brain

Let $H$ be an $mtimes n$ parity-check matrix over $mathbb{F}2$, and let $$f(x)= [Hx=0], qquad x=(x1,dots,xn)^T.$$ Fix a variable order $x1,dots,xn$.

We are given a timeline of $n$ days. Some days are fixed rest days (legal holidays). All other days are working days initially. We are allowed to convert additional working days into rest days, and these converted days are called paid leave.

We are given an $n times m$ grid. We may choose any subset of cells and place a camera in each chosen cell. Each camera placed at $(i, j)$ does not directly “cover” a fixed shape; instead, it can be configured by selecting another cell $(p, q)$, and this configuration turns…

Each test case describes a hidden “answer key” for a 10-question multiple choice quiz, where every question has exactly one correct option among A, B, C, and D.

We are given a binary string that is not written explicitly, but compressed as alternating runs of equal characters. Each run tells us how many consecutive zeros or ones appear, and runs always alternate between the two characters.

We are given several independent games, each consisting of multiple integer intervals. Each interval represents a set of currently “alive” integers, starting as all integers from $li$ to $ri$. There is also a fixed multiplier $p 1$.

Let $f = \text{COLOR}(x_1,\dots,x_n)$ be the Boolean function encoding proper 4-colorings of the US map, where each vertex variable $x_i$ takes values in ${0,1,2,3}$, represented in binary as in (73).

We are given a tree whose vertices are labeled from 1 to n. The task is to remove some edges so that the remaining connected components satisfy a strong ordering constraint: every component must correspond exactly to a contiguous segment in label order.

We are given an array of up to 21 positive integers and an initial value $x$. We are allowed to reorder the array arbitrarily. After fixing an order, we process the elements one by one, repeatedly updating $x$ by replacing it with $x bmod ai$.

Let $H$ be an $mtimes n$ parity-check matrix over $mathbb{F}2$, and let $$f(x)= [Hx=0], qquad x=(x1,dots,xn)^T.$$ Fix a variable order $x1,dots,xn$.

We are given a Tetris grid of width $w$ and a very small height, initially filled only in the bottom $n le 15$ rows. Above that, everything is empty. Some cells in these bottom rows are already occupied. No row is completely filled.

We are given a permutation of the numbers from 1 to n, and we run a modified selection sort on it. For each position i from left to right, we scan the suffix to the right of i and swap whenever we find a smaller element than the current value at position i.

We are given a short program consisting of two types of instructions stored in an array-like structure. The execution does not run this list directly in order once; instead, it builds a second structure, a queue, and executes instructions from it dynamically.

We are given a set of underground exits, each exit connects the cave to the outside world and has its own travel cost to a refrigerator.

Let $H$ be an $mtimes n$ parity-check matrix over $mathbb{F}2$, and let $$f(x)= [Hx=0], qquad x=(x1,dots,xn)^T.$$ Fix a variable order $x1,dots,xn$.

We are given a length-n array that starts completely zero. Instead of observing the array directly, we are told the final state only through a binary string: each position tells us whether the final value at that index is zero or not.

We are given an undirected graph where vertex 1 is the starting hub and vertex n is the final target we must eventually clear. Every vertex except 1 contains a boss with an initial strength. The player also has a strength value and evolves over time.

We are given several independent test cases. In each test case, there is an array of integers, and we need to count how many pairs of positions $(i, j)$ with $i < j$ satisfy a specific inequality involving multiplication and addition of the chosen elements.

Let $H$ be an $mtimes n$ parity-check matrix over $mathbb{F}2$, and let $$f(x)= [Hx=0], qquad x=(x1,dots,xn)^T.$$ Fix a variable order $x1,dots,xn$.

The prompt is missing essential information needed to write a correct editorial. The sample input and sample output in the statement are corrupted and do not line up, and there is no complete problem specification.

We are given a grid where each cell already has a fixed color, or is still undecided. The final goal is to assign colors to all undecided cells so that the resulting fully colored grid contains as many valid 2 × 2 “checker” blocks as possible.

We start from the binary representation of an integer $k$ with $n$ bits: k = (b_{n-1} b_{n-2} \dots b_0)_2,\quad b_j \in \{0,1\}, and we extend the notation by setting $b_n = 0$.

We are given a long sequence of days, each day having a numerical happiness value that can be positive, zero, or negative.

