permutation of 1 through n. The goal is to transform this permutation into the identity configuration, meaning vertex i must end up holding value i.

We are given a triangular grid whose rows grow as we go down, forming a total of roughly $n^2$ cells arranged in $n$ rows. Each cell contains a number from 1 to 4. We also have a tetrahedral die that moves on this grid.

We start with four lattice points forming an axis-aligned rectangle: the origin, the point on the x-axis at distance A, the point on the y-axis at distance B, and the opposite corner (A, B).

We are given a sequence of banknotes in a fixed order. Each banknote has a value, and we repeatedly perform an operation where we choose a contiguous block of currently remaining notes whose sum fits a specific arithmetic form, remove that block entirely, and then compress the…

We are given several containers of a solution. Each container has a fixed amount of liquid, but the amount of dissolved substance inside each unit of liquid is not known exactly.

We are given an array of positive integers and multiple queries, each query specifying a contiguous segment of that array. For each segment, we conceptually consider every pair of distinct indices inside it and compute the gcd of the two corresponding values.

We are standing on a corner of a rectangular box and are only allowed to move along its surface, not through the interior. From that starting corner, we want the point on the surface that is as far as possible in terms of shortest surface distance.

The grid describes a physical layout of components that behave like nodes in a computation tree. Each non-empty cell contains a unit, and every unit connects only to its orthogonally adjacent neighbors.

We are given a linear ribbon split into $n$ consecutive sections. Each section has a target dye level, and higher numbers correspond to darker shades. The ribbon starts completely white, and we need to transform it into the target pattern using dye operations.

We are given a set of cities connected by candidate undirected roads. Each city has a weight that represents its importance. Among these cities, city 1 is the capital, Yokohama.

We are given a number written in base 15, where digits can be 0-9 and A-E representing values 10-14. We are allowed to perform at most one swap between two positions in the digit string.

We are given a sequence of cards. Each card has a suit among four types and a number. Inside each suit, numbers are unique and come from the same range, so every suit behaves like a permutation of the same value set.

We are given a fixed sequence of operations applied to a single running value. Each operation in the sequence is one of three types: addition by a constant, subtraction by a constant, or multiplication by a constant. The sequence is applied in order from left to right.

We are given a pool of champions, each champion being described by a binary string over a fixed set of traits. A 1 at position j means that champion possesses trait j.

We are given a tree where each edge carries a weight that can be positive or negative. Between any two nodes there is exactly one simple path, so the “shortest path between two nodes” is not a choice among multiple routes, it is simply that unique tree path.

We are given a target number for each test case, and we need to decide whether it can be represented as a sum of a carefully constructed integer and the sum of its digits.

We are given a one-on-one fight between two minions. Each minion has health and attack. The fight proceeds in discrete rounds. In each round, Reborn’s minion strikes first, reducing the enemy’s health by its attack value.

We are given an array where every element is a power of two. So each value looks like $2^{ai}$, meaning the entire array is just a multiset of bit positions, each element contributing a single set bit in a binary number. We need to split this array into contiguous segments.

We are given a circle with an even number of points, and each point is paired with exactly one other point, forming a perfect matching. Each pair defines a chord inside the circle.

We are working with a one-dimensional sequence of cities arranged from left to right. Each city represents a starting point for a mercenary, and between consecutive cities there are shops.

We are given a sequence of integers and we want to split it into exactly $k$ contiguous segments. Each segment has an average value, computed as the sum of its elements divided by its length.

We are given two binary strings of equal length. Each position contains either 0 or 1. We are also given an integer k, and we are allowed to perform an operation that selects a cyclic segment of length k in the first string and another cyclic segment of the same length in the…

We are given a sequence of integers and we consider every possible way to split it into contiguous segments. Each segment contributes a value equal to the bitwise xor of its elements, and a partition’s score is the product of these segment xors.

We are given an $n times n$ matrix that must satisfy a strong monotonicity rule: values never increase when moving right or downward. In other words, every row is nonincreasing left to right and every column is nonincreasing top to bottom.

We are working with an infinite binary string constructed from the Fibonacci word recurrence. Instead of expanding it explicitly, we only rely on its recursive structure and the key property that any sufficiently long segment contains both characters and behaves “mixed” in…

We are given a tree where some vertices initially contain tokens. The twist is that we must evaluate the same movement process for every possible choice of root independently, and for each root compute how many moves can be made under an optimal strategy.

We are given a directed acyclic graph with a fixed start vertex 1 and a fixed end vertex n. The task is not to find a single path, but to construct as long a sequence of valid 1-to-n paths as possible, with a constraint that makes each new path “bring something new” compared…

We are given an $n times n$ grid. Every cell has a non-negative cost, and placing a radar in that cell covers a large square region of fixed size that depends on $n$.

We are given an array $b1, b2, dots, bn$. We are allowed to choose an integer shift $x$ between 1 and $k$, and apply it to every element, forming a new array $ai = bi + x$.