We are dealing with a three-player card game involving Alice, Bob, and Prof. Pang. Each player has a private hand of cards, and all cards are distinct. Cards are ranked only by their face value, with Ace being highest and 2 being lowest, while suits are irrelevant.

We are given a rooted tree with node 1 as the root. A depth-first search is run on this tree, and the only constraint is that each node can visit its children in any order.

We are given a digit string and asked to consider every substring of it. For each substring, we look at ways to split it into exactly six consecutive nonempty parts.

Let $H$ be an $mtimes n$ parity-check matrix over $mathbb{F}2$, and let $$f(x)= [Hx=0], qquad x=(x1,dots,xn)^T.$$ Fix a variable order $x1,dots,xn$.

The problem defines a sequence of values $f1, f2, dots, fn$ that must be computed in increasing order. Each $fi$ depends on a direct contribution term $ci$, a scaling term $bi$, and a history-dependent quantity $ai$.

We are given a sequence of values on cities, and we want to travel from city 1 to city N. Moving from a city i to a later city j has a cost that depends on the bitwise xor of the values between them, specifically the xor of the segment from i+1 to j, followed by a fixed shift…

Let $H$ be an $m times n$ binary matrix and let $$f(x) = [Hx = 0],$$ where arithmetic is over $mathbb{F}2$. The BDD for $f$ is constructed under a fixed ordering of variables $x1,dots,xn$ as in Section 7.1.4.

We are given a set of horizontal or slanted “roads” that can be thought of as intervals on the x axis, each equipped with a y coordinate representing its height.

An $n$-tuple $(a1,dots,an)$ is admissible when it satisfies the alternating constraint $$a1 le a2 ge a3 le a4 ge cdots .$$ Let $mathcal{A}n$ denote the set of all such binary $n$-tuples.

Let $n$ be fixed and let each array location be indexed by an $n$-bit integer.

We are working on a grid where a fixed “intended path” is already given implicitly, and the task is to place obstacles so that this path becomes the only viable way to traverse under the movement rules.

We are working with a weighted undirected graph where some vertices are marked as “cool”. The goal is not to compute ordinary shortest paths, but something stronger: for any pair of cool vertices, we consider all possible paths between them and look at the largest edge…

We are given a graph with labeled edges, and two entities moving on it simultaneously: Waymo and Thomas. Each state of the system is described by a pair of positions, one for Thomas and one for Waymo.

We are looking at a stochastic elimination process built around repeated rounds of a three-choice game where each participant independently picks one of three options.

The task revolves around the text that would be produced if we expand a piece of code that represents a matrix-style computation.

We construct a mixed-radix decrementing analogue of Algorithm M, operating in the same state space 0 \le a_j < m_j \quad (1 \le j \le n), but traversing it in reverse lexicographic order by repeatedly...

We are given a sequence of students, each associated with a trust value that changes as we process the system. At any moment, some students are considered to have positive trust, and we are interested in two things: how many students currently have positive trust, and the sum…

We are given a rectangular grid of integers with n rows and m columns. Each cell contributes a value, and we can compute the sum over any subrectangle using a function f(a, b, c, d), which means summing all cells in rows a through b and columns c through d.

I can’t produce a correct editorial for Codeforces 103870G - XOR Fun yet because the actual problem statement is not available from standard archives or mirrors, and it is not identifiable from known Codeforces contests or gym indexes with certainty.

The flaw in the previous solution is not in the identification of trailing maximal components, but in how step M4 is counted inside a single transition.

We are given a line of positions of some length, together with several constraints, each constraint describing a contiguous segment. The task is to decide whether it is possible to assign two types of symbols across the whole line so that all constraints are satisfied.

We are given a sequence that represents spending events over time, where each event is associated with a person identifier. The same person may appear multiple times, forming contiguous segments of activity.

We are working with a simplified calendar of a non-leap year, where the year has 365 days and each day can either contain an event or be empty. The input ultimately describes which specific days have events, and everything else is implicitly empty.

The task is unusually minimal: there is no meaningful structured input to process, and the output is expected to be produced directly based on the problem’s statement rather than any computation over data.

We are tracking a simple repeating process over time. Each day contributes to a running counter that measures how many days have passed since the last reset event.

An $n$-tuple $(a1,dots,an)$ is admissible when it satisfies the alternating constraint $$a1 le a2 ge a3 le a4 ge cdots .$$ Let $mathcal{A}n$ denote the set of all such binary $n$-tuples.

Algorithm M visits n-tuples $(a_1,\dots,a_n)$ in lexicographic order induced by the nested loops in (3), so the tuple index corresponds to a mixed-radix expansion with radices $m_1,\dots,m_n$.

I can’t write a correct editorial yet because the problem statement for Codeforces 103886Q - Cereal Trees II isn’t included in your message, and I don’t have enough reliable context to reconstruct it.

I can’t write a correct editorial yet because the actual problem statement for Codeforces 103886O - Vista (Cereal Mountains II) is missing from your message.

We are given a collection of intervals, where each interval represents a “shopping group candidate” occupying a range on a line. Two intervals are considered connected if their ranges overlap at least at one point.

We are given an $n times n$ grid and a number $k$. We need to place $k$ identical objects (called $w$ in the statement) onto distinct cells of the grid. The objective is to maximize a score that depends on how these chosen cells interact with their neighbors in the grid.

An $n$-tuple $(a1,dots,an)$ is admissible when it satisfies the alternating constraint $$a1 le a2 ge a3 le a4 ge cdots .$$ Let $mathcal{A}n$ denote the set of all such binary $n$-tuples.

Introduce shifted variables $b_j = a_j - l_j$.

We are working on a grid where only a small number of cells actually matter: a base location and several fossil locations. The grid itself can be large and mostly empty, but movement is only relevant through shortest paths on the grid.

Let $f = \text{COLOR}(x_1,\dots,x_n)$ be the Boolean function encoding proper 4-colorings of the US map, where each vertex variable $x_i$ takes values in ${0,1,2,3}$, represented in binary as in (73).

We are given a sequence of boxes arranged in a line, where each position may contain a box with some property that affects whether it can be shifted left through the line.

We are given a one-dimensional landscape represented by an array of heights. As a water level rises from low to high, positions become submerged once the water level strictly exceeds their height.

We are given a rectangular grid with multiple rows and columns. Each cell in this grid either contains a balloon or a bomb. Two agents move through the grid row by row, and in each row they choose positions in that row to collect balloons.

An $n$-tuple $(a1,dots,an)$ is admissible when it satisfies the alternating constraint $$a1 le a2 ge a3 le a4 ge cdots .$$ Let $mathcal{A}n$ denote the set of all such binary $n$-tuples.

The problem statement you provided is incomplete. It only contains the construction idea and does not include the actual input/output specification or sample tests. Because of that, I cannot produce a correct editorial or Python solution.

We are given an array of integers and a requirement to split it into exactly k contiguous subarrays. For any such partition, each subarray has an OR value computed across its elements.

The problem asks us to compute a value for each given integer $x$, where each $x$ represents a “project size” or target sum.

We are working on a geometric optimization problem on a discrete grid. Imagine a 2D classroom-like layout where some cells contain “hall monitors” and the outer boundary of the grid acts like a hard wall that also restricts movement.

We are given a sequence of red pandas sitting in a line, where each panda has an integer ID. The process we care about repeatedly looks at these IDs in increasing order of value, and whenever a particular ID appears in the current line, all pandas with that ID contribute to…

An $n$-tuple $(a1,dots,an)$ is admissible when it satisfies the alternating constraint $$a1 le a2 ge a3 le a4 ge cdots .$$ Let $mathcal{A}n$ denote the set of all such binary $n$-tuples.

An $n$-tuple $(a1,dots,an)$ is admissible when it satisfies the alternating constraint $$a1 le a2 ge a3 le a4 ge cdots .$$ Let $mathcal{A}n$ denote the set of all such binary $n$-tuples.

Let $G(x,y)$ be the Boolean function defined in (70), representing adjacency in an undirected graph on vertices ${1,\dots,n}$, where $G(x,y)=1$ iff there is an edge between $x$ and $y$.

I can’t write a correct editorial for that problem yet because the actual problem statement is missing. Right now I only see the title “Tyrannosaurus Typing”, but no description of the rules, input format, or what needs to be computed.

I can’t reliably write a correct editorial for this problem yet because the actual problem statement is missing from your prompt. Right now I don’t know what “Carrot Thief” is asking about, what the input represents, or what the required output is.

An $n$-tuple $(a1,dots,an)$ is admissible when it satisfies the alternating constraint $$a1 le a2 ge a3 le a4 ge cdots .$$ Let $mathcal{A}n$ denote the set of all such binary $n$-tuples.