A grid is filled with numbers, and these numbers describe an order in which cells will be marked. You can think of the process as reading a permutation of all grid cells and activating them one by one. After each activation, some subset of cells becomes marked.

We are given a sequence of days, where each day produces a certain amount of experience if we play a character on that day. A character can only be played on a continuous segment of days, and once we stop using that character, it is discarded.

We are assigning labels to two collections of cards. One collection contains n1 character cards and the other contains n2 music cards. Every card receives an integer value between 1 and m, and repetition is allowed, so the final state of each side is a multiset rather than a set.

We are given a partition of the numbers from 1 to n into k consecutive segments. Each segment is meant to represent a heavy chain in some heavy-light decomposition of a rooted tree, where inside a chain every vertex is connected to the next one, and the last vertex of the…

We are given a cycle graph with vertices labeled from 1 to n in circular order, so each vertex i is connected to i−1 and i+1 modulo n. On top of this cycle, exactly one extra edge is added between two vertices that are not neighbors on the cycle.

We are given a directed graph on n nodes where every node has exactly one outgoing edge, defined by an array a. If we stand at node i, we deterministically move to node a[i].

We are given a sequence and we must split its elements into two subsequences, called $B$ and $C$, without changing the original order inside either subsequence. Every element of the original array goes to exactly one of them.

Each task represents a person who must be picked up from a starting floor and dropped at a higher floor using a single elevator. The elevator begins at some initial floor and can only carry one person at a time.

We are given several complete rankings of the same set of universities. Each ranking is a permutation, so it defines a strict order from best to worst. From these rankings, we build a derived notion of “superiority”.

We are given a fixed horizontal segment on the x-axis, from $x=l$ to $x=r$, but the endpoints are forbidden, so we only care about positions strictly inside this interval.

We are given an array of pig ratings, where each value is a 63-bit integer. The core operation that defines all behavior is bitwise AND, so every rating can only lose bits over time and never gain new ones unless explicitly assigned. The system supports three types of operations.

We are given three strings over the lowercase English alphabet, and we are allowed to define a mapping from characters to characters. This mapping is not required to be bijective, multiple letters can map to the same letter, but every character must map to exactly one character.

The task is a constructive number theory problem disguised in a knapsack-style encoding system. We are given a target value and must construct a selection of special items whose combined contribution encodes that target through modular arithmetic constraints.

We are working with a system where a large modulus $D$ is built from several prime factors, and the task is to construct a controlled linear combination of specially structured values so that the resulting sum matches a target residue modulo $D$.

We are given a permutation and we are asked to decide whether it is possible to extract a subsequence of length $K$ with a strong structural restriction: inside that chosen subsequence, the elements must be decomposable into a small number of strictly decreasing sequences.

We are working with integers that can “transform” into smaller integers through a digit-based reduction rule. Starting from a number, you are allowed to repeatedly replace it by dividing it by one of its digits, but only if that digit actually appears in its decimal…

We are given a set of our own monsters, each described by two strength values. The first value represents how strong that monster is when fighting type 0 enemies, and the second value represents its strength against type 1 enemies.

We are given a sequence of colored blocks, represented as an array where each position contains a color identifier. The task is to split this sequence into the minimum number of subsequences such that each subsequence satisfies a monotonic consistency condition on colors.

We are given a graph where the task is to decompose its edges into simple paths, each of length exactly three edges. In other words, every edge must belong to exactly one path, and every path must consist of four distinct vertices connected consecutively.

The structure described in this problem is a rooted tree where each vertex represents a room and each edge represents a corridor with a travel cost. From the root, every vertex has a well-defined depth equal to its distance in edges from the root.

We are working on a grid where each cell is either filled or empty, and each query gives us a rectangular subgrid. For every query, we need to decide whether the filled cells inside that rectangle form a valid structure called a snake.

We are given a system of $N$ containers, each initially holding a single marble. Every container has a target container where its marble is supposed to end up. That target is given by an array $c$, where $ci$ tells us the destination of the marble currently in container $i$.

We are given a collection of containers, each initially holding a marble that is labeled by a target container index. Every container has exactly one marble at the start, but marbles are not necessarily in the correct place.

We are working on a rectangular grid where each cell is either empty or belongs to a connected structure. The grid is fixed, and for each query we are given a sub-rectangle.

We are given a grid of integers. A move in the game repeatedly modifies the grid until no move is possible anymore. The key structural property is that while values may change through the game, the relative ordering between cells never changes.

We are given an array that is being updated over time, and we must answer queries about what value can safely be used from a subarray under a specific rule involving “missing” numbers.

We are given several screenshots of a scoreboard, where each screenshot records the scores of the same set of teams at some moment in time. Each screenshot is an array of length $K$, and the $i$-th value represents the score of team $i$ at that moment.

We are given a single integer, and we must classify it into a very small set of time labels that depend on which range the number falls into.