I can’t write a correct editorial for this yet because the actual problem statement is missing. Right now we only have the title “Penguin Problems”, but none of the essential information: what the input represents, what operation or computation is required, or what…

The problem statement for “Codeforces 103895F - Rats Rats” is missing from your prompt, so I can’t reliably derive the solution or write a correct editorial without inventing details. Please paste the full problem statement (input, output, constraints, and any samples).

I don’t have the actual statement of Codeforces 103895E - Feed Worm, so I can’t reconstruct the mechanics of the problem reliably.

The problem statement for Codeforces 103895D - Owl Defense is missing from your prompt, so I can’t correctly reconstruct the solution or write a valid editorial.

The problem statement for Codeforces 103895C - Ellie the Elephant isn’t included in your message, so I don’t have the actual rules, input format, or required computation to base an editorial on.

I can’t write a correct editorial for Codeforces 103895A - Love Your Llama without the actual problem statement.

I don’t have the actual problem statement for Codeforces 103896K - Fish Exercise in your prompt (the section is empty), so I can’t reliably reconstruct the task or produce a correct editorial.

I can’t write a correct editorial for Codeforces 103896L - Dog and Friends because the problem statement is missing from your prompt.

An $n$-tuple $(a1,dots,an)$ is admissible when it satisfies the alternating constraint $$a1 le a2 ge a3 le a4 ge cdots .$$ Let $mathcal{A}n$ denote the set of all such binary $n$-tuples.

The problem statement for “Codeforces 103896I - Tyrannosaurus Typing” is missing from your prompt, so there isn’t enough information to derive an algorithm, write an editorial, or construct solutions and test cases.

I can’t write a correct Codeforces editorial for “103896J - Dragon Buffs” without the actual problem statement. Right now the prompt only contains formatting instructions and no description of the task (no input/output definition, constraints, or rules).

I can’t write a correct, detailed editorial without the actual problem statement. “Codeforces 103896H - Penguin Problems” isn’t enough on its own, and I don’t have the input/output description, constraints, or even the core task to reconstruct the solution reliably.

I can’t write a correct editorial for this problem yet because the actual problem statement is missing. “Codeforces 103896G - Carrot Thief” isn’t enough on its own to reconstruct the rules, constraints, or required output, and guessing would very likely produce a…

I don’t have the actual problem statement for Codeforces 103896F - Rats Rats, so I can’t reliably write a correct editorial yet.

We are given a line of cows, each with a weight. A “raid” is defined by two integers, a starting position and a step size. Starting from position a, we repeatedly jump forward by b positions and collect all cows we land on, stopping once we go past the end of the line.

I can’t write a correct editorial for CF 103896C - Ellie the Elephant without the actual problem statement. Right now the prompt contains only the title and metadata, but no description of the task, inputs, or outputs.

We are given a collection of lines in the plane. Each line acts like a reflection axis, and we are interested in sets of points that remain consistent under reflection across every one of these lines.

We are given not one graph but a sequence of graphs that evolve from each other. The first graph is explicitly constructed, and every subsequent graph is obtained from an earlier one by a single edge insertion or deletion.

We are given an undirected tree, and the task is to place each vertex at an integer grid point so that when edges are drawn as straight line segments, the drawing behaves like a clean planar tree embedding with no crossings or unintended intersections.

Equation (65) computes existential quantification over a Boolean variable by combining cofactors of a function $f$ with respect to that variable, typically using the structure of a BDD node whose low...

We are given a strictly convex polygon with vertices listed in counterclockwise order. From this polygon, we look at a growing sequence of prefixes: the polygon formed by the first 3 vertices, then the first 4, and so on until all n vertices.

We are not asked to compute anything for the input. The input is only a seed that the checker uses to generate test expressions. Our task is to output a fixed 1024-word program, i.e.

We are working with two evolving permutations over a set of positions from 1 to n. Initially, both permutations are identical to the identity permutation. One permutation, call it a, can be modified by swapping values at two positions.

We are given two sequences of integers, but neither sequence is provided directly. Instead, both are produced by repeatedly applying the same quadratic recurrence modulo a fixed value.

We are given two players, Alice and Bob, each starting with exactly two known cards from a standard 52-card deck. In addition, six community cards will be chosen from the remaining deck.

We are given a complete poker situation involving ten known cards. Alice starts with two private cards, Bob starts with two private cards, and there are six shared community cards on the table. The players do not draw from an unknown deck, everything is already revealed.

We are dealing with a very rigidly structured Sudoku-like construction, but the actual task is not to solve a Sudoku.